improving the e ciency of selection in a plant breeding

Improving the efficiency of

selection in a plant breeding

program using information on

correlated traits, ancestry and

environments.

Aanandini Ganesalingam

Bachelor of Science (Agriculture) (Hons) & Bachelor of Economics

This thesis is presented for the degree of

Doctor of Philosophy

of

The University of Western Australia

School of Plant Biology & The UWA Institute of Agriculture

2013

mailto:[email protected]

Abstract

This thesis presents how information on correlated traits, ancestry and envi-

ronments can be used within a mixed model framework to improve selection

in plant breeding. The motivating example is canola (Brassica napus L.).

Plant survival data in blackleg disease of canola are often composed of

multiple measures used to form a derived variable, such as percent survival

values, which is then subject to analysis. Instead, a bivariate linear mixed

model approach is proposed in which the two variables are the initial and

final plant counts. This approach is demonstrated using data from blackleg

disease nurseries in the 2009 growing season in Australia. The counts were

considered as two ‘traits’, which are affected by different biological, genetic

and environmental influences. The bivariate mixed model approach for the

analysis of plant survival data not only provided a more detailed picture

but also a more accurate assessment of the impact of disease resistance

compared with the univariate analysis of percentage survival data.

The release of new cultivars onto the market is preceded by extensive test-

ing of varieties across target environments and growing seasons in multi-

environment trials (METs), which is a core process in plant breeding. An-

other related objective is the selection of parents for the next cycle of breed-

ing. The inclusion of pedigree information in the MET analysis satisfies

both objectives.

Using the 2011 subset of data from a canola breeding program, this thesis

demonstrates the use of spatial analysis of individual trials and then extends

this to an across site analysis using a MET and factor analytic (FA) mixed

model framework. The efficiency of this process is demonstrated in the

iii

spatial analysis of individual trials to control within trial environmental im-

pacts when pedigree information is included. The study demonstrates that

pedigree information aids in the modeling of spatial errors and identification

of outliers by adding information for entry performance from relatives. The

study concludes that base-line non-genetic modeling should always include

pedigree information for the determination of site-specific spatial models,

especially in the case of p-rep trial designs, which are commonly used in

plant breeding programs for the testing of early generation entries.

The extension of the single site pedigree analysis to a MET/FA analy-

sis examines how environments impact on entry performance (genotype by

environment interaction) within a breeding program. The MET/FA with

pedigree information not only enables independent estimation of additive

and non-additive genetic effect of entries, but also the impact of GxE on

these genetic effects. This study also derived total genetic variance for

hybrid and non-hybrid entries, to observe the impact of GxE on these dif-

fering entry types. While the estimated genetic correlations resulting from

MET/FA analysis did not indicate different patterns of GxE for hybrid or

non-hybrid entry types, it is a more accurate selection tool given the dif-

ferences in inbreeding levels between entry types. In other plant breeding

datasets that jointly trial hybrid and non-hybrid entries it may indicate

broad insights into the basis of possible sources of GxE on trial groupings.

Finally a topic of interest that arose during the research of this thesis is

the extensive time to analysis completion in MET/FA with pedigree model.

This chapter investigated the algorithm employed in ASReml-R and the

time required for the completion of a single iteration for different genetic

variance models alongside different lengths of data sets with correspond-

ing pedigree files. While it was observed that iteration completion times

increased substantially when pedigree information is included in MET/FA

analysis, the findings of this chapter also indicate that a so-called Reduced

Rank + diagonal formulation of the FA model took a third of the time

for the completion of the second iteration completion than the standard

formulation.

iv

The outcomes of research from this thesis have implications for all plant

breeding programs whether hybrid or self-pollinated crops.

v

Acknowledgements

“Not all those who wander are lost.”

-J.R.R Tolkien

To my supervisor Alison, I am indebted to you for your patience with my

wandering given that I didn’t really know what I signed up for. I am most

grateful for your ability to make any mixed model analysis look ‘simple’ and

for teaching me most of the mixed model theory from scratch. I would also

like to acknowledge Brian Cullis, who deserves the credit of a supervisor,

with his steady input, ideas and discussions during the length of this PhD.

Alison and Brian, your guidance, motivation (and perseverance!) with me

for the last three and a bit years shaped this thesis. I owe both of you my

deepest gratitude, as this thesis wouldn’t have been completed without such

an excellent supervisory tag-team.

To my two supervisors at UWA. Thanks Wallace for providing me with

the CBWA data set, reading the numerous drafts of this thesis and co-

ordinating my thesis. Thanks Cameron for putting me on this path, when

you unwittingly hired to me to do casual work at CBWA all those years

ago.

I would also like to thank Dr. Ed Roumen for providing me with the oppor-

tunity to undertake this PhD in the first place and for your numerous and

stimulating discussions in the initial stages. I would also like to acknowl-

edge Bayer Crop Science for providing me with the scholarship to undertake

this PhD and the Mike Carroll travel fellowship for providing the financial

assistance and opportunity to undertake research at Rothamsted Research.

vii

To my friends at UWA who shared this journey with me, Annaliese, An-

nisa, Caroline, Christine and Maggie, the biggest thank-you is owed. You

girls were not only my support group but ensured that I was motivated

(caffeinated) for work on a daily basis. Special mention and thanks here to

Emily for making the stay at Rothamsted and trips to Tumut an absolute

blast.

Thanks to papa and mame for your love and support, especially for putting

up with me being a perpetual (and often absent) student. Last but not least,

I would also like to thank my husband, Hari for his unwavering support,

understanding and patience during the ups and downs of this journey; I

could not have done this without you, I dedicate this thesis to you.

viii

Statement of original contribution

This thesis has been completed during the course of enrollment in a PhD

degree at the University of Western Australia, and has not been used pre-

viously for a degree or diploma at any other institution. To the best of

my knowledge and belief, this thesis does not contain material previously

published or written by another person, except where due reference is made

in the text of the thesis.

Aanandini Ganesalingam

May, 2013

ix

Contents

List of Figures xvii

List of Tables xxi

Glossary xxiii

1 Introduction 1

2 Literature Review - Methods of measurement and analysis of plant

survival data sets 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Measures of disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Blackleg disease incidence . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Bivariate analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1 Biological motivations . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.2 Statistical motivations . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 A bivariate mixed model approach for the analysis of plant survival

data 15

3.1 Data set description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Measuring disease incidence . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Univariate analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.1 Statistical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.2 Checking the adequacy of the spatial model . . . . . . . . . . . . 20

3.3.3 Estimation and Fitting . . . . . . . . . . . . . . . . . . . . . . . 20

xi

CONTENTS

3.3.4 Univariate analysis results . . . . . . . . . . . . . . . . . . . . . . 21

3.3.4.1 York disease nursery . . . . . . . . . . . . . . . . . . . . 21

3.3.4.2 All disease nurseries . . . . . . . . . . . . . . . . . . . . 25

3.4 Bivariate analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4.1 Statistical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4.2 Model Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4.3 Bivariate analysis results . . . . . . . . . . . . . . . . . . . . . . 30

3.4.3.1 York disease nursery . . . . . . . . . . . . . . . . . . . . 30

3.4.3.2 All disease nurseries . . . . . . . . . . . . . . . . . . . . 34

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Further applications of bivariate analysis for plant breeding data 45

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Breeding for disease resistance . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 Adjustment for seedling emergence . . . . . . . . . . . . . . . . . 47

4.2.2 Adjustment for heading date . . . . . . . . . . . . . . . . . . . . 47

4.2.3 Adjustment for fungal mould levels . . . . . . . . . . . . . . . . . 48

4.2.4 Adjustment for plant stand and days from planting . . . . . . . . 49

4.3 Breeding for grain yield . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.1 Adjustment for plant stand . . . . . . . . . . . . . . . . . . . . . 50

4.3.2 Adjustment for grain moisture levels . . . . . . . . . . . . . . . . 51

4.4 QTL analysis - adjusting for other traits . . . . . . . . . . . . . . . . . . 52

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Literature Review - Pedigree information in plant breeding METs 57

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Analysis of MET trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2.1 Linear Mixed Model Approach . . . . . . . . . . . . . . . . . . . 61

5.2.1.1 Prediction models and relationship matrices . . . . . . 63

5.3 Heterosis and GxE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4 Relationship Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.4.1 Pedigree based estimators of COF . . . . . . . . . . . . . . . . . 65

xii

CONTENTS

5.4.2 Molecular marker based estimators . . . . . . . . . . . . . . . . . 67

5.4.3 Higher order interactions . . . . . . . . . . . . . . . . . . . . . . 70

5.5 Conclusion and further research . . . . . . . . . . . . . . . . . . . . . . . 71

6 Canola multi-environment trial data set 73

6.1 Data set description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2 Pedigree Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7 Spatial analysis (N-gen modelling) of trials with pedigree information 81

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.2 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.2.1 Data set description . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.2.1.1 Superblock design component . . . . . . . . . . . . . . 84

7.2.2 Single Trial analysis . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.2.2.1 Standard statistical model . . . . . . . . . . . . . . . . 85

7.2.2.2 Pedigree statistical model . . . . . . . . . . . . . . . . . 86

7.2.3 Ngen variance modeling . . . . . . . . . . . . . . . . . . . . . . . 87

7.2.4 Outlier detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.2.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.2.6 Estimation and Fitting . . . . . . . . . . . . . . . . . . . . . . . 89

7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.3.1 Ngen variance modeling - York trial . . . . . . . . . . . . . . . . 89

7.3.1.1 Model parameters . . . . . . . . . . . . . . . . . . . . . 97

7.3.2 All trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8 MET analysis of trials with pedigree information 105

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8.2 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8.2.1 Description of data . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8.2.2 Statistical models . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.2.3 Model fitting and examination of GxE . . . . . . . . . . . . . . . 112

8.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

xiii

CONTENTS

8.3.1 N-gen variance modeling . . . . . . . . . . . . . . . . . . . . . . . 113

8.3.2 Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.3.3 FA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.3.4 GxE for additive effects . . . . . . . . . . . . . . . . . . . . . . . 116

8.3.5 GxE for non-additive effects . . . . . . . . . . . . . . . . . . . . . 119

8.3.6 GxE for total genetic effects . . . . . . . . . . . . . . . . . . . . 123

8.3.6.1 Total genetic effects: all entries . . . . . . . . . . . . . . 123

8.3.6.2 Total genetic effects: hybrid entries & non-hybrid entries 124

8.3.7 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.3.7.1 Commercial selection . . . . . . . . . . . . . . . . . . . 127

8.3.7.2 Selection for parents . . . . . . . . . . . . . . . . . . . . 130

8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

9 Analysis completion times: MET analysis with pedigree information 141

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

9.2 Computation background . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.2.1 Independent formulation . . . . . . . . . . . . . . . . . . . . . . . 143

9.2.1.1 Toy example . . . . . . . . . . . . . . . . . . . . . . . . 144

9.2.2 Dependent formulation . . . . . . . . . . . . . . . . . . . . . . . 147

9.2.2.1 Toy Example . . . . . . . . . . . . . . . . . . . . . . . 148

9.2.3 Reduced rank version - dependent formulation . . . . . . . . . . 150

9.2.3.1 Toy Example . . . . . . . . . . . . . . . . . . . . . . . . 150

9.2.4 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

9.2.5 Sparsity and ordering . . . . . . . . . . . . . . . . . . . . . . . . 153

9.3 Example: Analysis completion times . . . . . . . . . . . . . . . . . . . . 158

9.3.1 The data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

9.3.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

9.3.3 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

9.3.4 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 159

9.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

10 General Discussion 163

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

xiv

CONTENTS

10.2 Correlated traits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

10.3 Ancestry & Environments . . . . . . . . . . . . . . . . . . . . . . . . . . 166

10.4 Future directions of research: correlated traits, ancestry and environments170

10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Appendices 173

A Published paper based on Chapter 3 175

B ASReml-R Code 189

B.1 ASReml-R Code for fitting the univariate trait models in Chapter 3 . . 189

B.2 ASReml-R Code for fitting the bivariate trait models in Chapter 3 . . . 190

Bibliography 191

xv

CONTENTS

xvi

List of Figures

1.1 Canola production regions across southern Australia . . . . . . . . . . . 2

2.1 The lifecycle of blackleg disease . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 Location of blackleg disease nurseries across Australia . . . . . . . . . . 16

3.2 York disease nursery initial plot of residuals and sample variogram . . . 23

3.3 York disease nursery final plot of residuals and sample variogram . . . . 24

3.4 Plot of predicted entry means at maturity against emergence. . . . . . . 32

3.5 Plot of the difference between predicted entry means at maturity and

emergence against emergence . . . . . . . . . . . . . . . . . . . . . . . . 33


emergence against emergence at Shenton Park disease nursery . . . . . . 36


emergence against emergence at Wagga Wagga disease nursery . . . . . 37

5.1 Schematic representation of two entries (blue = entry 1 and pink =

entry 2) and their performance across two environments: (a) no GxE;

(b) GxE due to heterogeneity of variance between the environments but

not lack of genetic correlation; (c) GxE due to lack of genetic correlation

but not heterogeneity of variance between environments; (d) GxE due

to heterogeneity of variance between the environments and the lack of

genetic correlation. This diagram has been reproduced from Cooper

et al. (1996). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.1 Location of multi-environment trials across Australia . . . . . . . . . . . 74

xvii

LIST OF FIGURES

7.1 Initial plot of residuals and sample variogram for N-gen models fitted for

standard and pedigree models for the York trial. . . . . . . . . . . . . . 91

7.2 Initial plots of faces of the sample variogram (solid line) and the simu-

lation mean (dotted line) as banded by 95% coverage intervals (dashed

lines) for standard and pedigree models at the York trial. . . . . . . . . 92

7.3 Plot of residuals and sample variogram for N-gen models fitted for stan-

dard and pedigree models after the addition of linear regression on row

number at the York trial. . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.4 Plots of faces of the sample variogram (solid line) and the simulation

mean (dotted line) as banded by 95% coverage intervals (dashed lines)

for standard and pedigree models after the addition of linear regression

on row number at the York trial. . . . . . . . . . . . . . . . . . . . . . . 94

7.5 Plot of residuals and sample variogram for N-gen models for standard

and pedigree models after the addition of random column effects for the

York trial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.6 Plots of faces of the sample variogram (solid line) and the simulation

mean (dotted line) as banded by 95% coverage intervals (dashed lines)

for standard and pedigree models after the addition of random column

effects at the York trial. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.7 Outliers detected under standard and pedigree models . . . . . . . . . . 100

8.1 Dendrogram of the dissimilarity matrix (It−Cea) of additive effects for

yield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.2 Heatmap of the REML estimate of the additive genetic correlation ma-

trix (Cea) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.3 Dendrogram of the dissimilarity matrix (It −Cei) of trial non-additive

genetic effects for yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.4 Heatmap of the REML estimate of non-additive genetic correlation ma-

trix (Cei) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

8.5 Heatmap of the REML estimate of the total genetic correlation matrix

(Ceg, where a = 1.82) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.6 Total genetic C-BLUPs for hybrid entries from Cluster 2 plotted against

Cluster 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

xviii

LIST OF FIGURES

8.7 Total genetic C-BLUPs for non-hybrid entries from Cluster 2 plotted

against Cluster 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.8 Additive genetic C-BLUPs for non-hybrid entries from Cluster 2 plotted

against Cluster 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

9.1 Toy example of the independent formulation . . . . . . . . . . . . . . . . 146

9.2 Toy example of dependent formulation . . . . . . . . . . . . . . . . . . . 149

9.3 Toy example of RR version of dependent formulation . . . . . . . . . . . 152

9.4 Sparsity after absorption in a toy example of the dependent formulation

with correct ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

9.5 Sparsity after absorption in a toy example of the dependent formulation

with incorrect ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

9.6 Second iteration completion times . . . . . . . . . . . . . . . . . . . . . 161

xix

LIST OF FIGURES

xx

List of Tables

3.1 Location based summaries of the 2009 blackleg disease nurseries . . . . 17

3.2 Description of 2009 blackleg disease nursery experiments . . . . . . . . . 17

3.3 Spatial modeling in univariate analyses of emergence and maturity trait

data for each experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4 REML estimates of error variance from the univariate and bivariate mod-

els at each disease nursery. . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 REML estimates of entry variance from the univariate and bivariate

models at each disease nursery. . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Accuracy of prediction for univariate and bivariate models . . . . . . . . 38

6.1 Details of multi-environment trials in the canola data set . . . . . . . . 75

6.2 Summary of individual trial details from the canola multi-environment

trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.3 Commonality of entries across the canola multi-environment trials . . . 77

6.4 Summary of the canola breeding program pedigree data . . . . . . . . . 78

6.5 Parent concurrence matrix for the canola multi-environment trials . . . 78

6.6 Example extract of the CBWA Pedigree file . . . . . . . . . . . . . . . . 79

6.7 Summary of entry details within the canola multi-environment trials . . 79

6.8 Depth of pedigree information with varying data set length . . . . . . . 80

7.1 Description of the 2011 CBWA motivational data set . . . . . . . . . . . 84

7.2 Overview of the sequence of models fitted for the York trial. . . . . . . . 98

7.3 Spatial modeling of the 2011 growing season trials . . . . . . . . . . . . 101

8.1 Location based summaries of the 2011 METs . . . . . . . . . . . . . . . 108

xxi

LIST OF TABLES

8.2 Concurrence of entries across the 2011 MET. . . . . . . . . . . . . . . . 109

8.3 Outliers detected at the MNGN6 site. . . . . . . . . . . . . . . . . . . . 113

8.4 Spatial modeling for the 2011 METs . . . . . . . . . . . . . . . . . . . . 114

8.5 REML estimates of percent of variance accounted for by each factor of

the the FA(2) model for the additive and non-additive genetic effects. . 116

8.6 Genetic variance models fitted for the MET . . . . . . . . . . . . . . . . 116

8.7 REML estimate of the genetic correlation matrix for additive and non-

additive genetic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.8 Levels of inbreeding for entries in the 2011 MET data set . . . . . . . . 123

8.9 REML estimates of total genetic correlation matrix for hybrid and non-

hybrid entries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.10 REML estimates of proportion of additive (%), non-additive (%) and

total genetic variance from the FA(2) model. . . . . . . . . . . . . . . . 130

8.11 Summaries of trials obtained from each cluster group . . . . . . . . . . . 139

9.1 Time taken for completion of an iteration for two algorithms . . . . . . 157

9.2 Summary information on CBWA data subsets. . . . . . . . . . . . . . . 158

9.3 Sequence of models fitted for genetic variance structures . . . . . . . . . 160

xxii

Glossary

AFLP Amplified Fragment Length Poly-

morphism

AI average information algorithm

AIS alike in state

ANCOVA analysis of covariance

ANOVA analysis of variance

AOMM Alternative Outlier Mixed Model

AR1 Autoregressive process of order 1

CAA Canola Association of Australia

CBWA Canola Breeders Western Australia

Pty Ltd

C-BLUP cluster - Best Linear Unbiased Pre-

dictor

COF coefficient of co-ancestry

DH Double Haploid

DTF Days to flowering

E-BLUE empirical-Best Linear Unbiased Esti-

mator

E-BLUP empirical-Best Linear Unbiased Pre-

dictor

FA factor analytic model

FHB Fusarium Head Blight

FP flour protein

FPC flour protein content

FY flour yield

GCA general combining ability

GPC grain protein content

GxE Genotype by environment interac-

tion

GYS grain yield per spike

HD heading date

IBD identity by descent

KNS kernel number per spike

MET multi-environment trial

MME Mixed model equations

NBG National Blackleg Group

N-gen non-genetic

NVT National Variety Trial

PH plant height

p-rep replicated plots for a percentage (p)

of the test lines

PSI particle size index

QTL Quantitative Trait Loci

REML Residual Maximum Likelihood

RFLP Restriction Fragment Length Poly-

morphism

RILs Recombinant Inbred Lines

RR reduced rank model

SCA specific combining ability

Scres Studentised conditional residuals

SSR Simple Sequence Repeat

TKW thousand-kernel weight

xxiii

GLOSSARY

xxiv

Chapter 1

Introduction

The two main species fo oilseed rape, that is Brassica napus L. and Brassica rapa L.

provide 13% of the worlds oilseed supply, and form the second largest oilseed crop

(Raymer, 2002). In Australia, canola (Brassica napus L.) is the most important oilseed

crop. In global production rankings, Australia is the second largest exporter of canola

(Wang et al., 2009), accounting for a total value of $1.7 billion in the years 2011-2012

(ABARES, 2012). Besides its cash crop value, canola has various on-farm benefits when

grown in rotation with cereal crops, including the control of root diseases in ensuing

cereal crops and additional weed management options (Norton et al., 1999). The most

valuable component of canola is the oil, which has the added nutrition benefits of low

erucic acid (less than 2%) and meal with less than 30µmol of aliphatic glucosinolates

per gram (Raymer, 2002).

Broad acre crops such as canola face a challenging future, due to an increasing global

population and higher demand for food production, while simultaneously facing large

scale challenges from global environment change (Tester and Langridge, 2010). As a

result, broad acre crop production needs to increase with less reliance on greater inputs

for production. This is where plant breeding has a major role to play. It is important to

recognise that there is scope to improve the efficiency of breeding and selection methods

of crops through research into the statistical analysis of plant breeding trials.

Breeding is a series of procedures that aims to change (genetically) the phenotype of

a potentially economic species of plant and animals (Comstock et al., 1996). As such,

1

1. INTRODUCTION

plant breeding is defined by Allard (1999) as consisting of three main ideas: “1) the

expression of genes, 2) the behaviour of genes in populations and 3) the evolution of

breeding populations by allelic substitutions under natural selection supplemented by

artificial selection imposed by breeders”’ (p. 48). Ultimately, the aim is to use these

ideas to produce new varieties that are superior to those already in the market, in terms

of traits of economic importance such as yield and quality etc. As a result the success

of a plant-breeding program is based on the efficiency of selection methods.

In canola breeding, selection is undertaken for traits such as grain yield, blackleg dis-

ease resistance, oil content, protein content, vigor, maturity and plant height amongst

other traits (Salisbury and Wratten, 1999). Selection is based on the phenotype, which

is an observable/measurable trait of an individual, and is composed of two components,

the sum of the total genetic effects of all loci for the trait (G) and an environmental

deviation (E) (Lynch and Walsh, 1998). This recognises that most traits of interest are

the result of the combined action of many genes and non-genetic influences. In terms

of E, the target environments for canola in Australia have vastly different growing con-

ditions. Canola is predominantly grown across southern Australia (Fig. 1.1), from the

sand-plain agriculture with winter dominated rainfall conditions in Western Australia

to the clay loamy soils of Eastern Australia which are characterized by equi-seasonal

rainfall (Kirkegaard et al., 2011). Such growing environments have been previously

reported as extremely variable between locations as well as between seasons (Chapman

et al., 2003).

Figure 1.1: Canola production regions across southern Australia - Shaded areasindicate potential geographic regions where canola is grown across Australia. This diagramis reproduced from ABS (2008).

2

The principal objectives of a plant breeding program are to select new combinations

of genotypes/entries for such target population of environments (Comstock et al., 1996),

for release as new commercial varieties and also as parents for the next cycle of breeding

and selection. Selection is based on measurements on variety plots from designed trials

across multiple locations, termed Multi-environment trial(s) (METs). The standard

for selection in these programs is based on Best Linear Unbiased Predictions (BLUPs)

of variety effects from mixed model analysis (Bauer et al., 2009, Piepho et al., 2008).

Such trials and analysis methods not only enable an estimate of genetic value, but

also breeding value when pedigree (ancestry) information is included (Oakey et al.,

2006, 2007). Note that here and in other parts of the thesis genetic value refers to

the total genetic effect of an individual which is composed of component additive and

non-additive genetic effects, and breeding value refers to the additive component only

and represents the ability of an individual to pass on their alleles to their progeny

(Bauer et al., 2009). The inclusion of pedigree information in mixed model analysis is

an attempt to model the gene to phenotype relationship previously reviewed by Cooper

et al. (2005).

A brief introduction is only provided here, as there are two literature reviews in this

thesis, which provide an in depth discussion on the literature concerning components

of research: correlated traits, ancestry (pedigree information) and environments.

The first half of the thesis focuses on correlated traits. While selection is usually

undertaken on several traits within a breeding program, plant breeding programs rarely

use multivariate methods which are common place in animal breeding programs (Com-

stock et al., 1996, Piepho et al., 2008). Selection on multiple traits avoids any bias

in selection especially when traits are highly correlated (Lin et al., 1985). Using the

motivational data set comprising plant survival data from the National Blackleg trials

across Australia, Chapter 2 provides a literature review on the analysis and measure-

ment of plant survival data. Following on from this, Chapter 3 describes and applies

a bivariate mixed model approach for the analysis of plant survival data. Chapter 4

presents a literature review, extending the applications of the bivariate mixed model

approach to other plant breeding selection experiments.

The promotion of new cultivars in a breeding program is based on a large set of

3

1. INTRODUCTION

potential genotypes tested across a set of target environments, so the estimation of

genetic value is the core of a breeding program (Piepho et al., 2008). The inclusion

of pedigree information by Oakey et al. (2007) in mixed model analysis of MET data

has resulted in plant breeding programs increasingly using breeding values for parental

selection (Atkin et al., 2009, Beeck et al., 2010, Crossa et al., 2006, Cullis et al., 2010,

Kelly et al., 2009). A literature review on the inclusion of pedigree information in plant

breeding trials is covered in Chapter 5. This is followed in Chapter 6 by the presentation

of a background review of the motivating data set of the second half of the thesis, an

actual plant breeding program data set kindly provided by Canola Breeders Western

Australia Pty Ltd, coded for anonymity.

In the second half of the thesis, Chapter 7 focuses on another method of improving

gain from selection, that is, through the control of environmental effects using spatial

analysis within the mixed model framework. Data from field trials exhibit spatial

variation, which arises from the physical location of plots within a field (Smith et al.,

2002b). If not accounted for, the presence of extraneous variation can complicate the

analysis, as well as reduce the efficiency of selection (Stefanova et al., 2009). This

is first addressed through observing the inclusion of ancestry (pedigree) information

in mixed models in Chapter 7. Chapter 8 then uses these spatial models within a

MET framework to observe how environment impacts on entry performance (genotype

by environment interaction) of entry types (hybrid and non-hybrid) within a breeding

program. The last chapter covers a topic of interest that arose during research, which is

a potential barrier to adoption of mixed model analysis with pedigree information - the

extensive time to analysis completion. While this thesis focuses canola, the outcomes

of this research has implications for all plant breeding programs whether hybrid or

self-pollinated crops.

4

Chapter 2

Literature Review - Methods of

measurement and analysis of

plant survival data sets

2.1 Introduction

This chapter presents details for the measurement and analysis of plant survival counts.

An overview of the current methods of analysis for plant survival counts is presented

before the introduction of the bivariate method of analysis for blackleg (Leptospheria

maculans) disease incidence data for canola (Brassica napus L.) varieties. In Chapter

3 the bivariate analysis methodology is developed and applied to plant survival counts

obtained from a set of Australian blackleg disease resistance trials. The bivariate mixed

model approach is readily applicable to designed field experiments and can be applied

to various selection experiments. The discussion in this and the following chapter is

limited to the scope of disease incidence, as Chapter 4 discusses further applications of

this method in current plant breeding literature.

In this thesis the motivational data set consists of two sets of plant survival counts

taken at different sampling times, emergence and maturity, to determine disease in-

cidence. Historically such data have been analyzed using a derived variable, percent

survival, which are the maturity counts divided by the emergence counts, multiplied

5

2. LITERATURE REVIEW - METHODS OF MEASUREMENT ANDANALYSIS OF PLANT SURVIVAL DATA SETS

by a hundred. This chapter instead explores the analysis of such data using a bivariate

framework of analysis, arguing that each trait (that is, counts at emergence and at

maturity) has different biological, environmental and genetic factors and thus should

be treated as individual traits. This chapter commences with a description of measures

of disease in plant breeding experiments and is followed by a discussion of the bivariate

analysis in the context of the blackleg disease resistance data set in terms of biological

and statistical arguments.

2.2 Measures of disease

Plant disease infection can range from mild symptoms to large scale crop destruction.

Biologically, the main method of plant pathogen control is the breeding of host plant

resistance (Waller et al., 2002). Accurate measures of disease are critical not only

for the identification of disease, but also for selection against disease resistance in the

field (Rempel and Hall, 1996). There are numerous methods of measuring disease, a

common method is disease incidence, defined as the number of plants infected out of the

total number of plants assessed (Parlevliet, 1979). In some cases, measures of disease

incidence can be taken across time, resulting in multiple observations (Parlevliet, 1979).

This is the case with the measurement of blackleg disease of canola in Australia, where

incidence is measured in terms of counting the number of seedlings that have emerged

and then recounting the number of plants present at maturity. These data have been

used to compute a derived variable, percentage survival of established plants, which

is analyzed as the trait of interest. These measures are undertaken in designed field

trials across Australia (Li et al., 2008, Marcroft et al., 2012) and in France (Pilet et al.,

1998). In the proposed study however, the analysis of blackleg disease incidence is

considered within a bivariate framework where the plant survival counts are treated as

two separate traits.

2.3 Blackleg disease incidence

Blackleg is a fungal disease of Brassica napus (rapeseed or canola) (Punithalingam

and Holliday, 1972), which causes severe yield losses in Australia and worldwide (Fitt

6

2.3 Blackleg disease incidence

et al., 2006, West et al., 2001). Grain yield losses associated with blackleg have been

reported to range from less than 10% to greater than 50% (Hall, 1992, West et al.,

2001, Zhou, 1999). Blackleg disease is of special interest in Australian agriculture, as it

destroyed most rapeseed crops soon after their introduction to Australia in the 1960’s

and 1970’s (Gugel and Petrie, 1992, Khangura and Barbetti, 2001) and discouraged

further attempts to grow the crop for several years. The industry however recovered,

stimulated primarily by the release of canola varieties in 1993 with increasing levels of

resistance to blackleg (Khangura and Barbetti, 2001). Since then, the acreage planted

to canola has increased dramatically due to crop profitability (ABARES, 2011), and

agronomic benefits associated with canola in crop rotations, which include the control

of cereal root diseases and flexibility in weed management (Kirkegaard and Sarwar,

1999, Turner, 2004). However, blackleg disease remains an ongoing threat to Australian

canola production due to favorable conditions for epidemics in Australian environments.

Figure 2.1: The lifecycle of blackleg disease - Reproduced from Howlett et al. (2001).

The lifecycle of L. maculans comprises a single sexual generation of ascospores and

multiple asexual generations of pycnidiospores (Hayden et al., 2007). Ascospores, the

7


primary inoculum, are discharged from pseudothecia formed on stubble remnants, on

which the fungus survives over the summer period in Australia (Gladders and Musa,

1980, Hall, 1992, McGee and Emmett, 1977, West et al., 2001). Ascospore release

occurs after rain events (McGee and Emmett, 1977). Hence, in Australian agricul-

tural systems seedling establishment and ascospore release coincide, providing ideal

conditions for severe crown canker epidemics (Barbetti and Khangura, 1999). Seedling

infection occurs when the fungus enters the cotyledons through stomata or wounds in

which the hyphae extend (Hammond and Lewis, 1987, Hayden et al., 2007, Howlett

et al., 2001). The fungus then grows internally from leaf infections through petiole and

stem tissue to the crown of the plant, where it causes cell necrosis and the girdling

of the stem. The crown rot is accompanied by black or purple staining of the stems,

which is characteristic of the disease (Howlett et al., 2001).

To date, the main method of controlling blackleg disease is through breeding for im-

proved cultivar resistance (Kirkegaard et al., 2006, Rimmer and Van den Berg, 1992).

This is confirmed on a national scale in Australia by the annual testing and publication

of the National Blackleg Resistance ratings for commercial canola varieties. These resis-

tance ratings are determined by measuring disease incidence from designed field trials

at multiple locations across southern Australia, coordinated by the National Blackleg

Group (NBG) with individual trials managed by public researchers and private plant

breeding companies. The importance of these disease ratings to farmers is reflected by

the recent publication of the resistance ratings on the National Variety Trial (NVT)

data base (http://www.nvtonline.com.au/home.htm).

While there are numerous methods for quantifying blackleg disease infection, see

Rimmer and Van den Berg (1992) for full listing, the rating of designed field experiments

necessitates the use of a measure that is not only quick, but also relatively easy and

accurate to undertake. Plant survival counts, which compares counts at emergence

and maturity, are relatively easy to measure on large scale field trials. Further such a

measure reflects the economic losses associated with blackleg disease (Fitt et al., 2006).

Plant survival counts have been successfully used for measuring disease resistance in

many Australian plant breeding programs for the past 30 years (Marcroft et al., 2002).

As a result, the annual National Blackleg Resistance ratings, published by Canola

Association of Australia (http://www.australianoilseeds.com) and some researchers use

8

http://www.nvtonline.com.au/home.htm

http://www.australianoilseeds.com/commodity_groups/canola_association_of_australia/pests__and__disease

2.4 Bivariate analysis

this method as a measure of disease incidence (Li et al., 2008, Marcroft et al., 2002,

2012).

The studies that have used plant survival counts as a measure of blackleg disease

incidence, (Li et al., 2008, Marcroft et al., 2002, 2012), all use a univariate analysis of

the percent survival values within an analysis of variance (ANOVA) framework. The

only other study in blackleg disease to use a variation of this disease incidence measure,

percentage of plants infected per plot, is the study by Rempel and Hall (1996). In this

case, they also used a univariate analysis of percent infected plants, however this was

within a repeated measures ANOVA framework.

Traditionally, the NBG have analyzed ‘percent survival’ values, which are calculated

by dividing maturity counts by emergence counts and multiplying this by a hundred.

This derived variable is then subjected to a univariate analysis using the spatial mixed

model approach of Gilmour et al. (1997). This enables a single site analysis of each of

the disease nursery trials to determine spatial models for errors as well as to diagnose

and remove outliers. These single sites are then combined across sites in a second stage

of analysis, known as a Multi Environment Trial (MET) and analyzed using a factor

Analytic (FA) variance structure of (Smith et al., 2002b). The MET analysis enables

an individual genetic variance for each site and a genetic covariance between pairs of

sites (Smith et al., 2001b). To distinguish this analysis from the proposed bivariate

approach, the univariate analysis will be referred to as the ‘historical analysis’. This

chapter proposes the use of a bivariate method of analysis where the two analyzed

‘traits’ are the plant survival counts at emergence and at maturity.


2.4.1 Biological motivations

The main motivation for a bivariate analysis of plant survival data is that each sam-

pling time, emergence and maturity, constitutes an individual ‘trait’, and each may

be effected by different biological, genetic and environmental factors. Hence it would

not only be statistically but also biologically more accurate to determine trait specific

spatial models, outlier detection and error and genetic variance. This section discusses

9


the biological reasons for using a bivariate framework for the analysis of plant survival

data for blackleg disease.

There are two types of varietal resistance to blackleg disease, quantitative (poly-

genic) and qualitative (major gene) (Leflon et al., 2007). Quantitative resistance is

evaluated in adult plants in field nurseries and results in the reduced severity of disease

symptoms, however it is known to be partial, and can succumb under high disease pres-

sure resulting in significant yield losses (Khangura and Barbetti, 2001, Sivasithamparam

et al., 2005). Further, quantitative resistance is also known to be strongly affected by

environmental conditions (Balesdent et al., 2001, Delourme et al., 2008, Fitt et al.,

2006). Qualitative resistance however, is controlled by major genes, and can provide

complete resistance to disease symptoms and infection (Ansan Melayah et al., 1997).

This is race-specific resistance, that is, provides resistance against races of blackleg and

as a result exerts a higher selection pressure on the blackleg population.

In addition to variation in the type of varietal resistance, studies have also shown

that these types of resistance are different based on the stage of plant growth (Ballinger

and Salisbury, 1996, Rempel and Hall, 1996, Roy, 1984). Ballinger and Salisbury (1996)

demonstrated that there is a differential response in seedling and mature plant resis-

tance to blackleg and in some cases resistance improves with age. This was recognized

by the above study of Rempel and Hall (1996) who attempted to observe the differential

biological factors associated with the sampling time in field evaluation trials for black-

leg disease using a repeated measures ANOVA framework. Bivariate analysis allows for

changes in genetic variance across sampling times and this could provide insight into

the different mechanisms of disease resistance present at the particular plant growth

stage.

The epidemiology of blackleg also differs at the two sampling times. The focus of

attention on blackleg infection is usually at the mature plant stage, as this is when

economic losses occur due to reduced seed production. However, studies have also

demonstrated that blackleg infection at the seedling stage can result from soil borne

ascospores and pycnidiospores (Li et al., 2007, Sosnowski et al., 2006). The study by

Li et al. (2007), found that infection at the seedling stage can result in seedling death.

10


A bivariate analysis of the two plant counts may provide insight into the differential

impact of blackleg disease at the different sampling times.

In addition to the above, counts at emergence are also affected by different environ-

mental and biological factors that arise at seedling emergence, which could be caused

by seed source differences. Seedling emergence is a factor that cannot be controlled

in disease nurseries, as it is affected by soil fertility, salinity, compaction, tillage and

surface residues (Forcella et al., 2000). Seed source differences on the other hand arise

in seed lot variations, from factors such as age of seed (Finch Savage, 1986), the storage

environment of the seed (Ellis and Roberts, 1980), and seed production environment

(Ellis et al., 1993). While variation in seed source is a known issue across Australian

blackleg disease nurseries, these issues have been confounded in the past with disease

effects in the derived variable, percentage survival.

Thus there are biological, genetic and environmental differences between the two

sampling times and this necessitates the treatment of them as individual traits. The

bivariate analysis is able to accommodate this, hence it will enable a discussion on such

sampling time factors, unlike the historical analysis in which such effects are masked.

2.4.2 Statistical motivations

Statistically, the bivariate approach is preferred as it allows for (i) the modeling of

error, such as spatial field trend for each trait (ii) the identification of outliers for each

trait and (iii) the examination of individual trait genetic effects. For points one and

two, these may be masked when using the derived variable of the historical approach.

With the third point, an examination of the genetic effects for each trait may reveal

greater insight into plant pathogen interactions.

The modeling of spatial trend for each trait is a valuable component of a bivariate

framework. Previous studies have demonstrated that improved estimates of treat-

ments effects are obtained after correcting for environmental effects in designed field

experiments, for both agriculture or forestry experiments (Dutkowski et al., 2006). In

agricultural field trials this is achieved through the use of spatial analysis (Cullis et al.,

1998, 2006, Gilmour et al., 1997). Until recently, spatial analysis has mainly focused on

11


annual crops or forestry trials where only one measure is taken on a plant (de Resende

et al., 2006). Other than the study by de Resende et al. (2006) there are very few

examples of studies which evaluate the impact of spatial analysis on repeated measures

or multivariate data.

A forestry based study by Dutkowski et al. (2006) indicated that spatial analysis

improved genetic response predictions by more than 10% for 20 out of the 216 traits,

tested. Most importantly this study demonstrated that some traits (growth) responded

better to spatial analysis than others (fungus damage) resulting in significant model

improvement. Dutkowski et al. (2006) also showed that many measures of model fit,

such as in error variance, prediction accuracy and standard error of genetic variance

estimates improved from the modelling of individual trait spatial error. This study

demonstrated that spatial analysis can lead to modest to large improvements in selec-

tion. In the blackleg data set the plant counts are taken on the same plot, however the

type of error at each sampling time could possibly reflect a different spatial modelling

for each trait, as they are known to be affected by different environmental, biological

and genetic components.

The measurement of plant counts at two sampling times on the same variety plot

is essentially a form of a repeated measures experiment. An important feature of

repeated measures experiments is that the measures on the same experimental unit

or sequence of measures in time are likely to be correlated (Gurevitch and Jr, 1986,

Littell et al., 1998). Hence it is important to model such variance covariance structure

in mixed model analysis (Littell et al., 1998, Piepho and Mohring, 2006). Under the

historical analysis this was ignored by using a derived variable, which does not require

the modeling of this covariance. Given that the aim of testing entries in the blackleg

disease nurseries is to accurately determine disease resistance ratings of commercial

canola varieties, a bivariate mixed model methodology enables such data to be more

accurately modeled.

Further, the variances of the repeated measures may often change with time (Littell

et al., 1998), which was demonstrated to be the case in blackleg by the study of Rempel

and Hall (1996). Of particular interest is the variance attributed to each sampling time

as it may be a reflection of the different genetic, biological and environmental impact

12

2.5 Summary

of the disease. As a result it will be an important component of the bivariate analysis

to be able to model the individual variances and covariance between sampling times.

Selection is improved when based on multiple traits, so that any bias in selection

due to correlated traits is avoided (Kerr, 1998). Even slight improvements of accuracy

can result in large economic effects in large populations (Pollak et al., 1984), which is

often what is encountered in breeding programs. As a result, multi-trait analyses are

commonly utilized in animal breeding programs, (Henderson and Quaas, 1976, Mrode

and Thompson, 2005) yet there are very few plant breeding programs in annual crops

that utilize this (Piepho et al., 2008).

Theoretically, multivariate methods can result in increases in the accuracy of eval-

uations as it utilizes information from phenotypic and genotypic correlations between

traits (Mrode and Thompson, 2005). In addition, the studies by Thompson and Meyer

(1986) and Villanueva et al. (1993) have shown that a bivariate analysis can result in

gains in accuracy in evaluation for a trait when using other correlated traits. Further,

multivariate analysis would eliminate any potential bias that occurs from the selection

of a correlated trait (Pollak et al., 1984, Kerr, 1998), that is, any bias in the evalua-

tion that arises due to disregarding covariance structures between traits is avoided (Lin

et al., 1985).

2.5 Summary

Plant survival data are often composed of multiple measures used to form a derived

variable such as percent survival values. These derived values are then subject to a uni-

variate analysis. Chapter 3 will develop and apply a bivariate mixed model approach

where the multiple measures are realized as individual traits. This is demonstrated us-

ing the motivational example of a set of designed field trials of blackleg disease incidence

data from the Australian National Blackleg Resistance trials. This literature review

has discussed how such counts can be subject to different biological, environmental

and genetic factors, and the bivariate framework can be statistically more accurate in

accommodating this with trait based spatial modeling, outlier detection and genetic

variance.

13


14

Chapter 3

A bivariate mixed model

approach for the analysis of plant

survival data

The motivating data set for this chapter consists of a series of blackleg disease nurs-

ery trials, kindly provided by the National Blackleg Group (NBG). These trials are

used to determine the annual disease resistance ratings for canola varieties and are co-

ordinated by the NBG and published annually by the Canola Association of Australia.

The NBG is responsible for deciding the published blackleg rating for each entry, which

by convention is based on an analysis of the previous three years of blackleg plant sur-

vival data (square root percentage survival) under high disease pressure. This chapter

presents a bivariate mixed model methodology for the analysis of such a plant survival

data set. The chapter commences with an overview of the current protocol for the run-

ning of the National Blackleg Disease nurseries and the measurement of plant disease

is presented. This is then followed by a section on the description of the mixed model

approaches (univariate and then bivariate) each followed by an example using the York

disease nursery site and summarized for the other disease nursery sites. This chapter

concludes with a discussion on the bivariate mixed model approach. The methodology

and analysis presented in this chapter has been published in the journal Euphytica,

and a reprint of this submission is attached in the Appendices (Appendix A).

15

3. A BIVARIATE MIXED MODEL APPROACH FOR THE ANALYSISOF PLANT SURVIVAL DATA

3.1 Data set description

The data comprises 140 commercial and unreleased entries (varieties) of B. napus from

the 2009 growing season disease nurseries. These disease nurseries were located at 6

sites across southern Australia in canola producing areas of medium to high rainfall

(Table 3.1 and Fig. 3.1). Disease nursery sites were managed and run by public

researchers and private breeding companies. The sites were composed of designed

experiments with varieties from all the herbicide groups, i.e. conventional, Clearfield R©

and Triazine Tolerant (Table 3.1). In this data set all the trials were designed as

randomized complete block designs, sometimes with extra replicates of control entries.

All experiments were laid out as a rectangular array indexed by rows and columns

(Table 3.2). The design and implementation of these trials were left up to the discretion

of the breeding companies or research groups managing them. However the NBG

coordinated trial management and ensured quality assurance through the use of unified

protocols, see Marcroft (2009) for a full listing of disease nursery protocols. High disease

levels at all nurseries were maintained by growing entries alongside or on disease stubble

obtained from the previous season.

Bakers Hill●

Clear Lake●

Shenton Park●

Wagga Wagga●

Wonwondah●

York●

Figure 3.1: Location of blackleg disease nurseries across Australia - Geographiclocations of the six blackleg (Leptospheria maculans) disease nurseries across southernAustralia during the 2009 growing season.

16

3.2 Measuring disease incidence

Table 3.1: Location based trial details: state, stubble type, entry herbicide type andaverage plant counts at emergence (eme) and maturity (mat) for each of the 2009 blacklegdisease nurseries.

Location State Stubble Type Herbicide Type? AverageEme Mat

Bakers Hill WA Bravo TT C, Cl, TT 60 35Clear Lake VIC 45Y77 C, Cl, TT 50 33Shenton Park WA CB Telfer C, Cl, TT 75 57Wagga Wagga NSW Bravo TT C, Cl, TT 34 10Wonwondah VIC AV-Garnet C, TT 37 14York WA ATR-Cobbler C, TT 59 13

?Herbicide type acronyms: C=Conventional, Cl=ClearfieldR© and TT=Triazine Tolerant

Table 3.2: Details of blackleg disease nursery experiments during the 2009 growing season.The number of entries, columns, rows and blocks are listed for each experiment in this dataset.

Location Location Code Entries Columns Rows Blocks?

Bakers Hill BH 57 3 57 3Clear Lake CL 18 4 20 4Shenton Park SP 65 22 9 4Wagga Wagga WA 74 15 16 3Wonwondah WO 31 12 10 3York YK 78 3 79 3

?Note that “Blocks” correspond to biological replicates.

3.2 Measuring disease incidence

Plots of entries were sown with a targeted minimum of 100 seedlings per plot. Plant

counts were first taken at emergence, which corresponds to the open cotyledon stage of

plant growth and occurs approximately 4− 6 weeks after plant emergence. Plants are

then recounted at maturity that is the windrowing stage. Disease nursery sites were

only included in the analysis if there was less than 30% survival on susceptible control

entries. Historically, plots with less than 20% emergence were deemed unreliable and

defined as missing, however for this analysis the data from these missing plots were

obtained from trial managers and included in the data set.

17


3.3 Univariate analysis

The count data were first log-transformed before analysis. This ensured that the resid-

uals approximated a Gaussian distribution with a constant variance. This has been

historically appropriate for this data set and also ensures that the predicted counts are

non-negative, which is of biological significance to this analysis.

The component traits of the bivariate analysis, plant survival counts at emergence

and maturity, were first each subjected to a univariate analysis to enable appropriate

spatial model selection using the approach of Gilmour et al. (1997). Field experiments

often have spatial variation due to the physical location of individual plots in the

field. The approach of Gilmour et al. (1997) enables modeling of spatial trend for field

trials, which accounts for three sources of variation namely global, local and extraneous

variation. Global trend refers to variation that occurs across the field, local represents

short-term trend such as soil fertility and extraneous variation is often the result of

experimental procedures that are aligned with rows and columns (Gilmour et al., 1997).

Local trend is accommodated within the mixed model by an appropriate covariance

structure of which the separable autoregressive process of order 1 (denoted AR1×AR1)

is the most commonly used (Gilmour et al., 1997). Models for non-genetic variation

encompass model terms for both experimental design and spatial variation.

3.3.1 Statistical model

Each disease nursery is comprised of a rectangular array of plots with r rows and c

columns, so that the number of plots in an experiment is given by n = rc. Additionally,

m is the number of entries and b is the number of blocks in the experiment.

The base-line spatial mixed model for the (log transformed) plant survival counts for

each sample time (j = 1, 2), with j corresponding to 1 at emergence and 2 for maturity

can be written as,

yj = Xτ j +Zvuvj +Zbubj + ej (3.1)

; yj is a n×1 vector of plant survival counts for individual plots within an experiment,

ordered as rows within columns; X, Zv and Zb are design matrices for fixed effects,

18


random entry effects and random block effects respectively; τ j is the vector of fixed

effects; uvj is the m×1 vector of random entry effects; ubj is the b×1 vector of random

block effects and ej is the vector of residuals ordered as per the data vector. There are

no sub-scripts associated with the design matrices since, for the base-line model, they

are the same for both sampling times.

The assumptions for the univariate base line model (Equation 3.1) are,

E(yj) = Xτ j

E(uvj) = E(ej) = 0

The variance assumptions for the entry effects in Equation 3.1 are:

var(uvj

)= σ2

vjIm

where σ2vj is the entry variance at sampling time j and Im is an identity matrix of

order m.

For block effects, the variance assumptions are:

var(ubj

)= σ2

bjIb

where σ2bj is the block variance at sampling time j and Ib is an identity matrix of order

b.

The variance matrix for the errors (Rj) assuming a separable AR1 process is:

var (ej) = Rj = σ2jΣcj ⊗Σrj

where σ2j is the error variance at sampling time j, and Σcj and Σrj are correlation

matrices of dimensions c×c and r×r for columns and rows respectively of AR1 processes

in the column and row directions. Each matrix is a function of a single autocorrelation

parameter ρcj and ρrj for the column and row dimensions respectively. Note that in

some experiments where there were four or less columns, it was assumed that there was

independence for errors in the column dimension, so that Σcj=Ic.

The var(yj)

is then,

var(yj)

= σ2vjZvZv

T + σ2bjZbZb

T +Rj

19


3.3.2 Checking the adequacy of the spatial model

Following the univariate analysis, an examination of the adequacy of the spatial models

was undertaken. This involved using two diagnostics, a 3D sample variogram and a plot

of residuals against row/column numbers (termed as residual plots) from Gilmour et al.

(1997). Residual plots are used to observe for local trend and possible outliers. The

sample variogram, enables for the visualization of extraneous variation/global trend as

well as to check the adequacy of the variance structure for local trend. If additional

terms were needed to accommodate any observed extraneous variation, they were added

to the initial base mixed model. For example, τ j would include additional terms for

linear regression across rows, or additional random effects terms would be added to the

base-line model.

While the residual plot is used to visualize possible outliers, the Alternative Outlier

Mixed Model (AOMM) in ASReml-R (Smith et al. unpublished), was used to pro-

duce Studentised conditional residuals as part of the outlier identification diagnostics.

Studentised conditional residual values greater than 3.5 where identified as outliers,

however the plant breeder was still consulted to confirm these.

3.3.3 Estimation and Fitting

The fitting of mixed models involves two processes, firstly the variance parameters

(σ2vj ,σ

2bj , ρcj , ρrj and σ2

j ) are estimated using the REML method of Patterson and

Thompson (1971) and secondly these estimates are then used to solve the mixed model

equations (Henderson, 1975) (Equation 3.2). This results in (empirical) Best Linear

Unbiased Estimates of fixed effects (E-BLUEs), and (empirical) Best Linear Unbiased

Predictions of the random effects (E-BLUPs). The term ‘empirical’ is used as the

variance parameters are unknown and are estimated form the data.

20


The mixed model equations for the base line univariate model (Equation 3.1) are, XTR−1j X XTR−1

j Zv XTR−1j Zb

ZvTR−1

j X (ZvTR−1

j Zv + (σ2vj)

−1Im) ZvTR−1

j Zb

ZbTR−1

j X ZbTR−1

j Zv (ZbTR−1

j Zb + (σ2bjIb)

−1)

τ j

uvj

ubj

(3.2)

=

XTR−1j yj

ZvTR−1

j yj

ZbTR−1

j yj

where τ j is the E-BLUE of the fixed effects and uvj and ubj are the E-BLUPs of

the random effects for entries and blocks respectively.

All models in this chapter and the thesis were fitted in the software package ASReml-

R (Butler et al., 2009).

3.3.4 Univariate analysis results

3.3.4.1 York disease nursery

The univariate analysis is described in detail for the disease nursery at York. The York

disease nursery had n = 237 plots, with r = 79 rows and c = 3 columns, b = 3 blocks

and m = 78 entries, see Table 3.2. There should be 79 entries, however due to lack of

seed for one of the entries, an extra plot of another entry was sown. The initial base

line model, equation 3.1 was fitted, with independence assumed in the column direction

for the spatial model, so that var (ej) = Rj = σ2j I ⊗Σrj .

First the emergence model is considered. The resulting plot of residuals and the

sample variogram can be seen in Fig. (3.2). The residual plot indicated the presence

of three outliers. These were confirmed by checking AOMM statistics, which indicated

unusually large studentised conditional residuals. These were omitted from the analysis

by setting the plots to the missing value qualifier. In addition, the sample variogram

indicated the presence of extraneous variation in the row direction, observed by the

up/down pattern. This was accommodated by fitting random row effects in the model.

Having removed the outliers and included a term for random row effects, the model for

21


emergence was then refitted,

y1 = X1τ 1 +Zv1uv1 +Zb1ub1 +Zr1ur1 + e1 (3.3)

where the dimensions of the respective matrices are as follows, y is a 237×1 data vector;

τ is the grand mean with corresponding design matrix X with dimensions 237× 1; uv

is a 78 × 1 vector of random entry effects with corresponding design matrix Zv of

dimensions 237 × 78; ub is a 3 × 1 vector of random block effects with corresponding

design matrix Zb of dimensions 237 × 3; ur is a 79 × 1 vector of random row effects

with corresponding design matrix Zr of dimensions 237× 79; and lastly, e is a 237× 1

vector of residuals

The resulting REML estimates of variance parameters for this equation are:

σv21 = 0.4

σb21 = 0.04

σ12 = 0.382

ρr1 = 0.72

The re-fitting of Equation 3.3 resulted in the sample variogram in Fig. 3.3, which

indicated a more adequate spatial model.

In terms of REML estimates, the genetic variance component was non-zero for emer-

gence (0.400), however the block variance component was almost zero at 0.041. The

row autocorrelation value was large at 0.72 indicating strong smooth spatial variation.

The error variance for the emergence mixed model was 0.382.

Now consider the maturity model at this nursery site. There appeared to be no

extraneous variation, and only a single outlier was detected and set to a missing value.

Similar to the emergence mixed model, the REML estimate of the genetic variance

component was non-zero for maturity. The entry variance estimate was larger for

maturity (0.511) than for emergence (0.400) (Table 3.5). The block variance component

were almost zero 0.066 for maturity as well. The error variance for the maturity model,

was smaller than that of emergence model (Table 3.4) and the autocorrelation for trend

in the row direction was much stronger for emergence (0.72) than maturity (0.22) (Table

3.3) .

22


Figure 3.2: York disease nursery initial plot of residuals and sample variogram- Initial plot of residuals and sample variogram from the univariate emergence model atthe York disease nursery.

23


Figure 3.3: York disease nursery final plot of residuals and sample variogram -Plot of residuals and sample variogram from the univariate emergence model at the Yorkdisease nursery after the addition of random row effects and removal of outliers.

24


3.3.4.2 All disease nurseries

For each trial and sampling time, the base line univariate model was fitted (Equation

3.1). Non-stationary trend and extraneous variation components were needed for 5

out of 6 disease nursery sites. None of these sites had the same extraneous variation

components for both traits (Table 3.3). Overall there were more extraneous variation

terms included for the emergence model than the maturity mixed model. Stationary

trend differed between the trait models, for column and row AR1 values (Table 3.3).

There were two instances (Shenton Park and Wonwondah) where spatial correlation was

modeled for the column dimension as well as the row dimension. For these two disease

nurseries, the AR1 values for each dimension differed largely for each trait (Table 3.3).

The row AR1 values ranged from −0.13 to 0.72 for the emergence model and −0.07

to 0.28 for the maturity models. The greatest difference between traits for row AR1

values was for the York and Clearlake disease nurseries (Table 3.3). Further, at 4 out

of the 6 disease nurseries these row AR1 values were larger for the maturity model than

the emergence model. Overall the largest AR1 value was observed for the emergence

trait at York (0.72) and the maturity trait at Wagga (0.28). Additionally, the outliers

removed from the analysis differed for each trait across disease nurseries with only one

disease nursery having the same number of outliers removed for each trait (Table 3.3).

REML estimates of entry variance components across all disease nursery sites were

non-zero, ranging from 0.033 at Wonwondah to 0.400 at York for the emergence models

and 0.127 at Bakers Hill to 0.768 at Wagga Wagga for the maturity models (Table 3.5).

Thus there was entry variation observed for each trait at all disease nursery locations.

Additionally, the entry variance components for maturity were always substantially

larger than those of emergence, except for Bakers Hill where the variance components

appeared similar at 0.108 and 0.127 for the emergence and maturity models respectively.

Across all disease nurseries except Wonwondah, REML estimates of error variance

components were larger for the maturity trait than the emergence model (Table 3.4).

For the emergence models these ranged from 0.015 at Shenton Park to 0.382 at York.

For the maturity models these ranged from 0.031 at Wonwondah to 0.317 at Bakers

Hill (Table 3.4). REML estimates of block variances at all disease nurseries however,

appeared close to 0 (Table 3.3).

25


Tab

le3.3

:S

patial

mod

eling

inu

nivariate

analy

sesof

emerg

ence

(eme)

an

dm

atu

rity(m

at)trait

data

foreach

exp

erimen

t:term

sad

ded

forglob

al

trend

or

extran

eou

sva

riation

,R

EM

Lestim

ates

of

au

toco

rrelatio

np

arameters

(forcolu

mn

san

drow

s,w

here

fitted

)an

dnu

mb

erof

ou

tliersrem

oved.

Exp

erimen

tG

lob

al

trend

&E

xtran

eous

Au

tocorrelation

Blo

ckN

um

ber

ofvariation

terms?

Colu

mn

Row

outliers

Em

eM

atE

me

Mat

Em

eM

atE

me

Mat

Em

eM

at(ρ

c1 )

(ρc2 )

(ρr1 )

(ρr2 )

(σb21 )

(σb22 )

BH

rd(R

)0.19

-0.030.01

0.021

CL

00.26

00

1S

Plin

(R)

0.02-0.17

-0.050.17

00

22

WG

rd(R

)&

rd(C

)lin

(C)

0.240.28

00

2W

Ord

(R)

&rd

(C)

0.160.03

-0.13-0.07

00.1

YK

rd(R

)0.72

0.220.04

0.073

1?lin

(R)

and

lin(C

)in

dica

tesa

fixed

linea

rreg

ression

on

rowor

colu

mn

num

ber;

rd(R

)and

rd(C

)in

dica

tera

ndom

rowand

colu

mn

effects.

26



3.4.1 Statistical model

For the bivariate analysis, the spatial modeling terms from each of the univariate trait

mixed models were carried over to the bivariate model. The mixed model for the

bivariate analysis is given by:

y = X∗τ +Zv∗uv +Zb

∗ub +Zo∗uo + e (3.4)

The response variable, y = (yT1 , yT2 )T , is the combined vector of log transformed plant

survival counts ordered by sampling times (trait), where (yT1 )T and (yT2 )T are the

vectors of log transformed plant survival counts at emergence and maturity respectively.

uv = (uvT1 , uv

T2 )T is the 2m × 1 vector of random entry effects and Z∗

v = I2 ⊗ Zv

is the associated design matrix; ub = (ubT1 , ub

T2 )T is the 2b × 1 vector of random

block effects and Zb∗ = I2 ⊗Zb is the associated design matrix; e = (eT1 , e

T2 )T is the

vector of errors ordered as for the data vector. The vector of fixed effects, τ , includes

an overall mean for each sampling time and any other fixed effects as identified in the

spatial modeling (e.g. linear regression on rows) from the univariate analyses. Any

random effects identified in the univariate analyses are included in the vector uo.

The variance assumptions for the genetic effects in Equation 3.4 are:

var (uv) = var

(uv1

uv2

)=

[σ2v1 σv12

σv12 σ2v2

]⊗ Im (3.5)

where σ2vj (j = 1, 2) is as previously defined, that is the variance of entry effects for each

of the sampling times and σv12 is the covariance between the entry effects at emergence

and maturity. For ease of interpretation, the covariance between entry effects will be

reported as a correlation, namely

ρv12 =σv12√σ2v1σ

2v2

(3.6)

The variance for block effects was similar to entry effects, however the covariance

27


was omitted between the sampling times, as the variances of blocks for both traits were

close to zero at all disease nursery sites. The variance assumptions for the vector uo

were chosen appropriately for the terms involved.

In terms of the errors a separable spatial correlation model was assumed.

var (e) = var

(e1

e2

)=

[σ2

1 σ12

σ12 σ22

]⊗Σc ⊗Σr = R (3.7)

As in the univariate analysis, σ2j (j = 1, 2) are the error variances for each of the

sampling times and σ12 is the covariance between the errors at emergence and matu-

rity. Similar to the entry covariance, the error covariance was converted to a correlation

between the traits, see Equation 3.6. The spatial correlation matrices Σc and Σr cor-

respond to autoregressive processes of order one, that is, functions of single parameters

ρc and ρr respectively. The separability assumption implies that the same spatial cor-

relation parameters are applicable for both sampling times. It may be desirable to

allow different parameters, but such models are not yet available and are the subject

of current research.

After the bivariate model was fitted E-BLUPS of entry means were obtained for

each sampling time. Note that the difference between the predicted entry means for

emergence and maturity corresponded to the percent survival scale of the historical

approach, when back-transformed. To see this, let πjk denote the predicted entry

mean for entry k at sampling time j, then

exp(π2k − π1k) =exp(π2k)

exp(π1k)(3.8)

This transformation enables entries to be assessed on the same scale as the historic

approach, percent survival. The E-BLUPs were used to produce two plots. In the

first entry means at maturity were plotted against entry means at emergence (Fig.

3.4). This plot also included a regression line of maturity against emergence, which

corresponded to the regression of the true entry effects for maturity (i.e uv2) on the

28


true entry effects for emergence (i.e uv1). The slope is given by

β = ρv12 ×

√σ2v2

σ2v1

(3.9)

The second plot, was of the difference between E-BLUPs of entry means for emergence

and maturity plotted against the E-BLUPs of entry means at emergence (Fig. 3.5).

3.4.2 Model Comparisons

There is a potential gain in accuracy of analysis that results when using a multivariate

analysis over a univariate analysis. Prediction error variance is a measure of the gain

in accuracy that results from multiple trait analysis (Henderson, 1973, Thompson and

Meyer, 1986). The accuracy of prediction, as defined by Mrode and Thompson (2005)

is the square of the correlation between the true (uij) and predicted effects (uij) of a

variety (Equation 3.10). In the bivariate analysis of plant survival data the correlation

between true and predicted effect for entry i and sampling time j is given by,

rij = cor(uvij , uvij) =√

1− PEVvij/σ2vj (3.10)

where PEVvij is the prediction error variance, and σ2vj is the estimated genetic variance

for the sampling time (j = 1, 2). Prediction accuracies were obtained for each variety

for each disease nursery for the traits:

1. (log) emergence counts

2. (log) maturity counts

3. the difference in log counts ie. log maturity - log emergence

from three univariate analyses (ie. one for each trait) and a single bivariate analysis

(with log emergence and log maturity counts being the two variables).

To eliminate any effects of variance parameter estimation from the comparisons,

variance parameters for the univariate analyses were constrained to be the same as

those obtained from the bivariate analysis. For example, consider the disease nurs-

ery at Bakers Hill, where the REML estimates of genetic variance from the bivariate

29


analysis were σv21 = 0.109 and σv

22 = 0.131 for the emergence and maturity traits re-

spectively and the estimate of the genetic correlation was ρv12 = 0.68 (Table 3.5). In

the comparable univariate analyses the genetic variances were constrained to be equal

to

1. 0.109 for the analysis of log emergence counts

2. 0.131 for the analysis of log maturity counts and

3. σv21 + σv

22 − 2ρv12σv1σv2 = 0.078 for the analysis of the differences.

Non-genetic components were constrained in a similar manner.

3.4.3 Bivariate analysis results

3.4.3.1 York disease nursery

For the York disease nursery, REML estimates of entry effects for emergence and ma-

turity from the bivariate mixed model were close approximations to those obtained

from the individual univariate models (Table 3.5). Similarly, the error variance compo-

nents from the bivariate model were close approximations to those obtained from the

individual univariate trait analyses (Table 3.4). The AR1 row correlation value under

the bivariate model was 0.362, which was close to the average of the row correlations

obtained under the univariate trait analyses (Table 3.3).

Inherent in the bivariate model structure is a correlation between the traits for the

entry effects and the errors. For the York disease nursery, the correlation between entry

effects was 0.71 and the correlation between errors was 0.59.

The plot of predicted maturity means against emergence (Fig. 3.4) showed large

variation in emergence, with plant counts (on the back-transformed scale) ranging from

10 to 100. The majority of entries were clustered towards the center of the graph with

emergence counts between 20 and 50 and maturity counts between 5 and 20. The

regression slope for this disease nursery was 0.84, indicating a strong linear relationship

between (log) maturity and (log) emergence counts.

30


The plot of the difference between maturity and emergence means against emer-

gence showed that the control entry Surpass501TT had a very low emergence count,

with less than 20 plant counts, and an average percentage survival value of 25% (Fig.

3.5). The highly resistant entry Hyola50 had average emergence, and the highest per-

centage survival value at 65%. The entry 46Y20(J) had the highest plant emergence

and maturity counts (Fig. 3.4) but only an average percentage survival value of 25%

(Fig. 3.5).

In terms of prediction accuracies, higher prediction accuracies were obtained under

the bivariate model for the emergence trait only (1.21%) and there was no percent

improvement under the bivariate model for the maturity trait. For the difference trait

there was a 0.93% improvement in prediction accuracy under the bivariate model com-

pared with the univariate (Table 3.6).

31


Emergence

Ma

turi

ty

1

2

3

4

2.5 3.0 3.5 4.0 4.5 5.0

Surpass501TTHyola5046Y20(J)

10 20 30 40 50 60 70 80

10

20

30

40

50

60

70

80

Figure 3.4: Plot of predicted entry means at maturity against emergence. -Predicted entry means at maturity plotted against predicted entry means at emergencefrom the bivariate model for the disease nursery at York. A regression line of maturityagainst emergence was included, with the slope having a value of 0.84. The axes are ona log scale (as for the analysis) with the back-transformed scale (i.e. plant counts) showninside each axis.

32


Emergence

Ma

turi

ty −

Em

erg

en

ce

−2.5

−2.0

−1.5

−1.0

−0.5

2.5 3.0 3.5 4.0 4.5 5.0


10 20 30 40 50 60 70 80

10

20

30

40

50

60

70

Figure 3.5: Plot of the difference between predicted entry means at maturityand emergence against emergence - The difference between predicted entry meansat maturity and emergence (corresponds to percentage survival when back transformed,these values are shown on the inside of the y-axis) plotted against predicted entry meansat emergence from the bivariate model for the disease nursery at York.

33


3.4.3.2 All disease nurseries

The REML estimates of the variance of entry effects for emergence and maturity from

the bivariate analyses were similar to the estimates obtained from the individual trait

univariate analyses for all sites (Table 3.5). The correlation between entry effects was

high across the 6 disease nurseries, averaging 0.74 with a range of 0.71 to 0.94. This

was also reflected in the regression coefficients of maturity on emergence, which were

positive and ranged from 0.75 at Bakers Hill to 3.18 at Wonwondah.

Under the bivariate analysis the error variance component for the maturity trait was

always larger than that of the emergence trait (Table 3.4). The correlations between

traits were moderate, ranging from 0.23 (Wagga Wagga) to 0.59 (York), with an average

of 0.46.

The plots of the difference between maturity and emergence counts against emer-

gence, differed substantially across nurseries. The plot for Shenton Park (Fig. 3.6),

indicated a majority of entries clustered in the top right hand corner of the plot. This

cluster represented a majority of entries having greater than 60 counts at emergence

and percent survival values greater than 55%. This was the only disease nursery in

this data set that had such a distribution of entries for emergence and percent survival

values. In contrast to this, the Wagga Wagga disease nursery had a large variation for

emergence counts with the maximum emergence count less than 60 counts and corre-

sponding large distribution of percent survival values ranging from less than 10 to 50

(Fig. 3.7).

Overall the accuracy of prediction under the bivariate analysis were always greater

than or equal to the accuracies under the univariate analysis. The emergence trait,

when analyzed under the bivariate analysis, always resulted in a percent accuracy im-

provement across all sites. These improvements ranged from 0.08% at Shenton Park

to 10.61% in Wonwondah, with an average improvement of 2.28% (Table 3.6). Addi-

tionally, the maturity trait resulted in an accuracy improvement in five out of the six

sites for the bivariate analysis ranging from 0.02% at Clear Lake to 14.13% at Bakers

Hill, with an average of 2.48%. There was only one instance, for the maturity trait at

the York disease nursery where there was no change in accuracy of prediction under

the bivariate model (Table 3.6). Considering the difference trait; there was always an

34


improvement using the bivariate analysis over the univariate analysis and these ac-

curacy improvements ranged from 0.14% at Clear Lake to 5.63% at Bakers Hill and

the mean improvement was 1.59%. The gains were smallest for those nurseries where

the univariate accuracies were high (the maximum possible accuracy value being 1),

whereas more substantial gains were observed for those nurseries where the univariate

accuracies were lower.

Table 3.4: REML estimates of error variance from the univariate and bivariate modelsat each disease nursery location. The correlation between trait errors from the bivariatemodel is also shown.

Location Univariate BivariateEme Mat Eme Mat Correlation(σ2

1) (σ22) (σ2

1) (σ22) (ρ12)

Bakers Hill 0.04 0.317 0.04 0.315 0.29Clear Lake 0.017 0.064 0.017 0.058 0.39Shenton Park 0.015 0.054 0.015 0.055 0.35Wagga Wagga 0.029 0.265 0.029 0.261 0.23Wonwondah 0.163 0.031 0.059 0.278 0.54York 0.382 0.299 0.329 0.334 0.59

Table 3.5: REML estimates of entry variance from univariate and bivariate models ateach disease nursery. The correlation between entry effects and the slope of the regressionline of maturity against emergence from the bivariate model is also shown.

Location Univariate BivariateEme Mat Eme Mat Correlation Slope(σv

21) (σv

22) (σv

21) (σv

22) (ρv12)

Bakers Hill 0.108 0.127 0.109 0.131 0.684 0.75Clear Lake 0.042 0.232 0.047 0.259 0.682 1.6Shenton Park 0.191 0.657 0.194 0.636 0.935 1.69Wagga Wagga 0.053 0.768 0.053 0.765 0.729 2.78Wonwondah 0.033 0.687 0.034 0.691 0.728 3.18York 0.4 0.511 0.354 0.493 0.71 0.83

35


Emergence

Mat

urity

− E

mer

genc

e

−2.5

−2.0

−1.5

−1.0

−0.5

2.0 2.5 3.0 3.5 4.0 4.5

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●●

●

●●●

●

●

●

10 20 30 40 50 60 70 80

10

20

30

40

50

60

70

80

Figure 3.6: Plot of the difference between predicted entry means at maturityand emergence against emergence at Shenton Park disease nursery - The differ-ence between predicted entry means at maturity and emergence (corresponds to percentagesurvival when back transformed, these values are shown on the inside of the y-axis) plot-ted against predicted entry means at emergence from the bivariate model for the diseasenursery at Shenton Park.

36


Emergence

Mat

urity

− E

mer

genc

e

−3.5

−3.0

−2.5

−2.0

−1.5

−1.0

3.2 3.4 3.6 3.8 4.0

●●

●

●

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●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

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●

●●

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●

●

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●

●

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●

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●

●

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●

●

●

●

●

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30 40 50

10

20

30

40

50

60

Figure 3.7: Plot of the difference between predicted entry means at maturityand emergence against emergence at Wagga Wagga disease nursery - The differ-ence between predicted entry means at maturity and emergence (corresponds to percentagesurvival when back transformed, these values are shown on the inside of the y-axis) plot-ted against predicted entry means at emergence from the bivariate model for the diseasenursery at Wagga Wagga.

37


Tab

le3.6

:T

he

accura

cyof

pred

iction

foreach

trait;(lo

g)

emerg

ence,

(log)

matu

rityan

dth

ed

ifferen

ce(log

matu

ritym

inu

slog

emergen

ce)avera

ged

for

allva

rietiesat

each

sitefor

the

un

ivaria

tean

aly

ses.T

he

evencolu

mn

sin

dicate

the

percen

tage(%

)in

creasein

accu

racy

observed

un

der

the

bivariate

mod

elfor

the

trait.

Location

Em

ergen

ce%

Imp

rovemen

tM

aturity

%Im

provem

ent

Diff

erence

%Im

provem

ent

Un

ivariateE

mergen

ceU

nivariate

Matu

rityU

nivariate

Diff

erence

Bakers

Hill

0.8

40.19

0.5414.13

0.425.63

Clear

Lake

0.870.16

0.890.02

0.870.14

Sh

enton

Park

0.9

70.08

0.960.15

0.910.79

Wagg

aW

agga

0.841.40

0.890.30

0.861.14

Won

won

dah

0.5

810.61

0.850.27

0.850.89

York

0.7

61.21

0.830.00

0.730.93

38

3.5 Discussion

3.5 Discussion

One of the main features of the bivariate approach is the ability to model spatial

variation for each trait. From the results, the components of spatial variation (Gilmour

et al., 1997) differed between the two traits with global trend and extraneous variation

components being added in 5 out of 6 disease nurseries. Local stationary trend also

varied across disease nurseries for each trait with the largest difference between trait

models observed for the York disease nursery (Table 3.3). Another component of the

bivariate analysis is the ability to determine trait-based outliers. It was found that

the number of outliers removed from the analysis differed between traits, with only

one disease nursery having the same numbers of outliers removed for each trait (Table

3.3). These results clearly show that each trait has its own spatial trend and hence

should be modeled individually. Previously such spatial trend differences were not

observed under the historical approach as the use of a ratio of the two plant counts

(percent survival) would have confounded the sources of error of each of the traits. The

differences in spatial distributional properties and outliers between traits is expected as

the error associated with each set of the plant counts/sampling times arises from human

counting error as well as biological and environmental differences that are specific for

each sampling time. In terms of error, it is stated in the NBG protocol that the

emergence counts are to be taken after ‘total germination’, that is the open cotyledon

stage, and before plant death from blackleg disease. However this ‘window’ of time is

not perfect, as plant counts may be taken before total germination and so not all the

plants are counted. The maturity counts according to the protocol, are taken before

the windrowing stage, as it is difficult to determine plant death due to blackleg or

senescence. In addition plants are not counted at maturity if they lean more than 45◦

and if they are not infected by blackleg. As both these counts are based on visual

scores, which are prone to human error there is a different type of error associated with

each sampling time, and this is more accurately modeled under the bivariate analysis.

The error correlation between traits in the bivariate analysis was moderate, averag-

ing at 0.46 across disease nurseries. This represents the repeated measures nature of

the data set, where counts at the different sampling times are taken on the same plot.

Hence while the errors associated with the two sampling times might arise from different

39


sources, they are still moderately correlated. The univariate analysis effectively ignores

this covariance that arises between repeated measures. The bivariate framework is the

preferred method of analysis for plant survival data sets, as it considers each measure

(sampling time) as a realization of an individual trait, thereby enabling the modeling

of covariances between errors.

The current disease nursery protocols require plots with less than 20 counts at

emergence to be omitted and plots with greater than a 100% survival to be truncated

to 100% (Marcroft, 2009). The main reason for the former was to account for entry

plots sown with a poor seed source, and the latter ensured that survival values were

not over 100% as these had no biological meaning. Such values arise due to error, and

under the historical approach these protocols would have resulted in a loss of data from

21% of the total number of plots at the York disease nursery - a substantial loss of data.

Under the bivariate approach such protocols are avoided, as the analysis accommodates

error variation for each of the traits, enabling all data points to be included in the

analysis. Additionally, if the researcher still wants to discount entries with less than

20% emergence this can be done after the analysis. This is a more informed approach

than deletion of the raw data under the historical approach.

The bivariate analysis also allowed for examination of the entry variance for each

of the traits. It was observed across all disease nurseries and traits that the entry

variance was non-zero. Further, the entry variance for maturity was always larger than

that of emergence (Table 3.5). More importantly the bivariate analysis demonstrated

that there is variation across entries for emergence and that this differed across the

disease nurseries, see plots of Wagga Wagga (Fig. 3.7) and Shenton Park (Fig. 3.6).

The entry variance for maturity counts can be safely attributed to the effects of

resistance to blackleg disease, as all other impacts of pests and disease were minimized in

the disease nursery management protocols (Marcroft, 2009). The variation in emergence

counts however could be due to either resistance to early blackleg infection or differences

attributed to variable seed sources across disease nursery locations. Early blackleg

disease infection has been demonstrated to impact on seedling emergence (Li et al.,

2007, Sosnowski et al., 2006). The study by Li et al. (2007) demonstrated that soil

borne ascospores and pycnidiospores of Leptospheria maculans caused seedling death

40

3.5 Discussion

from early infection, resulting in a seedling death rate as high as 59% of seedlings after

sowing in infested soil. Hence the differences in entry emergence attributable to early

infection would constitute genetic effects of resistance.

The existence of genetic variance for emergence counts also raises issues with the

use of the historical method of analysis, which is similar to analyzing maturity counts

with emergence counts as a covariate, commonly known as an analysis of covariance

(Cochran, 1957). In this particular data set, such an analysis will result in maturity

counts being adjusted to a common emergence value, often the average value across all

entries. This not only has the potential to ‘create’ varieties that don’t exist, but this

effectively ‘hides’ the impact of blackleg at emergence. This will be covered in more

detail in Chapter 4.

Seedling emergence is known to be affected by environmental factors such as soil

fertility, salinity, compaction, tillage and surface residues (Forcella et al., 2000). It can

also be affected by seed lot factors such as age of seed (Finch Savage, 1986), the storage

environment of the seed (Ellis and Roberts, 1980), and seed production environment

(Ellis et al., 1993). An illustration of this variance in emergence at a disease nursery

site is evident from the plot of percent survival plotted against emergence at Shenton

Park (Fig. 3.6). At this site, the emergence counts observed were the highest across

all disease nurseries in the data set. When this was queried with the disease nursery

manager it was found out that the disease stubble was distributed on top of plots

after emergence (Dr. Cameron Beeck, pers.comm.), so plots were not sown into disease

stubble. Hence at this particular disease nursery the plots with poor emergence counts

can be directly attributed to poor seed sources and not to early infection of blackleg

disease. Seed source variation is a known issue for Australian blackleg disease nurseries,

however the impact of this variation has not been previously quantified.

A key component of the bivariate analysis is the inclusion of a correlation between

entry effects at emergence and maturity. This correlation was strong across the 6 disease

nurseries, averaging 0.74 with a range of 0.71 to 0.94 (Table 3.5). A high correlation

between entry effects indicates a strong agreement between entry rankings for both

the traits. That is, regardless of the different causes of the variation at emergence

and maturity, they are still strongly correlated at most disease nurseries. Thus the

41


bivariate analysis enables further insights into plant pathogen interactions, which would

otherwise not be observed under the historical analysis.

Another key (statistical) motivation of the bivariate analysis is the increases in

accuracy afforded by multi-trait predictions. The results from the 6 disease nurseries

show that there are improvements in prediction accuracies under the bivariate model

for the emergence and maturity traits and also for the difference which is particularly

important since it is analogous to the trait of percent survival as used in the historical

approach. While the improvements are modest within this data-set, the gains for any

particular data-set are obviously unknown prior to an analysis but may be larger than

reported here and are worth pursuing given that there is little extra cost or difficulty

involved in conducting the bivariate analysis.

For entry selection, the bivariate approach provides a more detailed picture; a two

dimensional representation of disease impact not provided by the historical approach.

The analysis enables the prediction of entry means at emergence and maturity, which

can be used to generate three sources of information for selection: emergence counts,

maturity counts and percentage survival values. Additionally, this study demonstrates

that if percent survival values are preferred as the trait for selection, this should not

be done without reference to emergence counts. This is because biologically each set of

plant counts is affected by different biological, genetic and environmental impacts.

3.6 Summary

This chapter presents an approach for a bivariate mixed model analysis of plant survival

data from designed field trials. In the motivating data set the two variables subject to

analysis are the two ‘traits’; plant survival counts taken at emergence and at maturity

sampling times. This method is not only an improvement over the historical method for

analyzing a derived variable ‘percent survival’ but demonstrates how entries can still be

assessed according to the historical selection basis (percent survival) in a more accurate

manner. Additionally this analysis method encompasses the differences between traits,

which are clearly affected by different biological, genetic and environmental influences.

The bivariate approach provides a more detailed picture for entry selection for blackleg

42

3.6 Summary

resistance than the historical approach. E-BLUPs of entry means at emergence and

maturity can be used to generate three sources of information for the basis of selection,

namely emergence counts, maturity counts and percentage survival values. The next

chapter will discuss other potential applications of bivariate analyses for plant breeding

data.

43


44

Chapter 4

Further applications of bivariate

analysis for plant breeding data

4.1 Introduction

Plant breeding programs rarely carry out selection for one trait at a time, as breeding

objectives commonly involve the selection of multiple traits concurrently. However there

are very few examples of multivariate trait analysis in plant breeding trials (Piepho

et al., 2008), yet it is commonly utilized in animal and forestry trials (Balestre et al.,

2012). Instead, plant breeding trials commonly utilize either a multi-trait index for

selection or covariance analysis to analyze for a single trait while ‘adjusting’ for the

presence of another trait. In Chapter 3 it was briefly outlined how covariance analysis

may be biologically and statistically inferior in comparison to a bivariate analysis. In

this chapter the pitfalls of covariance analysis are discussed by reviewing a select set of

previous studies in plant breeding which utilize such an analysis, and contrasting these

with the bivariate analysis. This review also highlights further selection applications

in plant breeding where bivariate methods may be beneficial.

Analysis of covariance, also referred to as ANCOVA, was introduced by Fisher in

1934. The most common uses of ANCOVA in agricultural studies include the removal

of extraneous variation that is not controlled by experiment design or the adjustment

of treatment means by a covariate value for ‘suitable comparisons’ (Yang and Juskiw,

45

4. FURTHER APPLICATIONS OF BIVARIATE ANALYSIS FORPLANT BREEDING DATA

2011). Since the introduction of ANCOVA in 1943, there have been numerous papers

published that warn against the misuse of this analysis method; see discussions by

Cochran (1957), Smith (1957), Urquhart (1982). Nevertheless there are still a large

number of plant breeding studies that routinely use this form of analysis.

Using the notation of Smith (1957), ANCOVA is commonly summarized in the form

of

yij = µ+ τi + βxij + εij (4.1)

Here, yij and xij are the jth observation on the ith treatment of the dependent variate

and covariate (Smith, 1957). τi is the effect of the ith treatment, µ is the grand mean,

β is the slope of the regression on the covariate and εij is the random error. The

main assumptions for use of ANCOVA are (i) the covariate must be measured without

error, (ii) the covariate must not be affected by any treatment and (iii) the slope of the

regression line is the same for each treatment (Elashoff, 1969). In most of the studies

reviewed in this chapter, the misuse of covariance analysis is discussed with respect to

assumption (ii), which a majority of studies fail to meet.

This chapter consists of a review of plant breeding studies in the areas of disease

resistance and grain yield, which both utilize covariance analysis. Also considered is

a section on Quantitative Trait Loci (QTL) studies in the areas of disease resistance,

grain yield and protein content, which also commonly use covariance adjustment.

4.2 Breeding for disease resistance

Breeding for disease resistance is an important component of cultivar production, as it

is an effective method of ensuring yield stability. The studies reviewed in this section

consider experiments, which select for disease resistance. For the disease resistance

trait, covariance analysis is often used to adjust for other traits such as emergence

levels, heading date and mould levels.

46


4.2.1 Adjustment for seedling emergence

Disease incidence data is commonly expressed as the proportion of disease units out

of a total amount, and is often represented as a percentage. The blackleg data set of

Chapter 3 is a form of disease incidence data as plant counts are taken at emergence

and maturity and used to construct the variable percent survival,

maturity

emergence× 100 = %survival (4.2)

In the blackleg plant breeding experiment, the aim is to determine the treatment im-

pact (variety resistance) under disease pressure. In this case the historical analysis is

analogous to an analysis of covariance where the response variable, maturity counts,

are analyzed using the emergence counts as a covariate, but with the slope fixed at one.

However, in Chapter 3 it was demonstrated that the individual plant counts have dif-

ferent biological, environment and genetic differences, constituting two separate traits,

which would be more accurately analyzed under a bivariate framework of analysis. It

was also demonstrated for the blackleg data set of Chapter 3, that there exists genetic

variance for emergence. This could either be due to differential blackleg at emergence

(Li et al., 2007, Sosnowski et al., 2006) or due to differences that arise in seed lots

(Finch Savage, 1986). If it is due to the former, it implies that the covariate is af-

fected by the treatment, which invalidates a key assumption of covariance analysis.

As a result, covariance analysis could in fact misrepresent the real treatment effect

by adjusting out the part of the treatment effect which results in the covariate in the

first place (Urquhart, 1982). Another limitation with covariance analysis is that, the

analysis adjusts maturity means to a common emergence level, typically the site mean

(Smith, 1957). This not only eliminates the impact of blackleg at emergence, it also

implies that it is possible for each variety to attain this ‘mean’ level of emergence. This

effectively creates ‘new’ varieties, which may not exist (Smith, 1957).

4.2.2 Adjustment for heading date

The study by Emrich et al. (2008) on Fusarium head blight resistance in wheat cultivars,

was based on covariance analysis to adjust for heading date (HD). This aimed to avoid

47


the selection of late heading genotypes, which tend to develop less head blight due

to a much shorter vegetative period. The covariate and the response variable were

measured on the same experimental unit; field plots of a variety. This poses issues

for the interpretation of the results, as the adjustment to a common heading level is

meaningless - HD is a variety specific characteristic that cannot be adjusted. The

authors acknowledge this, but argue that such an adjustment is acceptable as they test

varieties from similar earliness classes, resulting in similar heading dates anyway.

The use of HD as a covariate in the study by Emrich et al. (2008) is not biologically

valid for two main reasons. The first is that heading date is a major trait of selection

critical for regional and seasonal based adaptation of wheat cultivars (Zhang et al.,

2009). Covariance adjustment creates a common HD, which is not only biologically

incorrect but also impedes the selection of this trait. The second reason arises because

the genetic background of the cultivar and environment can affect the heading date

(Stelmakh, 1992). HD is known to be under polygenic gene control, controlled by

three categories of genes: vernalization response, photoperiod response and earliness

per se (Stelmakh, 1992, Zhang et al., 2009). Further it is also affected by environmental

conditions such as day length and temperature (Zhang et al., 2009). Considering these

factors, HD is clearly a trait with its own genetic variance and thus should not be used

as a covariate.

4.2.3 Adjustment for fungal mould levels

The study by Atlin et al. (1983) developed a method for selecting corn hybrids for

resistance to the fungal pathogen Gibberella zeae, which is known to be the cause of

ear rot. Six corn hybrids were tested for two factors; the first was the level of ear

rot and the second is the mycotoxin levels that result from Gibberella zeae produced

metabolites. The latter was important in this study, as mycotoxins are toxic when

consumed by cattle. Covariance analysis was used to adjust mycotoxin accumulation

for the level of Gibberella zeae mould. However, the analysis resulted in no significant

differences in toxin accumulation in either of the years the trials were run.

This study is also an example where there are differences of the covariate imposed

with treatments. For example, the treatments in this experiment are the six corn

48


varieties, with each having a different response to disease infection. So it is unknown

if the variation in mould levels can be directly attributed to the corn variety or a

combination of biological, environmental and genetic factors. Adjustment to a common

Gibberella mould level is in general an adjustment to an environment condition specific

to infection, i.e it is impossible to associate a set of factors to enable a particular level

of fungal infection. This is mainly due to varieties having different genetic responses

to mould infection and level of toxin production. As a result the use of mould level as

a covariate is not accurate for the selection objective of this study.

4.2.4 Adjustment for plant stand and days from planting

In the study by Littley and Rahe (1987), disease levels in onions from white rot (Scle-

rotium cepivorum Berk.) were analyzed using plant density level as a covariate. The

motivation for this being that plant density is known to impact on host plant disease

levels. Like some of the above examples, the covariate could be affected by the treat-

ment (varieties), as some varieties may have a higher density than others for many

reasons. As a result adjustment of disease levels for the covariate, might adjust out

the impact of disease levels on varieties. The authors state that they only used means

adjustment for trials that were significant under an ANCOVA, which also did not show

significant differences for slopes between treatments. However, this was only the case in

3 out of the 6 trials. For one of the three trials where ANCOVA was significant, means

adjustment led to no significant difference among varieties, when there were significant

differences under an ANOVA. Thus, the use of covariance analysis may not have been

statistically valid.

In this study, as well as the study by Atlin et al. (1983), there are differences in

the covariate for the imposed treatments. In particular, consider that the treatments

are the different varieties tested, with each having different plant density based on the

variety morphology. In this case adjustment to a common plant density is misleading

as it is not possible to know if plant density alone impacts on disease levels, or if there

is a combination of genetic, environmental and biological factors that has an impact

on disease levels. Plant stand can be affected by any number of factors. Environmen-

tal conditions in the field can impact the number of plants emerged (Forcella et al.,

49


2000), and seed characteristics such as seed source differences can impact stand number

achieved (Finch Savage, 1986). In addition plant stand can also be viewed as a genetic

effect, as it is also impacted by disease. Littley and Rahe (1987) state that the disease

has been known to directly cause damping off and seedling loss, especially at the Grand

Forks site. The use of covariance adjustment to a common stand level in this case, is

biologically meaningless.

4.3 Breeding for grain yield

Grain yield is the main trait for selection in field crops, with the aim to surpass the

current levels of commercial varieties. Studies reviewed in this section examine the

selection of grain yield, while using covariance analysis to adjust for factors such as

plant stand and grain moisture content.

4.3.1 Adjustment for plant stand

The study by Kamidi (1995) aimed at selecting high grain yielding maize varieties, while

accounting for incomplete plant stands in agronomy field trials. Covariance analysis was

used to analyze grain yield with plant stand number as a covariate, thereby obtaining

treatment effects that are comparable across varying numbers of plot stand. While the

author states that this adjustment is often satisfactory, he found that this adjustment

might not be acceptable when plants are missing at germination and before maturity

as competition effects invalidate linear covariance adjustment. The authors instead

suggest the use of an exponential model to correct for plot stands and contrast the

results from this analysis with those obtained from covariance adjustment.

Kamidi (1995) used covariance analysis in an identical way to the example outlined

by Smith (1957) where maize grain yield was analyzed with stand number as a covariate.

In his review on the interpretation of adjusted means, Smith (1957) argues against the

use of plant stand to adjust for yield variations, arguing that stand form is an integral

part of the treatment effect. This is primarily because it is not possible to determine

if variations in grain yield are directly attributable to plant stand number or due to

variations in fertility in the field as well (Smith, 1957). Hence if covariance analysis

50

4.3 Breeding for grain yield

is used, the adjustment no longer removes a component of the experimental error and

more importantly distorts the real treatment effect measured (Cochran, 1957). The use

of covariance analysis to obtain adjusted means for grain yield effectively assumes that

all the varieties tested can achieve this plant stand number. Like the above examples,

it is biologically incorrect and results in the creation of varieties that do not exist.

Pixley and Bjarnason (2002) screened a series of quality protein maize cultivars

for the assessment of the traits grain yield, protein content, quality and endosperm

modification (translucent or near normal phenotype). These cultivars consisted of

three-way cross, double cross and open pollinated varieties which were grown across

three tropical locations in four continents. Covariance analysis was used in this study to

adjusted grain yield for plant stand at two of the sites in Thailand, Tak Fa and Suwahn.

This was mainly because plant stand at Suwahn was affected by water logging, and at

Tak Fa was severely affected by downy mildew, resulting in diseased seedlings. Across

location analyses used these covariance-adjusted means from these two sites along with

the raw, lattice-adjusted means from the other sites. The use of covariance analysis

may have been inappropriate in this study because at both the sites plant stand could

be considered a trait in its own right.

4.3.2 Adjustment for grain moisture levels

The majority of above ground dry matter of corn is commonly referred to as stover

and is used as animal feed. Stover is usually allowed to dry above ground, however

weather conditions in Ontario prevent drying to prescribed moisture levels required for

dry feed storage (Leask and Daynard, 1973). Cultivars used for stover production are

thus selected to have a high rate of moisture loss before and after harvest - referred

to as stover quality. The study by Leask and Daynard (1973) tested the variability

among commercial corn hybrids in stover quality and yield. The traits grain yield and

dry matter yield were covariance adjusted to 15.5% moisture and 30% grain moisture

respectively.

The use of grain moisture as a covariate may not be acceptable as it is a trait that

is selected for, which in turn implies that it will have its own genetic variance. Further

Leask and Daynard (1973) and another study by Pordesimo et al. (2004) state that

51


differences in stover moisture content results from the variations in the date at which

the cultivars started drying below the initial moisture content. Clearly this implies that

drying below initial moisture content is a genetic characteristic of a particular variety,

which is affected by maturity date. Thus, adjusted means for grain moisture has little

biological meaning, as not all varieties can achieve this level.

4.4 QTL analysis - adjusting for other traits

A number of studies that routinely use covariance analysis are found in the area of

Quantitative Trait Loci (QTL) studies. A subset reviewed in this section includes

studies in the areas of disease resistance, grain yield, and end use quality traits in

wheat and barley breeding.

Klahr et al. (2007) studied a population of wheat recombinant inbred lines (RILs),

to determine QTLs for FHB resistance in wheat. The additional traits plant height

(PH) and heading date (HD) were also measured to determine correlations with FHB

resistance. Disease infection was measured in terms of percent of infected spikelets

(visual score), which were scored over multiple day intervals. These scores were then

used to calculate the area under the disease progress curve (AUDPC) for each plot

and environment. In cases where a significant correlation between FHB and PH or

HD existed the visual scores were adjusted using a covariance analysis. These adjusted

scores were then used for scanning FHB QTLs.

As discussed previously for the study by Emrich et al. (2008), HD is not a valid

covariate as it is a trait itself impacted by different genetic and environmental factors.

Similarly PH is another important trait of selection for wheat breeders, as it represents

a compromise between plant density requirements and lodging resistance (Zhang et al.,

2008). PH is also a morphological characteristic of a particular variety, and known to

be under polygenic control (Cadalen et al., 1998). Thus the adjustment for PH and

HD is not biologically meaningful, as these are inherent characteristics of a particular

genotype, and so any resulting adjustment has the potential of creating varieties that

do not exist. Further the use of these traits as a covariate also ‘covers’ the impact of

52

4.4 QTL analysis - adjusting for other traits

these on FHB resistance, which will have a detrimental impact on selection for FHB

resistance.

In rice QTL studies it is common to use days to flowering (DTF) as a covariate for the

analysis of grain yield. This is the case in both the studies Venuprasad et al. (2009) and

Vikram et al. (2011). Venuprasad et al. (2009) used rice RILs from the cross of varieties

Apo and Swarna to detect QTL’s for grain yield under drought stress. The effect of

days to flowering (DTF) was adjusted for grain yield for the different marker classes,

based on covariance analysis. Vikram et al. (2011) similar to Venuprasad et al. (2009)

identifies QTLs for grain yield in rice under stress however during the reproductive

period in a F3 mapping population produced from crosses of N22 with IR64, Swarna

and MTU1010. To eliminate DTF and as well as PH effects on grain yield, grain yield

was analyzed using covariance analysis with DTF and PH as covariates. Predicted

grain yields after covariate analysis was then used for marker analysis.

The use of covariance analysis to adjust for DTF in Venuprasad et al. (2009) and

DTF or PH in Vikram et al. (2011) is inaccurate as the adjustment for such traits is

unrealistic given that these are inherent characteristics of a particular variety. As a

result any adjustment to a common level is not biologically meaningful, as it is clear

that not all varieties can achieve this. DTF is a key trait selected for in rice breeding,

as it indicates the maturity class of a variety. Both DTF and PH are known to be

under polygenic inheritance (Li et al., 1995). Further, both traits are impacted by

environment; with the study by Li et al. (2003) finding that genotype by environment

interaction (GxE) has more of an impact on HD than on PH when examining QTLs for

these respective traits. Hence DTF and PH are traits in their own right and will have

their own genetic variance implying their use under covariance analysis is inaccurate.

Breeding programs in wheat and barley include breeding objectives for the selection

of end use quality traits. For wheat, these traits include dough rheological characters,

and for barley these include malting quality attributes. In the following studies reviewed

these traits are often adjusted for protein content, whether it is grain protein content

(GPC) or flour protein content (FPC), due to negative correlations between these

traits with other end use traits. It is thus common in wheat and barley programs to

53


use covariance adjustment for GPC values while selection for yield and malting quality

traits are undertaken (Blanco et al., 2012, Emebiri et al., 2004).

Blanco et al. (2012) studied QTLs for GPC in a RIL population of 120 durum wheat

lines. These lines were derived from the cross between Svevo and Ciccio and trialed

across 5 environments in southern Italy. GPC data from this study was covariance

adjusted for each of the yield components: grain yield per spike (GYS), thousand-kernel

weight (TKW) and kernel number per spike (KNS) due to the negative correlation

between GPC and yield components. Ten genomic regions were identified as being

involved with GPC expression and 6 of these were associated with one or more grain

yield component QTL.

Kuchel et al. (2006) studied a Double Haploid (DH) population from the cross of

Trident and Molineux, to investigate end use quality traits and QTLs for dough rheo-

logical traits. These included flour protein (FP), particle size index (PSI), flour yield

(FY), and other baking quality traits (in total 14 traits). If there was a significant cor-

relation between two traits, an adjusted value was calculated using covariance analysis

and used for QTL mapping. FP was found to be correlated to grain yield, and so FP

was adjusted for grain yield (GY) using covariance analysis. This study identified QTL

associated with FP from the adjusted GY data, which were not identified when using

the unadjusted data.

Emebiri et al. (2004) studied QTLs impacting barley malting quality attributes,

across 180 DH lines produced by crossing parents with low GPC. Covariance analysis

was used to adjust malt extract and diastatic power attributes for GPC. Single QTL

scans were repeated for both these adjusted values. The authors found that covariance

adjusted GPC values resulted in an increased number of QTL identified for malt extract

and diastatic power

In the above QTL studies the issue with covariate analysis is that the covariate is a

trait in its own right, which is commonly selected for in a breeding program. As such

it will have its own genetic variation, and this is the main factor that precludes its use

in covariance analysis.

54

4.5 Discussion

4.5 Discussion

A key assumption of covariance analysis is that the covariate should not be affected

by the treatment. In the studies reviewed above, it is most likely that the covariate

is affected by the treatment as it is often a trait that is also selected for, which would

imply that it will have its own genetic variance. When this particular assumption

is violated the interpretation of results should be treated with caution, as covariance

analysis no longer reduces that component of experimental error, attributed to the

covariate and also alters the nature of the treatment effect measured (Cochran, 1957,

Urquhart, 1982). This is one of the main reasons why this form of analysis has been

historically warned against (Smith, 1957, Urquhart, 1982).

Additionally for all of the studies reviewed above, it could be argued that covariance

adjustment is not biologically accurate, as the adjustment in most cases led to a common

level which some varieties in the experiment could not achieve, hence ‘creating varieties’

(Smith, 1957).

Both the limitations mentioned above are easily avoided using a bivariate analysis.

A bivariate analysis would be more appropriate for these studies, as the covariate and

response variable are treated as two separate traits. This has additional benefits in

terms of selection, as the bivariate analysis enable a two-dimensional view for the selec-

tion of varieties based on the two predicted traits without the need for an adjustment.

Thereby it is a more flexible framework of analysis as under the bivariate analysis the

covariate can be incorporated into the selection process. This is especially the case with

the study by Emrich et al. (2008), where a bivariate analysis will enable predictions

for both FHB resistance and heading date, enabling conditional selection for FHB re-

sistance for a lower heading date. Thus, there is no need to specify analysis classes of

similar heading date genotypes under a covariance analysis.

Another assumption of the covariance analysis is that the covariate should be mea-

sured without any error. However in practice this is near impossible. The bivariate

framework however enables the modeling of spatial error for each trait, which is effec-

tively ignored under covariance analysis. As demonstrated in the blackleg motivational

data set in Chapter 3 the modeling of spatial error for emergence and maturity traits

55


encompasses the different sources of error that could arise from the measurement of

each trait. Furthermore, spatial modeling has been demonstrated to result in greater

accuracy in estimation of treatment effects (Cullis et al., 1998, Smith et al., 2002a,

2001b) and a reduction in error variance, leading to increases in prediction accuracy

(Dutkowski et al., 2006). Hence the treatment of the covariate and the response variable

as individual traits would allow for the fact that there are multiple factors contributing

to error variation in each.

The bivariate framework of analysis also allows for a genetic covariance structure

between traits, which is ignored under the covariance analysis. Hence the bivariate

analysis would avoid any bias in evaluation which could arise by ignoring covariance

structures between traits (Lin et al., 1985). This has been demonstrated by the study

of Korol et al. (1995), who found that joint treatment of correlated traits, may provide

better power of detection and higher precision of parameter estimation for linked QTLs

than single traits. In addition under a bivariate analysis, utilizing information from

genotypic correlations can often lead to increases in the accuracy of evaluation (Mrode

and Thompson, 2005, Thompson and Meyer, 1986). This is especially the case when

traits are known to be highly correlated, which is the case with FHB and PH or HD

in the study by Klahr et al. (2007) and GPC or FPC with other end use traits in the

studies by Blanco et al. (2012), Emebiri et al. (2004) and Kuchel et al. (2006).

4.6 Summary

There are numerous examples of plant breeding experiments that use ANCOVA ap-

proaches for adjusting one variable for another. In this chapter some of these studies

have been reviewed and it has been shown that in all cases ANCOVA was inappropri-

ate and that a bivariate analysis of the form described in Chapter 3 may have been

preferred.

56

Chapter 5

Literature Review - Pedigree

information in plant breeding

METs

5.1 Introduction

Plant breeding varieties are routinely evaluated in series of trials known as multi-

environment trials (METs) to evaluate variety performance at a range of locations

and years, where locations and years can be synonymous with variable growing seasons

(Frensham et al., 1997, Kelly et al., 2007). MET trial data, consisting of phenotype

and pedigree information, can be included in linear mixed models to obtain genetic

values including breeding values at each location (Beeck et al., 2010, Burgueno et al.,

2007, Crossa et al., 2006, Cullis et al., 2010, Oakey et al., 2007). In contrast to animal

breeding, very few plant breeding programs base their selection on breeding values,

and Piepho et al. (2008) indicated that this type of analysis is limited to research.

Recently, the benefits of MET data analysis with pedigrees was demonstrated inside a

commercial breeding program (Beeck et al., 2010, Cullis et al., 2010). Selection based

on breeding values is known to outperform other commonly used selection strategies

especially in cases where data sets are unbalanced, have large pedigrees, or low trait

heritability (Bauer et al., 2009).

57

5. LITERATURE REVIEW - PEDIGREE INFORMATION IN PLANTBREEDING METS

Animal breeders have used mixed model methodology to exploit large series of phe-

notypic and pedigree data and BLUPs for the prediction of breeding values as the basis

of selection (Henderson, 1973, 1975). Bernardo (1994, 1995) applied this to crop breed-

ing, using mixed models and Restriction Fragment Length Polymorphisms (RFLP)

estimations of relationships to obtain BLUP breeding values of hybrid maize entries.

Compared with animal breeding, plant breeding trials have the additional complexity

of varieties being tested in replicated plots which impacts on experimental design and

analysis, but allows for the exploration of non-genetic effects and non-additive genetic

effects when pedigree information is included. Further, crops have the added complex-

ity of trials being conducted across multiple environments which enables the testing of

GxE.

MET analysis provides estimates of the magnitude and patterns of GxE on additive

and non-additive genetic effects when pedigree information is included (Cullis et al.,

2010, Kelly et al., 2009, Oakey et al., 2007). GxE interactions can be differentiated

by (i) interactions that arise due to heterogeneity of genetic variance among environ-

ments (Fig. 5.1b) and (ii) the lack of genetic correlation among environments (Fig.

5.1c) (Cooper et al., 1996). Cross-over GxE represents the failure of varieties to rank

uniformly across environments for that trait (Figs. 5.1c & 5.1d) (Basford and Cooper,

1998). Biologically, GxE occurs when the contribution of gene expression for an entry

varies according to the environment (Basford and Cooper, 1998). GxE limits the re-

sponse to selection, as entries vary in their performance over environments (Argillier

et al., 1994, Cooper and DeLacy, 1994), thereby reducing the efficiency of plant breed-

ing. Alternatively, if patterns of GxE were better understood, it might be possible to

improve the efficiency of plant breeding, especially if genetic relationships were known

(Beeck et al., 2010, Cullis et al., 2010).

In the case of F1 hybrid breeding the most expensive step is the identification of

parental combinations that produce F1 hybrids with superior agronomic traits (Riaz

et al., 2001). It is difficult to assess the breadth of adaptation of a new variety without

exposing it to a wide range of environments. Thus, from a breeder’s perspective it is

important to be able to determine the GxE effects on parental combinations early in

the testing program, and to predict the adaptation range of new hybrid varieties.

58

5.1 Introduction

Environment

Gra

in y

ield

(t/h

a)

2

3

4

5

6

7

E1 E2

(a)

Environment

Gra

in y

ield

(t/h

a)

2

3

4

5

6

7

E1 E2

(b)

Environment

Gra

in y

ield

(t/h

a)

2

3

4

5

6

7

E1 E2

(c)

Environment

Gra

in y

ield

(t/h

a)

2

3

4

5

6

7

E1 E2

(d)

Figure 5.1: Schematic representation of two entries (blue = entry 1 and pink = entry 2)and their performance across two environments: (a) no GxE; (b) GxE due to heterogeneityof variance between the environments but not lack of genetic correlation; (c) GxE due tolack of genetic correlation but not heterogeneity of variance between environments; (d)GxE due to heterogeneity of variance between the environments and the lack of geneticcorrelation. This diagram has been reproduced from Cooper et al. (1996).

59


In hybrid breeding programs, parental selection has been based traditionally on

general combining ability (GCA), which predicts the average performance of a parent

in hybrid combinations in designed experiments (such as a diallel cross) and is often

assumed to be due to additive effects (Can et al., 1997, Lynch and Walsh, 1998). Wricke

and Weber (1986) indicate that the GCA is dependent on the tester population. In the

case of diallel crosses, the tester population is the population itself, and the GCA of the

parent is half the breeding value of the parent. With other tester populations, there

is a linear relationship between breeding values and GCA (Wricke and Weber, 1986).

Specific combining ability (SCA) is the deviation from prediction based on GCA, and

is assumed to be the result of non-additive gene effects that contribute to heterosis

(Virmani, 1994). It has long been acknowledged that GCA and SCA are subject to

GxE interaction (Kidwell, 1963). In studies of hybrid rice, heterosis for yield has

been shown to vary across parent combinations and across environments (Young and

Virmani, 1990). Most recently, Cullis et al. (2010) showed through pedigree analysis

that additive and non-additive components differed for each trait (oil and grain yield in

canola) and were subject to different GxE effects in both inbred entries and F1 hybrids.

In plant breeding programs, there are two concurrent objectives for the selection of

entries: to select the best parents for crossing, and to select superior future varieties

with improved performance (Liu and Wu, 1998). The estimation of additive and non-

additive components of entry performance in plant breeding trials potentially improves

parental selection and selection of future new varieties. The additive component or

‘estimated breeding value’ is the heritable portion of a entry (Lynch and Walsh, 1998),

which is the ability to pass on its genes (Burgueno et al., 2007). Historically, animal

breeders have focused on the estimation of breeding values, with less importance placed

on non-additive effects (Ovaskainen et al., 2008). Non-additive components however,

are important to estimate as ignoring the non-additive covariances can result in the

inaccurate estimation of the additive genetic variance (Du and Hoeschele, 2000, Misztal,

1997).

The purpose of this chapter is to review current methods and approaches for the

inclusion of pedigree data in mixed model approaches for the analysis of plant breeding

data sets. This chapter commences with a review on the applications of pedigree data

in MET data sets, followed by sections which consider pedigree-based and molecular

60

5.2 Analysis of MET trials

marker-based relationship matrices. It is then concluded with areas of current research

which will be examined in the subsequent experimental chapters.

5.2 Analysis of MET trials

5.2.1 Linear Mixed Model Approach

Linear models which jointly account for fixed and random effects are regarded as ‘mixed

models’ (Eisenhart, 1947). Mixed models have been applied to MET analysis of Aus-

tralian crop evaluation trials (Kelly et al., 2007, Smith et al., 2001b, 2005). Smith et al.

(2001a) were the first to develop and apply a factor analytic (FA) mixed model frame-

work for MET data. This particular mixed model approach allowed for heterogeneity

of genetic variance between trials; different patterns of genetic correlations among tri-

als and error variance structures of individual trials in the analysis of MET trial data

(Smith et al., 2001a). The benefits of this framework include the handling of large

unbalanced data sets, estimates of random entry and or environment effects, and esti-

mates of GxE interactions (Smith et al., 2005). FA mixed modeling can also be used

to assess patterns of genotypic adaptation and assist in the identification of groupings

of environments within MET data (Beeck et al., 2010, Cullis et al., 2010).

Historically, mixed model analysis of MET data has always assumed the indepen-

dence of entries (Piepho et al., 2008). This is not a realistic assumption, as in plant

breeding programs, entries tend to be related to each other, such as full sibs, half sibs,

sister lines etc. In a breeding program, the assumption of independence of entries does

not hold since there are often many generations of controlled crossing among selected

entries. Selection results in a genetic covariance from the common backgrounds of

entries within a program, and, by including the relationships between entries, mixed

model approaches can encompass this additional covariance (Malosetti et al., 2011).

Hence mixed model approaches which integrate pedigree information are usually supe-

rior to those that do not (Beeck et al., 2010, Crossa et al., 2006, Oakey et al., 2006,

2007).

Estimates of GCA and SCA are usually obtained through specialized mating de-

signs such as the diallel cross (Mather and Jinks, 1982), which separates the total

61


genetic effect of an entry into GCA and SCA. However, such mating designs are re-

source, time and cost intensive, as only a small number of parents and progenies can

be tested, and replication is limited (Oakey et al., 2006). These traditional designs also

assume that the parents are not related, which is not always the case in a breeding

program due to methods of selection (Balzarini, 2002, Bernardo, 1994). Henderson and

Quaas (1976) first utilized pedigree data for the derivation of a additive relationship

matrix within a mixed models setting for the prediction of breeding values (additive

component) in animal breeding. Oakey et al. (2006, 2007) used pedigree relationship

matrices with plant breeding MET trial data to independently estimate entry additive

and non-additive components. Oakey et al. (2006) incorporated mixed model analysis

with pedigree data in single trial analysis of pure line entries to determine additive

and non-additive effects, without the use of specialized mating designs. Both additive

and non-additive values were estimated for pure line entries, whereas SCA can only be

estimated for F1s in controlled mating designs - which demonstrates the limitations of

the GCA/SCA concept in plant breeding. Oakey et al. (2007) extended the MET/FA

mixed model framework of Smith et al. (2001a) to sugarcane; a cloned hybrid crop. In

this case, the non-additive component of entry effects included dominance and resid-

ual non-additive effects; the latter which can arise from inbreeding depression effects,

homozygous dominance effects, the covariance between additive and dominance effects

and epistatic effects (Oakey et al., 2006). This study however, did not distinguish

these latter sources of non-additive components, which may be important in the pre-

diction of heterosis in hybrid crops. Beeck et al. (2010) applied this framework to a

canola breeding program across southern Australia, and demonstrated improved model

fit with pedigree information, and Cullis et al. (2010) examined patterns of GxE for

yield additive and total effects and the further application of prediction and selection

indices for these components. These studies demonstrated that pedigree data can be

routinely included in the analysis of plant breeding trials to obtain breeding values for

parental selection and improved estimates of total genetic value for varietal selection.

Inbred parental entries have both additive and residual genetic components (the lat-

ter is presumably a restricted form of epistasis which results from interactions among

homozygous loci). In F1 progeny, on the other hand, the non-additive component of

genetic variance is due to dominance and epistatic effects (in this case, all types of epis-

62

5.3 Heterosis and GxE

tasis including interactions among heterozygous loci). Within a MET/FA framework

these genetic variances and correlations may need to be summarized separately for in-

bred entries and hybrid progeny. Combined analysis of parents and hybrids has been

accomplished through the use of marker and pedigree based data in maize entries by

Schrag et al. (2010). In this study they used mixed models to determine the prediction

ability of pedigree and molecular based data on hybrid performance, but they did not

use MET/FA framework to analyze the impact of GxE on hybrid performance, and

they ignored potentially large changes in additive and non-additive components across

environments.

5.2.1.1 Prediction models and relationship matrices

There is a great deal of interest in using relationship matrices in mixed models for

the prediction of heterosis in untested hybrids (Bernardo, 1994, 1995, Maenhout et al.,

2010, Schrag et al., 2006, 2009). In the studies by Bernardo (1994, 1995, 1996b,a)

phenotypic scores for each hybrid were obtained from the average of all the phenotypic

measurements, which were then subject to a combined analysis of variance across lo-

cations. Prediction models are not part of this review, as the focus of this research is

the analysis of plant breeding field trials and the partitioning of the field tested entries

into genetic components.

5.3 Heterosis and GxE

The term heterosis, as it applies to plant breeding, was developed by Shull in 1908 and

refers to “an F1 performance that exceeds the average parental performance” (Lynch

and Walsh, 1998). Heterosis is an observed characteristic (Bernardo, 2002), and histor-

ically has been based on observations of total genetic value. However, the underlying

genetic causes of heterosis are the subject of on-going research (Hochholdinger and

Hoecker, 2007). The main causes are assumed to be dominance, over dominance and

epistasis. The dominance hypothesis states that heterosis is based on superior dominant

alleles at multiple loci which mask the unfavourable alleles in the heterozygote (Lynch

and Walsh, 1998). The over-dominance hypothesis states that the heterozygous state

63


results in higher phenotypic values than either parental homozygous state (Lynch and

Walsh, 1998). Lastly, the epistasis hypothesis states that interactions between alleles at

different loci result in the manifestation of heterosis (Lynch and Walsh, 1998). Recent

reviews suggest that these three hypotheses are not mutually exclusive (Hochholdinger

and Hoecker, 2007, Lippman and Zamir, 2007). This limited understanding of the ge-

netic basis of heterosis has not limited the exploitation of heterosis for crop breeding,

however it is important in the long term to understand these components of heterosis

in order to predict hybrid performance in plant breeding.

It is difficult to assess the breadth of adaptation of a new hybrid variety without

exposing it to a wide range of environments. Thus, from a breeder’s perspective,

it is important to determine the GxE effects on parental combinations early in the

testing program, and to predict the range of adaptation of new hybrid varieties. This is

demonstrated in recent studies which have shown that entry additive components vary

across environments for wheat (Burgueno et al., 2007, Oakey et al., 2006). Additive

and non-additive components can differ for each trait (oil and grain yield in canola) and

vary across environments (Beeck et al., 2010, Cullis et al., 2010). MET/FA analysis may

assist in the estimation of additive and non-additive components across environments;

only if ancestry relationships are included in the analysis (see next section).

5.4 Relationship Information

The coefficient of co-ancestry (COF) (fij) is the main measure of genetic relatedness

of two varieties i and j. In other studies, it is also referred to as the coefficient of

consanguinity or the coefficient of kinship (Lynch and Walsh, 1998). The COF is used

to model the covariance between the additive genetic components of plants (Maenhout

et al., 2009). This measure if often used in breeding programs (Bernardo, 1993, 1994,

1995, 1996b,a) and association studies (Jannink et al., 2001, Yu et al., 2005). fij deter-

mines for any locus, if individuals i and j have alleles that are descended from a common

ancestor that are identical by descent (IBD) and alike in state (AIS) (Bernardo et al.,

1996). This measure can be determined from pedigree or molecular marker data. The

COF is utilised to form the additive relationship matrix (A) also known as the numer-

ator relationship matrix. The COF is used in plant breeding to model the covariance

64


between the genetic background of plants (Bernardo, 1993, 1994, 1995, 1996b,a), and

many studies (Beeck et al., 2010, Burgueno et al., 2007, Crossa et al., 2006, Oakey et al.,

2006, 2007) have found that pedigree-based BLUPs are superior to pedigree-excluded

BLUPs.

The calculation of fij from pedigree records is based on the assumptions that: (i)

entries must be traced back to a base population, (ii) the base population is unrelated

to each other and (iii) the base population is in Hardy-Weinberg equilibrium (Piepho

et al., 2008). The last assumption implies that there are no bottlenecks which could

limit genetic diversity in the data (Smith et al., 2004). The calculation of the COF also

assumes that the relatives are not inbred (Falconer, 1981). In most plant and animal

breeding programs, most if not all of these assumptions do not hold, which highlights

the limitations of pedigree-based estimators of genetic relatedness. In these programs,

intense selection can lead to deviations in actual parental contributions compared with

their COF-based expected values (Bernardo, 1996a). Thus many studies have promoted

the use of molecular markers to estimate fij, as they sample directly from the genome

and may account for deviations from parental expectations resulting from selection or

drift (Bernardo, 1996a, Piepho et al., 2008).

5.4.1 Pedigree based estimators of COF

The early studies by Bernardo (1994, 1995, 1996a) integrated pedigree records for

the prediction of yield performance in maize from single crosses. These studies used

a two stage approach where the entry means across environments were obtained in

the first stage and the A matrix was fitted in the second stage to the genetic main

effects. This however had implications on the differentiation between genetic main

effects and interactions (Piepho et al., 2008). Oakey et al. (2006) used pedigree data

to form an additive relationship matrix (A) to derive additive as well as non-additive

genetic entry effects within a single-stage mixed model framework for a set of wheat

breeding trials. This method incorporated spatial modeling of errors developed by

Gilmour et al. (1997) and also allowed for varying levels of inbreeding in the data

set. Oakey et al. (2006) showed that the pedigree model was superior to the standard

model (which did not partition genetic effects into additive and non-additive entry

65


genetic effects). Further the entry total genetic effect had a lower prediction error

variance in the pedigree model in comparison to that obtained from the standard model.

Oakey et al. (2007) extended Oakey et al. (2006)’s pedigree model to partition entry

genetic effects into additive, dominance and residual non-additive components for a

MET sugar cane crop. The derivation of the dominance and non-additive components

was necessary as the sugar cane entries under study were F1 hybrid entries. They

used the A matrix as in Oakey et al. (2006) and included a dominance relationship

(D) matrix. The D relationships in this case were summarized (Bernardo, 1994) in

two components: dominance relationships relating to between family effects and those

relating to within family effects, as derived by Hoeschele and VanRaden (1991), but

also included adjustments for varying levels of inbreeding. This study showed that the

MET mixed model which accounted for non-additive effects was superior to the models

which excluded non-additive effects.

Crossa et al. (2006) and Burgueno et al. (2007) demonstrated the use of different

models with pedigree information for MET data sets than those described in (Oakey

et al., 2007). Crossa et al. (2006) modeled the additive genetic effects alone, and

ignored non-additive, and Burgueno et al. (2007) modeled additive and additive by

additive (AxA) effects (ignoring non-additive effects) in CIMMYT international wheat

trials. Both these studies considered FA model covariance structure for additive effects,

with Crossa et al. (2006) concluding that FA models provided the best fit. However

Beeck et al. (2010) considered that these models were “simplistic models for non-genetic

effects”, due to the absence of design components and spatial correlation. Beeck et al.

(2010) analyzed a two year MET data set with pedigree data for oil and yield in canola

across southern Australia using the MET/FA approach of Smith et al. (2002a). This

analysis partitioned the total genetic effects into additive and non-additive, and the

mean degree of inbreeding was very high at 0.967. The variety effects were estimated

both for varieties with pedigree information but absent in the MET data set and vari-

eties with pedigree and present in the MET. Both the findings of Beeck et al. (2010)

and Oakey et al. (2007) suggest that, despite the deviations in a plant breeding pro-

gram from the assumptions of COF relationship matrices, the models which included

pedigree information were superior to those which excluded pedigree information.

66


5.4.2 Molecular marker based estimators

A pedigree based A matrix is derived from expectations of the proportions of genes that

two particular individuals have in common (Villanueva et al., 2005). These relationships

may be greater than those estimated by pedigree models, as pedigree data ignores the

effect of selection on entries, which can bias the estimates of additive genetic variance

(Oakey et al., 2007). Instead, molecular marker based genetic similarities can be sub-

stituted to relate entries. Marker-based data may provide a more accurate estimate of

genetic relationships, as it samples directly from the genome and may account for devia-

tions from parental expectations that result from selection or drift processes (Bernardo

et al., 1996, Melchinger et al., 1990, Piepho et al., 2008). The studies by Bernardo

(1994, 1995), Maenhout et al. (2009) have demonstrated how molecular marker based

similarities can be used to relate entries. Nevertheless, there appears to be very lim-

ited application of this in plant breeding programs. This is mainly because molecular

marker data are often only preferred when pedigree data is missing, are rarely available

for all individuals in the pedigree and when selection intensity is high or when there is

a bias from non-genetic effects of a trait (Bauer et al., 2006).

Many research papers have used genetic similarities to determine the COF (Fij)

between entries in a plant breeding program. The first among these was the study by

Bernardo (1993) which used the proportion of RFLP marker variants shared between

two individuals as a measure of genetic relationship (Sij), comparing this with the results

from a pedigree based COF (Fij) and an adjusted marker similarity COF (Fijm). As Sij

is an upwards biased estimator of the COF, especially between entries that are distantly

related, Fijm attempts to accommodate this bias by including a correction factor that

accounts for variants in common between unrelated entries in the data set. This study

showed that pedigree and molecular marker based COFs result in different estimates

of alleles that are IBD. Estimates of Sij and Fij between two entries were significantly

different in 76.3% of the pair wise comparisons. Further, 24.9% of the comparisons

between the estimates for Fijm and Fij were significantly different, thus demonstrating

that molecular maker based COF estimates are affected by the proportion of alleles

that are not IBD but alike in state (AIS) (Bernardo, 1993). Bernardo (1994) applied a

pedigree based COF and Bernardo (1993)’s marker adjusted COF in the prediction of

single cross yield performance in hybrid maize. RFLP-based estimates of COF resulted

67


in better predictions of hybrid yield than pedigree-based COF, which could be due to

limiting assumptions of pure pedigree-based COF. However, this study found that both

pedigree and RFLP genetic relationship estimations were highly correlated in their data

set.

Bernardo et al. (1996) developed a tabular analysis of RFLP marker data for the

estimation of the COF and compared this to a pedigree-based COF for a data set

consisting of inbred maize entries and their progenitors. Similar to Bernardo (1994), this

study found that marker and pedigree based COF’s were highly correlated (correlation

of 0.9, P< 0.01). They also found that there were large deviations in pedigree and

marker COF values for particular inbred pairs, which could be due to the effects of

selection and/or inbreeding. Bernardo et al. (2000) continued with this tabular analysis

procedure to compare the estimates of parental contribution and COF from RFLP, SSR

and pedigree data for a set of 13 maize inbred entries. Importantly, they found that

RFLP, SSR and pedigree-based estimates of COF were highly correlated to each other

(r = 0.87 − 0.97), although it was noted that pedigree and molecular marker data

resulted in significantly different estimates of COF. COF estimates for marker data

also differed based on the type of markers used, with SSR markers preferred to RFLP

markers in the estimation of genetic relationships (Bernardo et al., 2000).

The conclusions of Bauer et al. (2006) differed from the above studies, in a study

of a self-pollinating crop, spring barley. This study used genetic similarities from SSR

markers to determine genetic relationships among breeding entries for a simulated data

set and a MET data set. Bauer et al. (2006) showed that relationship information

improves BLUP breeding value estimates, but like the studies in maize (Bernardo,

1993, Bernardo et al., 1996, 2000), the COFs based on genetic similarities and pedigree

information were highly correlated to each other (Pearson’s correlation r = 0.95). In

all of the above studies, there was no spatial modelling of errors, however these non-

genetic models can be important components of the analysis (Beeck et al., 2010, Cullis

et al., 2010).

Recently, a study in maize hybrids by Maenhout et al. (2010) incorporated a marker-

based genetic relationship matrix into a MET/FA mixed model framework using SSR

markers and AFLP fingerprint data, instead of pedigree data, for the prediction of

68


hybrid performance. Marker similarity estimates for the COF between entries of the

same heterotic group were according to the method described in Bernardo (1993). The

MET data in this study consisted of 1, 280 trials from 110 locations in Europe during

the years 1989 to 2005. It was found that the COF estimates based on AFLP markers

had greater prediction accuracy than those based on SSR markers. This study found

large GxE effects for grain yield, but did not directly determine the impact of GxE

on GCA and SCA components, but concluded that SCA predictions were limited for

all traits due to uncertainty caused by GxE effects. There is very limited literature

available on the impact of GxE on GCA and SCA components (Kidwell, 1963), which

is an important factor in all hybrid breeding programs as it determines the selection of

parental entries and is the basis of hybrid progeny selection respectively (Liu and Wu,

1998).

A limitation to the usefulness of marker data arises from the estimation of the

A matrix. When the A matrix is formed from marker data, it may not be positive

definite, which is a requirement of many software packages (see the review by Piepho

et al. (2008)). It is a requirement of variance matrices that they are at least positive

semi-definite (psd) (Maenhout et al., 2009, 2010). In estimating the COF from RFLP

markers, Bernardo (1993) obtained some negative estimates, and these were assumed

to be zero. This could have arisen from errors in estimating molecular alleles which are

AIS but not IBD. Bauer et al. (2006) had a similar issue using genetic similarities from

SSR markers, which resulted in a singular genetic similarity matrix. The psd matrix

property is critical for the A matrix and in most cases is not fulfilled when using

marker data, however many studies and software packages have methods to circumvent

this requirement (Piepho et al., 2008). How this impacts on BLUP predictions has not

been outlined in the literature.

To resolve the issue of non-psd relationship matrixes, Maenhout et al. (2009) com-

pared pedigree based COF with 5 marker-based COF estimators using inbred entries

from a maize breeding program genotyped with SSR markers. Among these 5 marker-

based estimators was the proposed estimator of weighted alikeness in state (WAIS)

(Maenhout et al., 2009). For COFs that produced non-psd matrices, matrix bending

techniques were used. Interestingly this study demonstrated that pedigree-based esti-

mators were preferred to marker-based estimators of the COF in terms of the lowest

69


root mean squared error produced (RMSE). This is mainly because they found that the

bias from unequal parental contributions were insignificant compared to the bias that

resulted from marker-based estimators. Furthermore, pedigree-based COF was a bet-

ter model fit than marker-based models, based on the restricted log likelihood values.

Thus, while pedigree-based estimates of COF are restricted by their assumptions, they

still are preferred over marker-based estimates of COF due to their above limitations.

5.4.3 Higher order interactions

The modeling of higher order non-additive components, such as epistasis and additive

x additive interactions, is not common in plant breeding programs. Nevertheless, the

derivation of these components can aid in the selection of potential parent entries and

entries for release. The sum of the additive and additive x additive (AxA) epistasis

components determine the breeding value of an entry, as this determines its ability as

a parent to pass on its genes (Burgueno et al., 2007). Bernardo (1995) investigated

BLUP prediction from mixed models to estimate additive x additive (AxA) epistasis

in maize METs, and concluded that genetic models which include AxA epistasis did

not lead to better predictions of single cross performances compared to the intralocus

models which included additive and SCA effects only. They concluded that, while

there could be additive x additive effects, the estimation of this component is difficult

due to multicollinearity between AxA effects and test cross additive effects. Burgueno

et al. (2007) successfully modeled additive effects and additive x additive effects using

covariances of inbred relatives to form the A matrix, assuming no dominance in a wheat

breeding trials. They used FA covariance structures, however did not include spatial

modeling of errors. This study also mentions the complications of obtaining solutions

to these models due to the possibility of multicollinearity arising from the variance-

covariance matrix of the additive and AxA effects; however this did not limit their

ability to estimate these components on the models (Burgueno et al., 2007). Both these

studies show that while higher order interactions may be interesting in plant breeding

and can be modeled, in most cases they do not provide better BLUP predictions.

70

5.5 Conclusion and further research

5.5 Conclusion and further research

Recent studies have utilised a mixed model framework with pedigree information to

estimate additive (and sometimes dominance) values in plant breeding METs (Beeck

et al., 2010, Cullis et al., 2010, Oakey et al., 2006, 2007). Marker and pedigree based

COF, when contrasted for a data set, appeared to be highly correlated and in some

studies pedigree based COF was superior to marker based COF. Regardless, pedigree

information from plant breeding trials has resulted in vast improvements in selection.

Given the benefits of pedigree information in mixed model analysis; there are very

few examples in the current literature where it is routinely applied in plant breeding

programs as it is used in animal breeding programs. Why is this the case? There are

two main reasons to consider. The first is complexity, which arises from the fitting of

multiple models, such as spatial models for errors and FA models for GxE as well as

the fact that the time taken for analysis completion might be prohibitive. Additionally,

after obtaining the results from these analyses there is the added complexity of how to

interpret and apply the results. The second reason, is limited examples. There are few

worked examples published in technical journals outlining methodology and procedure

for such analyses.

To address these gaps in the literature Chapter 7 will illustrate on an individual

site basis the spatial modeling process and demonstrate the importance of pedigree

information in the spatial modeling of trials. Chapter 8 will complete the process of

model fitting by demonstrating the MET/FA genetic modeling of the trials in Chapter 7

as well as providing an interpretation of the results. Both these chapters use a data set

from a canola breeding program, described in the following Chapter 6. Lastly, Chapter

9 considers in detail the practical limitations of the use of pedigree information that

have arisen from the MET analysis.

71


72

Chapter 6

Canola multi-environment trial

data set

Breeding programs run extensive trials to achieve two traditional objectives, the first

being to promote entries for further testing or commercialization and the second being

the selection of entries as parents for the next cycle of breeding. The data set for

the subsequent chapters are based on a series of METs obtained from a private plant

breeding company. This chapter provides a description of their breeding program and

the pedigree origin of their breeding program material.

6.1 Data set description

The MET data set consists of a series of trials that form the basis of the company

Canola Breeders Western Australia Pty Ltd. (CBWA) canola (Brassica napus L.)

breeding program, which will be referred to as the ‘canola data set’. The canola data

set spans a four year period, including the 2008 to 2011 growing seasons, comprising

47 trials. These trials were located across major canola producing regions in Western

Australia, South Australia, Victoria and New South Wales (Fig. 6.1). There were

between 10 to 13 trials in each year, with at least 2 in NSW, 1 in SA, 1 in VIC and 4 in

WA (Table 6.1). While some of the location names are the same across the years, the

trials can be sown at different fields within this location, thus each trial is synonymous

73

6. CANOLA MULTI-ENVIRONMENT TRIAL DATA SET

to an environment. These broad locations are based on targeted growing areas which

encompass low to high rainfall production environments. Additionally, attempts are

also made to test at locations which could potentially be areas of future production.

While numerous traits are measured for these trials, the yield trait in tonnes per hectare

(t/ha) was the focus for this data set.

Albury●

Ardlethan●

Buntine● Croppa Creek●

Elmore●Horsham ●

Kellerberrin●

Kojonup ●

Lake Boloac●

Mingenew●

Nyabing●

Port Lincoln ●

Scaddan●

Stirling●

Wagga Wagga●

York ●

WA

SA

VIC

NSW

Figure 6.1: Location of multi-environment trials across Australia - Target envi-ronment locations of Canola Breeders Australia multi-environment trials across the 2008to 2011 growing seasons.

All trials were laid out as a rectangular array indexed by rows and columns, with

6 or 12 columns and between 32 to 99 rows across the data set (Table 6.2). Plot sizes

ranged from 3 m - 4 m x 1.8 m after spraying out pathways in the trial. A standard of 3

g of seed per plot was sown, in 4 rows of 5 - 6 m lengths, representing a standard seeding

rate of approximately 3 kg/ha. All trials were designed as p−rep designs (Cullis et al.,

2006) in DiGGer Coombes (2009) using the default, pre−specified spatial model. Each

trial was designed with a majority of entries either un-replicated or with 2 replications

(Table 6.2), and a standard of 2 blocks aligned in the column dimension.

A total of 2624 entries were tested across the 4 year data set, consisting of mainly

74

6.2 Pedigree Information

Table 6.1: Details of the canola multi-environment trials, including number of trials acrossyears and locations.

YearState 2008 2009 2010 2011 Total

NSW 2 3 3 3 11SA 2 2 1 2 7

VIC 1 1 1 1 4WA 7 7 7 4 25

Total 12 13 12 10 47

new test entries, retained promotions, and a subset of controls (commercial entries

and elite entries). Promoted entries were tested in the same trials as new lines and

there were no ‘stages’ of testing commonly seen in plant breeding programs. The only

material excluded was very early stage material which had not gone through enough

stages of selfing or which were eliminated as a result of prior testing in disease nurseries.

The number of entries at each trial ranged from 152 to 1045. Every trial had a subset

of commercial controls and elite entries, numbering 14 in total, which were common

across all trials. Entry concurrence within and across years was high; within years this

was highest with greater than a 100 entries. Minimum entry concurrence across years

and trials was 19 (See Table. 6.3).


The Australian Breeding Program (ABP) for canola consisted of a number of public

breeding programs established from 1970 with an initial founder population of 18 B.

napus entries (Cowling, 2007). From 1970 to 2000 this program was essentially a

closed breeding population. The pedigrees of entries produced from this program are

available in Salisbury and Wratten (1999). The CBWA program began in 2000 and

its founders included some of the ancestral entries from the earlier 1970 - 2000 public

breeding program. As a result of this, the pedigree information extends across two

phases of breeding; the first phase (ABP) includes 18 founders in 1970, and the second

phase (CBWA) includes 16 founders from the ABP used in 2000. No migrants appear

in the ABP breeding program, as it was a closed recurrent selection population, but

there are numerous migrants in the CBWA pedigree after the year 2000. There are

75


Table 6.2: Summary of individual trial details from the canola multi-environment trials,including total number of entries, replication levels, number of columns, rows, trial meanyield (t/ha), as well as missing yield and pedigree entries.

entries Columns Rows Trial Missing ValuesTrial Total r = 1 r = 2 r > 2 Mean Yield Yield Pedigree

CIA08ARDL2 153 55 93 5 6 43 1.31 1 5CIA08BUN6 152 55 93 4 6 43 1.90 0 6CIA08ELM3 153 55 93 5 6 43 0.27 14 5CIA08HOR3 153 55 94 4 6 43 0.90 0 5CIA08KEL6 152 55 92 5 6 43 1.14 0 7CIA08KOJ6 153 55 94 4 6 43 1.53 9 5CIA08MIN6 152 55 92 5 6 43 2.46 0 7CIA08NYA6 152 55 91 6 6 43 1.37 1 6CIA08PLI5 153 55 94 4 6 43 0.88 0 5

CIA08SCA6 153 55 94 4 6 43 0.94 0 5CIA08WAGG2 153 55 94 4 6 43 0.59 0 5

CIA08YOR6 153 55 95 3 6 43 1.64 0 5CIA09ARDL2 320 257 62 1 12 32 0.26 0 6

CIA09BUN6 304 247 55 2 12 32 0.99 24 6CIA09CCRK2 320 257 62 1 12 32 1.46 1 6

CIA09ELM3 321 258 63 0 12 32 0.58 6 6CIA09HOR3 320 257 62 1 12 32 1.52 0 6CIA09KEL6 321 258 63 0 12 32 1.09 0 6CIA09KOJ6 320 257 62 1 12 32 2.21 0 6CIA09MIN6 321 258 63 0 12 32 1.45 2 6CIA09NYA6 321 258 63 0 12 32 0.54 3 6CIA09PLI5 321 258 63 0 12 32 1.14 0 6

CIA09SCA6 320 257 62 1 12 32 0.55 1 6CIA09WAGG2 320 257 62 1 12 32 2.92 0 6

CIA09YOR6 716 653 62 1 12 65 1.80 2 6CIA10ALBR2 393 320 55 18 12 41 2.54 0 14CIA10ARDL2 394 319 58 17 12 41 1.93 0 13

CIA10BUN6 390 318 53 19 12 41 0.95 0 14CIA10CCRK2 395 322 62 11 12 41 1.47 2 12

CIA10ELM3 391 316 59 16 12 41 1.48 1 13CIA10KEL6 394 319 64 11 12 41 0.30 0 12CIA10KOJ6 394 321 57 16 12 41 1.30 0 13CIA10MIN6 393 318 61 14 12 41 0.59 2 12CIA10NYA6 399 325 62 12 12 41 0.30 0 12CIA10PLI5 395 323 57 15 12 41 1.67 1 13

CIA10SCA6 395 322 56 17 12 41 0.81 4 13CIA10YOR6 970 891 64 15 12 91 1.06 6 12

CTTA11ALBR2 426 349 72 5 12 44 1.47 4 53CTTA11BUNT6 354 242 89 23 12 44 1.49 2 56CTTA11CCRK2 423 348 69 6 12 44 2.03 0 59CTTA11ELMR3 424 346 72 6 12 44 2.15 0 53CTTA11LKBL3 419 340 73 6 12 44 0.96 12 57

CTTA11MNGN6 371 260 92 19 12 44 1.91 3 52CTTA11PTLI5 423 347 70 6 12 44 0.65 4 60CTTA11SSTL6 425 347 72 6 12 44 0.86 0 54

CTTA11WAGG2 423 346 71 6 12 44 0.95 3 57CTTA11YORK6 1045 945 91 9 12 99 1.31 0 61

76


Table 6.3: Entry commonality (concurrence) across trials within years in the canola multi-environment trials data set for the 2008 to 2011 growing seasons. Diagonal values indicatethe total number of entries at the sites within a year.

Year 2008 2009 2010 2011

2008 153 58 21 222009 58 717 113 622010 21 113 970 1022011 22 62 102 1084

up to 16 generations of pedigree information from 1970 to 2011 (Table 6.4). Pedigree

information has been used in CBWA MET analysis of yield and oil traits using the

method of Oakey et al. (2007) from 2008 onwards.

For the MET data set the pedigree information went back several generations to the

1970 founders (Table 6.4). Pedigree information was available for a total of 3208 entries

across the breeding program, with 22 entries having unknown pedigrees. In this case,

unknown pedigrees meant that they were either unknown filler entries or commercial

entries of other companies, for which pedigree information was not available. For

the complete MET data set, there were 146 unique mother entries and 700 unique

father entries. The same entry can be used as a male and/or female parent in a

cross due to canola being a self - pollinated crop. Hence there are multiple instances

of parental concurrence across the pedigree data set across and within years. The

maximum concurrence for parents (both male and female) between years was 169 (2010

and 2011) and the minimum concurrence for parents between years was 43 (2008 and

2010). Within a year the minimum number of parents was 92 and the maximum was

416 across the 4 year data set (Table 6.5).

The entries included in this data set resulted from a wide range of breeding methods,

including F1 hybrids, doubled haploidy (DH), single seed descent (SSD) and synthetic

entries (a type of composite derived from multiple entires). Hence there were various

levels of self-fertilization, that needed to be accommodated when forming the additive

genetic relationship matrix (A matrix). The A matrix in ASReml-R (Butler et al.,

2009) is calculated from information on genetic relationships supplied in a ‘Pedigree

file’. This file comprises four fields of information: Identity of the entry, Male parent,

Female parent and Fgen (see Table 6.6). Self-fertilization or inbreeding is quantified

77


Table 6.4: Number of generations of pedigree information available for entries in thecanola multi-environment trials data set.

Generations of No. ofPedigree entries

0 21871 6612 1003 884 435 176 157 128 149 1310 911 612 813 1114 915 1016 5

Table 6.5: Concurrence of parents (both male and female) across trials within years in thecanola multi-environment trials data set for the 2008 to 2011 growing seasons. Diagonalvalues indicate the total number of parents at all sites within a year.

Year 2008 2009 2010 2011

2008 92 54 43 552009 54 395 106 1392010 43 106 416 1692011 55 139 169 291

under the variable Fgen in this file. The pedigree file is also sorted to ensure that the

line of an entries’ pedigree will always precede any line where it appears as a parent.

In the CBWA data set values for Fgen are calculated from the last generation of single

plant selection when using pedigree selection methods. For competitor entries, Fgen

values were based on records obtained from Plant Breeders Right’s (PBR) data base

(http://www.ipaustralia.gov.au/), where available. In the special case of composite

varieties, Fgen values were derived from calculations outlined by Busbice (1969). Fgen

values varied from 0 corresponding to F1 hybrids and back cross intermediates, 0.74 to

0.9 (inbreeding values) for synthetic or composite entries, 1 to 5 (generations of selfing)

for pedigree entries, and a standard of 3 or 4 for SSD, and 10 corresponding to DH

populations (Table 6.7).

78

http://www.ipaustralia.gov.au/


Table 6.6: Example extract of the CBWA Pedigree file indicating entry, parents andFgen fields of information. Note that “0”’ represents no parent information, normally forfounder parents.

Entry Female Parent Male Parent FgenZephyr 0 0 4.00

Bronowski 0 0 4.00SV62-371 0 0 4.00

Ramses 0 0 4.00Oro 0 0 4.00

Haya 0 0 4.00Zephyr/Bronowski Zephyr Bronowski 4.00

Chisaya 0 0 4.00ATR-Tower 0 0 4.00

Norin20 0 0 4.00Chikuzen 0 0 4.00

SV62-371/Zephyr SV62-371 Zephyr 4.00BJ42 0 0 4.00

Wesway Ramses Oro 3.00

Inbreeding coefficients for entries in the pedigree were calculated using the A.inverse

function in ASReml-R, which uses the algorithm of Meuwissen and Luo (1992) with

adjustments for selfing. These inbreeding coefficients ranged from 0 (477 entries) to

> 0.99 (1797 entries) with an average of 0.68.

Table 6.7: Summary of entry details within the canola multi-environment trials, includingnumber of selfing cycles (Fgen) and their corresponding entry type.

Fgen Levels Entry Type Number of entries

0 Backcross derived, Hybrids 10830.74 Synthetic 20.9 Composite or Synthetic entries 11

0.932 F2 derived composite 10.941 F2 derived composite 1

1,2,3,4,5,6,7 Selections, Breeding entries 1353,4 SSD entries 581

5 Canola Breeders migrants 8210 DH entries 1312

Using the package ‘Pedicure’ (Butler, 2012) in R, a set of 12 pedigree files of varying

depth (in terms of generations) were generated for 4 data sets of varying length (in

terms of years of trial data) (Table 6.8). In the MET data set, the most recent data

was for the 2011 growing season so all combinations of years included 2011 with the

addition of an extra year to a maximum of 4 years of data. The minimum depth of

pedigree was 2 generations, that is the pedigree comprised parents of entries within the

data set.

79


As more pedigree information was added (as the depth of pedigree increased), the

number of new parents added to the ancestry decreased with each additional generation.

While there was pedigree information for 16 generations, additional parents were no

longer added after the 13th generation. Hence, the pedigree files were limited to that

generation. Within the generated pedigree files, there were 2 possible levels of founder

populations, up to 9 generations; which consists of the CBWA founders and up to 13

generations which includes the ABP founders (Table 6.8).

Table 6.8: The number of parents (and then grandparents) in the pedigree for varyinggeneration depth and years of data for current entries in the multi-environment trial dataset. The number in the brackets indicates the number of additional parents (and thengrandparents) that result with inclusion of an additional generation of pedigree information.

Length of data set (years of data)No. 2011 2010-2011 2009-2011 2008-2011

Generations

2 1417a 2499 3054 30843 1462b (45) 2549 (50) 3111 (57) 3139 (55)4 1484 (22) 2573 (24) 3131 (20) 3157 (18)5 1498 (14) 2585 (12) 3143 (12) 3169 (12)6 1507 (9) 2594 (9) 3152 (9) 3182 (13)7 1521 (14) 2608 (14) 3166 (14) 3193 (11)8 1527 (6) 2614 (6) 3172 (6) 3198 (5)9 1530 (3) 2617 (3) 3175 (3) 3201 (3)

10 1534 (4) 2621 (4) 3179 (4) 3203 (2)11 1542 (8) 2625 (4) 3183 (4) 3206 (3)12 1542 (0) 2629 (4) 3187 (4) 3208 (2)13 1544 (2) 2631 (2) 3189 (2) 3210 (2)

a For a single year of data this cell indicates that there are 1417 parents for the entries in 2generations of pedigree data, i.e 2 generations consists of entries and their parents.b For a single year of data this cell indicates that there are 1462 parents and grandparents for the

entries in 3 generations of pedigree data, i.e 3 generations consists of entries, their parents and their

grandparents.

80

Chapter 7

Spatial analysis (N-gen

modelling) of trials with pedigree

information

This chapter illustrates in detail the spatial analysis (or non-genetic, ‘Ngen’ modeling)

of the 2011 growing season trials, described in Chapter 6. The impact of including

pedigree information is also evaluated on the spatial mixed model analysis of plant

breeding trials. These spatial models are then used in Chapter 8 for a complete MET

analysis.

7.1 Introduction

Plant breeding programs utilize extensive METs across locations and years (synony-

mous with seasons) to select test entries for promotion, commercialization and use

as parents. Such data from field trials exhibit spatial variation, which arises from the

physical location of plots within a field (Smith et al., 2002a). Thus spatial variation can

be defined as the variable growing conditions encountered throughout a trial (Stringer

et al., 2011). If not accounted for, the presence of extraneous variation can complicate

the analysis, as well as reduce the efficiency of selection (Stefanova et al., 2009). In

81

7. SPATIAL ANALYSIS (N-GEN MODELLING) OF TRIALS WITHPEDIGREE INFORMATION

order to control error in field trials, spatial models are often included within the mixed

model framework.

Gilmour et al. (1997) developed an analysis, which encompasses the modeling of

spatial variation within a mixed model context. This approach for spatial analysis

accommodates three sources of variation, namely global, local and extraneous. Global

trend refers to variation that occurs across the field, local represents short-term trend

such as soil fertility and extraneous variation is the result of experimental procedures

that are aligned with rows and columns (Gilmour et al., 1997). Local trend is accom-

modated within the mixed model by an appropriate covariance structure of which the

separable autoregressive process of order 1 (denoted AR1×AR1) is the most commonly

used (Smith et al., 2002b). This effectively reflects the observation that plots, which

are closer together are more likely to be similar to ones that are further apart (Smith

et al., 2002b). Global trend and extraneous variation however are accommodated by

including additional design factors and random row/column components.

Spatial analysis has been applied to designed field experiments, for both agriculture

(Cullis et al., 1998, 2006, Gilmour et al., 1997) and forestry experiments (Dutkowski

et al., 2006) to correct for environmental effects. This approach has been demonstrated

to result in greater accuracy and precision for the estimation of treatment effects (Cullis

et al., 1998, Smith et al., 2001b,a) and thus leads to large reductions in effective er-

ror variance (Smith et al., 2006). However, the value of spatial analysis is especially

demonstrated in improving the reliability of varietal selection when trials are large and

have minimal replicates (Smith et al., 2001a, Stefanova et al., 2009). This is especially

the case with a class of designs commonly used in plant breeding programs called ‘repli-

cated plots for a percentage (p) of the test lines’, or p-rep designs (Cullis et al., 2006).

P-rep trial designs are often used for the testing of early generation entries. These de-

signs are useful as they are based on optimal spatial relationships, enable unbalanced

replication in check entries and test entries, to account for availability of seed, and

allows test entries to be the focus of the design (Cullis et al., 2006).

Since Gilmour et al. (1997) demonstrated the need to identify the sources and causes

of spatial variation within a mixed model framework of analysis, it has become a stan-

dard component for numerous plant breeding based studies (Cullis et al., 2006, Smith

82

7.2 Methods and Materials

et al., 2002a, Oakey et al., 2006, 2007, Kelly et al., 2007). Stefanova et al. (2009)

has since extended this process by including advanced diagnostics for the selection of

non-genetic variance models, with the aim of reducing the ambiguity in choosing ap-

propriate spatial models. However there are some instances where spatial models have

not been included when pedigree information is used in mixed model analysis. For

example Crossa et al. (2006) and Burgueno et al. (2007) modeled additive (A) and

additive by additive (AxA) effects respectively in CIMMYT international wheat trials,

with the omission of spatial models. Due to the absence of design components and

spatial correlation in these studies, the study by Beeck et al. (2010) considered that

these models were ‘simplistic models for non-genetic effects’.

In Chapter 5, it was highlighted that the selection of non-genetic variance models are

a source of complexity hindering the more widespread use of mixed model analysis with

pedigree information. This was also highlighted in the paper by Beeck et al. (2010),

where the authors state that such model identification it is a component of difficulty

for the analysis of MET data sets. This chapter addresses this by demonstrating the

process of non-genetic variance modeling for an actual plant breeding data set, coded

for anonymity. This process also aims at evaluating the differences that may arise in

the spatial analysis of plant breeding METs from the inclusion of pedigree information.

This chapter commences with a brief description of the CBWA data set, before under-

taking a series of analyses contrasting standard and pedigree models at a single trial

level. The findings of the spatial modeling with pedigree are then discussed within the

context of the breeding data.


7.2.1 Data set description

The data set for this chapter consists of the 2011 subset of the full CBWA MET yield

data described in Chapter 6. Briefly, the 2011 data set comprised 10 trials with a total

of 1084 varieties tested across locations (Table 7.1). Trials were individually designed

using a p−rep design (Cullis et al., 2006) with a ‘superblock’ (see Section 7.2.1.1 for

details) component with a majority of varieties sown once or with an extra replicate.

83


All trials except the one at York were composed of 12 columns by 44 rows and 2 blocks.

The full pedigree data set described in Chapter 6 was used in this chapter.

7.2.1.1 Superblock design component

The 2011 data set included a ‘superblock’ design component. This design element

consists of a ‘base-superblock’ trial and ‘superblocks’. The base-superblock was the

York trial, the largest trial with 12 columns and 99 rows, and comprised all the varieties

trialed within 2011. The superblocks consisted of a combination of 2 or 3 trials grouped

together, see summary in Table 7.1. Permitting sufficient seed, new entries were then

present in one of the four superblocks. Hence, each new entry had a maximum of four

replications. Superblocks were grouped on the basis of geographic locations that were

similar in climate and biotic factors.

Table 7.1: Details of the 2011 growing season trials, including the superblock the trialwas part of, number of entries, columns and rows, as well as trial mean yield in t/ha.

Trial Superblock Entries Columns Rows Mean yieldnumber (t/ha)

CTTA11ALBR2 2 426 12 44 1.47CTTA11BUNT6 3 354 12 44 1.49CTTA11CCRK2 4 423 12 44 2.03CTTA11ELMR3 4 424 12 44 2.15CTTA11LKBL3 2 419 12 44 0.96

CTTA11MNGN6 1 371 12 44 1.91CTTA11PTLI5 2 423 12 44 0.65CTTA11SSTL6 1 425 12 44 0.86

CTTA11WAGG2 4 423 12 44 0.95CTTA11YORK6 Base SB? 1045 12 99 1.32

?SB = superblock.

84


7.2.2 Single Trial analysis

Beeck et al. (2010) set out a process for mixed model selection, which consists of

two components, the first being the selection of the model for the genetic variance

structure and the second being the variance models for the trial non-genetic effects

(which they refer to as Ngen-variance models). At a single trial, a description of the

Ngen modeling with genetic models which exclude and include pedigree information

is first outlined. Following the process of Oakey et al. (2006, 2007) the models which

exclude pedigree information are referred to as Standard models and those which include

pedigree information are referred to as Pedigree models.

7.2.2.1 Standard statistical model

A single trial analysis without pedigree information for the jth (j = 1, ...., t) trial in the

CBWA data set is first described. Each trial is comprised of m entries in a rectangular

array of plots with rj rows and cj columns, so that the number of plots in a trial is

given by nj = rjcj . The spatial mixed model can be written as,

yj = Xjτ j +Zvjuvj +Zpjupj + ej (7.1)

where, yj is an n × 1 vector of entry yields in t/ha, ordered as rows within columns;

τ j is the vector of fixed effects, which in most trials includes an overall trial mean and

any additional trial specific spatial modeling terms, such as linear regression across

row; the associated design matrix is Xj ; uvj is a m× 1 vector of random entry effects,

with associated design matrix Zvj ; upj is a vector of random peripheral effects (non-

genetic), which includes block effects and other spatial modeling terms such as random

row effects; the associated design matrix is Zpj ; ej is the vector of residuals, ordered

as per the data vector.

The variance assumptions for random entry effects are:

var(uvj

)= σ2

vjIm

where σ2vj is the entry variance and Im is an identity matrix of order m.

85


The variance assumptions for random peripheral effects are:

var(upj

)= Gp = ⊕bj

l=1σ2pjlIqjl

where each trial has a maximum of bj random peripheral terms, and the lth term

(l = 1, ..., bj) has qjl effects and an associated variance component of σ2pjl

.

In terms of the errors, an AR1 model was used to model local trend. The AR1 model

has been previously demonstrated as the most commonly used for local spatial trend

(Smith et al., 2002b). Hence for a separable AR1 process for both rows and columns

the variance matrix of errors is:

var (ej) = Rj = σ2jΣcj ⊗Σrj (7.2)

where σ2j is the error variance, and Σcj and Σrj are correlation matrices of dimensions

c× c and r × r of AR1 processes in the column and row directions respectively. Each

matrix is a function of a single autocorrelation parameter ρcj and ρrj for the column

and row dimensions respectively. Given a set of ordered spatial coordinates (row or

column number) the correlation matrix has the form of:

Σ(ρ) =

1 ρ1 ρ2 · · · ρn−1

ρ1 1 ρ1 · · ·...

ρ2 ρ1 1 · · ·...

......

.... . .

...

ρn−1 · · · · · · · · · 1

Thus the models for non-genetic variation encompass model terms for both trial design

and spatial variation.

7.2.2.2 Pedigree statistical model

The mixed model with pedigree information is then fitted. This is an extension of

Equation 7.1 developed by Oakey et al. (2006) for a single trial analysis of a wheat

data set. Under the pedigree model, the model for uv is given by, uv = ua + ui,

that is the vector for random entry effects (uv) is partitioned into additive genetic

effects (ua(m×1)) and non-additive genetic effects (ui

(m×1)). Note that in the pedigree

86


statistical model, m is the number of entries in the pedigree. Thus the mixed model

for the jth (j = 1, ......, t) trial in the motivational data set can be written as,

yj = Xjτ j +Zvj(uaj + uij) +Zpjupj + ej (7.3)

The equation terms and the variance assumptions are as stated above for the standard

model. The variance assumptions for the additive and non-genetic effects are,

var (ua) = σ2aA

var (ui) = σ2i Im

Where the matrix A(m×1) = {aij} is the additive genetic relationship matrix which has

elements,

aii = 1 + Fi

aij = 2fij

where Fi is the inbreeding coefficient of entry i and fij is the coefficient of parentage

between entries i and j. Note that the total genetic effect was partitioned into additive

and non-additive effects, as the data set had a high level of inbreeding (1797 entries

had inbreeding coefficients of > 0.99).

7.2.3 Ngen variance modeling

This component involved choosing an appropriate model for the non-genetic effects,

through the use of graphical diagnostics. For each trial, the standard model was first

fitted and diagnostics used to determine if additional Ngen parameters were required

to accommodate global trend/extraneous variation and to check the adequacy of the

variance structure of local trend.

Diagnostics for Ngen examination, included the 3D sample variogram and plot of

residuals against row/column numbers (termed as residual plots) from Gilmour et al.

(1997) and the sample variogram augmented with coverage intervals obtained from

simulations from Stefanova et al. (2009). This latter approach was based on an approach

by Atkinson (1985), who used simulation to provide a reference for a set of fluctuations.

87


In Stefanova et al. (2009) this application consisted of plots of the ‘faces’ of the sample

variograms corresponding to zero row displacement (referred to as the column face)

and zero column displacement (referred to as a row face), which were plotted alongside

approximate 95% point-wise coverage intervals obtained from parametric bootstrap

simulations of the current model. The coverage intervals presented in this chapter were

based on N = 100 simulations.

7.2.4 Outlier detection

Erroneous data points were excluded based on formal tests of significance for outliers

in spatial analysis using the Alternative Outlier Mixed Model (AOMM) in ASReml-R

(Smith et al. unpublished). This process produces Studentised conditional residuals

(Scres) as part of the outlier identification diagnostics. For each trial, these Scres

from the pedigree model were plotted against those obtained from the standard model.

While AOMM diagnostics were used as a formal method of outlier identification, the

plant breeder was still consulted to determine which points were erroneous.

7.2.5 Analysis

For each of the trials, the Standard and Pedigree models were first fitted (these were

termed ‘base’ as they did not include any extraneous variation terms) followed by

AOMM diagnostics. Ngen diagnostics from both the base models were then used to

determine if additional global trend or extraneous variation terms were added. If this

was the case, Ngen diagnostics were repeated after each of the terms were added to

the model. Such stages of Ngen diagnostics enabled a comparison between spatial

covariance structures under the different models. A detailed description of Ngen model

fitting is first provided for the CTTA11YORK6 trial and then summarized for all other

trials. Note that for the sake brevity in this chapter and the next, the trial prefix

‘CTTA11’ will be dropped when reporting in the results and discussion. Trials will

instead be referred to by their location acronym and state number, i.e CTTA11YORK6

will now be referred to as YORK6.

88

7.3 Results

7.2.6 Estimation and Fitting

The estimation and fitting of the models are as previously described in Chapter 3. For

the pedigree models, the A−1 matrix was computed in R (R Development Core Team,

2012) package ASReml-R (Butler et al., 2009) using the A.inverse function, which uses

the algorithm of Meuwissen and Luo (1992) with adjustments for selfing.

7.3 Results

7.3.1 Ngen variance modeling - York trial

This section illustrates the N-gen modeling process for standard and pedigree models for

the York trial in detail. An overview of the series of Ngen models fitted are summarized

in Table 7.2.

Plots of residuals and sample variograms corresponding to the base model can be

seen for the standard model in Fig. 7.1a and for the pedigree model in Fig. 7.1b

respectively. For both these models, the corresponding variograms indicate the presence

of a strong linear trend, as both the sample variograms fail to reach a plateau as

expected in the theoretical sample variogram. This is also seen in both the residual

plots with a linear trend over row number for each column. These linear row effects

are also reflected by observing the row faces of both models, standard (Fig. 7.2c ) and

pedigree (Fig. 7.2d), of the augmented sample variogram. For both these row faces, the

sample variogram increases with increasing row separation exceeding the upper 95%

coverage interval before decreasing rapidly to below the mean of the simulations.

Model 1a and 2a (Table 7.2), which included a linear regression across rows for the

standard and pedigree models, were then fitted. The resulting sample variograms can

be seen in Fig. 7.3. The linear increase in residuals across rows for columns is no

longer observed. The presence of local spatial variation is seen by the smooth trend of

the residual plot. However, it is evident that there are random column effects, seen by

the jagged pattern in the column dimension of the sample variogram. The row face of

the augmented sample variograms is reviewed to confirm this. For both models, the

corresponding row faces of the augmented sample variograms (Fig. 7.4c and Fig. 7.4d),

89


indicates that the previous steep increase in the sample variogram is no longer seen,

instead a plateau is reached. However both sample variograms are also well below the

mean of the simulations and the sills are actually outside the coverage interval, indica-

tive of random column effects. The sample variogram for the pedigree model however,

appeared to be well below the lower confidence interval while the sample variogram

for the standard model was in-between the mean of the simulations and the bottom

confidence interval. Hence it was more obvious that the inclusion of column effects was

required under the pedigree model. The sample variogram from the column face of

the standard model (Fig. 7.4a) follows the mean of the simulations quite closely, and

no longer exceeds the upper or lower 95% confidence intervals. The sample variogram

from the column face of the pedigree model however, still exceeds the lower confidence

interval at around the 5th column displacement(Fig. 7.4b).

Models 1b and 2b (Table 7.2) were then fitted which included random column effects.

There is a large increase in REML log likelihood in comparison to the corresponding

previous models (Table 7.2). The resulting sample variograms can be seen in Fig, 7.5.

For the standard model, the smooth local trend observed previously (Fig. 7.3) is no

longer seen (ρr = 0.27 compared to ρr = 0.91, Table 7.2) however smooth local trend

is still evident for the pedigree model (ρr = 0.58, compared to ρr = 0.90, Table 7.2).

The augmented sample variograms (Fig. 7.6) both indicate that standard and pedi-

gree models show good agreement with the mean of the simulations and lie between

the 95% coverage intervals.

90

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91


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Figure 7.2: Initial plots of faces of the sample variogram (solid line) and the simulationmean (dotted line) as banded by 95% coverage intervals (dashed lines) for standard andpedigree models at the York trial.

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93


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94

7.3 Results

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95


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Figure 7.6: Plots of faces of the sample variogram (solid line) and the simulation mean(dotted line) as banded by 95% coverage intervals (dashed lines) for standard and pedigreemodels after the addition of random column effects at the York trial.

96

7.3 Results

7.3.1.1 Model parameters

REML estimates of variance parameters from the sequence of models fitted for the

YORK6 disease nursery are summarized in Table 7.2. REML estimates of error variance

were always lower under the pedigree model than the standard model. REML estimates

of row autocorrelation values were similar across the first two models fitted for standard

(1 and 1a) and pedigree (2 and 2a) models but differed between standard and pedigree

for the last model fitted (1b and 2b) with the correlations being much stronger for the

pedigree model. REML estimates of column autocorrelation values were always larger

and non-negative under the pedigree models than the standard models (Table 7.2). For

standard and pedigree models, the REML estimates of the random column components

were similar at 0.014 and 0.015 respectively.

97


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98

7.3 Results

7.3.2 All trials

Model terms to encompass extraneous variation and non-stationary trend were required

for 4 out of the 10 trials (Table 7.3). YORK6 was the only trial in the data set which

needed more than one term to encompass global and extraneous variation in both row

and column dimensions. Across all trials, the same extraneous variation components

were added to both standard and pedigree models.

In terms of stationary trend, row autocorrelation values were strong (> 0.3) at 4

out of the 10 trials and column autocorrelation values were strong at 2 of the trials

(Table 7.3). Overall, the largest row autocorrelation values was observed at LKBL3

at 0.63 for standard and 0.62 for pedigree models respectively. The largest difference

in row and column autocorrelation values between standard and pedigree models were

observed for the YORK6 and ELMR3 trials (Table 7.3).

An absolute Scres value of 3 was used to determine which data points would be

recognised as outliers. Out of the 10 trials, 6 trials had the same number of outliers

under both standard and pedigree models (BUNT6, LKBL3, MNGN6, PTLI5, SSTL6,

YORK6). However 2 trials, CCRK2 and ELMR3 had more outliers under the standard

model than the pedigree model. A plot of the Scres for the standard and pedigree

models at ELMR3 can be seen in Fig. 7.7. This plot indicates 5 outliers were identified

under the standard model and 3 outliers identified under the pedigree models, of which

2 outliers were in common between both models. Additionally it was observed that the

Scres values corresponding to single replicate entries (black dots) were furthest from

the line of equivalence (y=x), Scres values corresponding to entries with two replicates

were closer to this line, and other colours (i.e yellow dots) were even closer to this line.

This indicates poor agreement between models for single replicate Scres entries.

99


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Figure 7.7: Outliers detected under standard and pedigree models - Studentisedconditional residuals from AOMM diagnostics for the standard model plotted against Stu-dentised conditional residuals from the pedigree models at the Elmore trial. Scres valuescorresponding to entries with one or two replicates are shown as black and red coloureddots respectively, with other numbers of replicates have differing colours. The solid lineindicates the equivalence, that is y=x. Dashed horizontal and vertical lines indicate theabsolute Scres cut off value of 3.

100

7.4 Discussion

Table 7.3: Spatial modeling of the 2011 growing season trials. REML estimates of errorvariance, autocorrelation parameters for rows/columns, terms added for global trend andextraneous variation and outliers detected for standard (std) and pedigree (ped) models.

Trial Model Error Autocorrelation Global trend & Outliersvariance Row Column extraneous variation detected

ALBR2 std 0.059 0.15 0.10 3ALBR2 ped 0.061 0.24 0.13 4BUNT6 std 0.039 0.19 0.16 rd(R)# 4BUNT6 ped 0.037 0.09 0.14 rd(R) 4CCRK2 std 0.143 -0.01 0.13 11CCRK2 ped 0.126 0.14 0.21 10ELMR3 std 0.080 0.06 0.34 5ELMR3 ped 0.083 0.23 0.39 3LKBL3 std 0.085 0.63 0.14 2LKBL3 ped 0.082 0.62 0.15 2

MNGN6 std 0.091 0.26 0.07 2MNGN6 ped 0.084 0.32 0.09 2

PTLI5 std 0.003 0.35 0.02 4PTLI5 ped 0.003 0.34 0.03 4SSTL6 std 0.040 -0.02 0.25 rd(C)# 1SSTL6 ped 0.041 -0.02 0.33 rd(C) 1

WAGG2 std 0.027 0.05 -0.03 rd(C) 1WAGG2 ped 0.027 0.10 -0.11 rd(C) 2YORK6 std 0.030 0.27 -0.13 lin(R)+ + rd(C) 7YORK6 ped 0.024 0.58 0.06 lin(R) + rd(C) 7

+lin(R) indicates a fixed linear regression on row number; #rd(R) and rd(C) indicates random row

and random column effects respectively.

7.4 Discussion

This chapter illustrates the process of spatial mixed model analysis at a single trial stage

for a series of canola breeding trials from the 2011 growing season. Plant breeding trials

are commonly sown as partially replicated trials, which not only enables the testing of

a maximum number of entries, but also enables the testing of new test crosses with

minimal seed (Cullis et al., 2006). On average 80% of entries within a trial in this data

subset were sown as single replicates (Table 6.2 in Chapter 6). The importance of spatial

models in mixed model analysis is especially illustrated in such plant breeding trials.

With limited replication, these trials attempt to provide enough information for the

selection of entries for commercialization and parents for the next cycle of breeding. As

a result, there is a need to obtain accurate selections, which requires the minimisation

of error, such as environmental heterogeneity. This is evident, as environmental effects

are common in all designed field trials, and if not accounted for can lead to biased

estimates of treatment effects (Basford and Cooper, 1998). The estimation of genetic

merit in annual breeding trials is thus critical for the efficiency of a breeding program

101


and spatial mixed model analysis enables this. However, it has not been addressed in

any published research the impacts of pedigree information on spatial models in p-rep

trials.

The presence of local trend was evident in the 2011 growing season trials, with a

majority of trials indicating large autocorrelation values in the row dimension and a few

trials in the column dimension. In addition, terms for global trend/extraneous variation

components were detected at 4 out of the 10 trials with only the trial at YORK6

needing both linear row and random column components. Hence the majority of trials

presented from a single growing season exhibit different smooth spatial variation and

global trend/extraneous variation. This conforms to previous plant breeding studies

which have also identified the presence of extraneous variation in field experiments in

Australia (Cullis et al., 1998, Gilmour et al., 1997, Smith et al., 2001b,a, Stefanova et al.,

2009) and highlights the importance of detecting spatial variation for plant breeding

trials for improving the accuracy of selections in plant breeding experiments.

The addition of pedigree information to the model resulted in differences at 2 trials

for row autocorrelations and 2 trials for column autocorrelations. This differs from the

study by Oakey et al. (2006), who found that column and row autocorrelations were

similar under the standard and pedigree models. Such a difference can be attributed to

Oakey et al. (2006)’s study being based on wheat breeding trials, which were designed

as nearest neighbour designs with most entries sown in all trials. CBWA’s trial designs

on the other hand were p-rep designs with a majority of entries having 1 or 2 reps

within a trial, and a superblock component. Hence the relationships afforded through

pedigree information would be greater in the CBWA data set than those in Oakey et al.

(2006)’s, resulting in a larger impact on trial variance parameter estimates. Thus the

value of pedigree information is especially demonstrated by the ‘borrowing’ of additional

information from relatives to improve the modeling of entry genetic effects and thus

improving the accuracy of spatial modeling.

Six of the trials had the same number of outliers detected under the pedigree model

and standard models. However the two trials, ELMR3, and CCRK2, had more outliers

detected under the standard model than the pedigree model. For the outliers detected

for each trial (see Table 7.3), some were in common between the models but there were

102

7.4 Discussion

also some that corresponded to different data points unique to the model. From the

plot of Scres residuals under both standard and pedigree models at the ELMR3 trial

(Fig. 7.7), it was also evident that entries with single replicates had poor agreement

between the two models. This would be expected, as the impact of pedigree informa-

tion would be larger for entries with single replicates than for entries that have many

replicates. Hence the impact of pedigree inclusion is especially important for single

replicate entries, which explains entry performance better than the standard model,

which assumes independence of genetic relationships between trial entries. Pedigree

information is thus demonstrated as important in improving the accuracy of outlier

detection.

The spatial mixed model analysis was undertaken at a single trial, for all trials

within the 2011 growing season. It was not extended in this chapter to include dif-

ferences in spatial models that would arise within a MET/FA framework of analysis,

due to issues with the time taken for the analysis. This issue will be examined in more

detail in Chapter 9. An important point is that under a MET analysis, it would be

expected that greater relationships afforded through MET and FA modeling will result

in larger differences between spatial models for standard and pedigree models. This

would be furthered by the fact that p-rep design and superblock components enables

for replication to be balanced across trials, thereby contributing to more information

in the model.

The spatial mixed model analysis from this chapter will be extended in the next

chapter with a MET/FA analysis to obtain breeding program selections. However, it

is important to mention that each trial within a MET requires its own spatial model

since trial errors are characterised by heterogeneous variance or covariances (Crossa

et al., 2006). These result from (i) heterogeneous within site error variances resulting

from site to site variations among plots from properties that impact on the measured

traits, (ii) particular trials and or years showing more genotypic variation and (iii)

heterogeneous covariances among trials arising from similarities between trials based

on environmental factors (Crossa et al., 2006). Hence, the analysis of MET data must

encompass spatial structures to accommodate these sources of extraneous variation,

and the absence of this may result in large experimental error variance components.

103


7.5 Summary

This chapter demonstrated the stages of Ngen analysis and evaluated the impact of

spatial analysis under pedigree models in comparison to standard models at a single

trial, specifically for p-rep designs. From this study, it is evident that pedigree infor-

mation aids in the modeling of spatial errors, by adding information to the analysis

that would otherwise not have been included. Due to common relationships found in

breeding programs, it is evident that pedigree information aids in the explanation of

entry performance. As demonstrated by the differences between spatial models and

outlier detection under standard and pedigree models, it is recommended that base

line Ngen modeling should always include pedigree information for the determination

of trial spatial models.

104

Chapter 8

MET analysis of trials with

pedigree information

8.1 Introduction

The annual aims of CBWA’s canola breeding program are (1) to select entries for pro-

motion or commercialization and (2) to select parents for the next cycle of breeding.

Selections are based on a number of traits, including grain yield, blackleg disease re-

sistance, oil and protein quality, however the primary trait of selection is grain yield.

Such selection is undertaken annually from the analysis of METs located across a broad

range of Australian target environments. The objective is to produce open-pollinated

and F1 hybrid varieties for commercial release, or for use in crossing.

Historically additive and non-additive effects (often referred to as General Com-

bining Ability (GCA) and Specific Combining Ability (SCA) in the literature) are an

important basis for breeders decisions on hybrid breeding strategies (de la Vega and

Chapman, 2006). Additive genetic effects can be viewed as breeding values, as they rep-

resent the heritable component of genetic variation. The derivation of additive genetic

effects is important for breeding objectives and key to maximising breeding progress

(Falconer, 1981).

The performance of entries across locations, that is the magnitude of GxE, is an-

other important source of information. METs are critical in estimation of magnitude

105

8. MET ANALYSIS OF TRIALS WITH PEDIGREE INFORMATION

and patterns of GxE. GxE interactions can be the result of differences in genotypic

adaptation and/or due to heterogeneous environments within targeted areas for se-

lection (Fukai and Cooper, 1995). Cross-over GxE can limit response to selection,

as it complicates the comparisons of entry performance over environments (Argillier

et al., 1994, Cooper and DeLacy, 1994). However an understanding of GxE also allows

breeders to better exploit specific or general adaptation and even identify target envi-

ronment clusters (Bernardo, 2002). Hence an understanding of GxE is an important

component in maintaining genetic gain in selection in plant breeding programs. This

is especially the case where the target environments are diverse, such as in CBWA’s

breeding program, which develops varieties for low to high rainfall cropping zones of

southern Australia.

These selection aims and an understanding of the impact of GxE may be addressed

using the mixed model analysis of MET data developed by Smith et al. (2001b) with

an extension for pedigree information by Oakey et al. (2007). Having demonstrated

the application and process of N-gen modeling in the previous chapter, this chapter

commences with the next step in the mixed model process, that is the genetic mod-

eling (gen-modeling) process for MET data from p-rep trials. This is applied to the

motivating example of CBWA’s 2011 MET and followed by an interpretation of the

impact of environment on selection in CBWA’s breeding program.


8.2.1 Description of data

The motivating data set consists of a subset of the 2011 trial data with corresponding

pedigree file. The pedigree file was obtained using the package ‘Pedicure’ (Butler, 2012)

in R (R Development Core Team, 2012), which limited pedigree data to entries present

in the current data set. A total of 13 generations of pedigree information was available

for this pedigree/data set combination, see Table 6.8, Chapter 6. This consisted of 91

unique mothers and 226 unique fathers with a total of 1544 entries in the pedigree file.

The MET data set consisted of a total 5941 records on 1084 entries.

106


The trial details were as summarized in Chapters 6 and 7. The MET data set had a

total of 10 trials across 10 locations within a single year (growing season). These loca-

tions targeted canola production zones across four Australian states from low to high

rainfall. These ranged from Mingenew W.A (29◦19 S, 115◦16 E) to Croppa Creek NSW

(29◦71 S, 150◦18 E). Annual rainfall and growing season rainfall (May to November)

were obtained for each of the trial locations, from the closest weather station from online

records at the Australian Bureau of Meteorology (http://www.bom.gov.au/climate)

(Table 8.1).

There was good concurrence (commonality) of entries across trials in the MET

data set. A minimum of 141 entries were in common between any pair of trials (upper

triangle Table 8.2). Concurrence of parent entries, that is both males and females, were

high as well with a minimum concurrence of 116 (lower triangle Table 8.2). Overall

there was greater concurrence of parents across trials than entries.

107

http://www.bom.gov.au/climate/data/?ref=ftr


Tab

le8.1

:L

oca

tionb

asedtria

ld

etails:state,

latitu

de,

lon

gitu

de,

sowin

gan

dh

arvest

dates,

trialm

eanyield

and

rainfa

llfor

CB

WA

’s2011

ME

Ts.

Tria

lT

rial

Sta

teL

atitu

de

Lon

gitu

de

Sow

ing

Harv

estT

rial

mea

nR

ain

fall

(mm

)C

od

ed

ate

date

yield

An

nu

al

Gro

win

gsea

son

AL

BR

2A

lbu

ryN

SW

35◦90

S146◦92

E17/05/11

06/12/11

1.4

7874

420

BU

NT

6B

untin

eW

A29◦95

S116◦29

E11/05/11

01/11/11

1.4

9354

286

CC

RK

2C

rop

pa

Creek

NS

W29◦71

S150◦18

E13/05/11

04/11/11

2.0

3171

433

EL

MR

3E

lmore

VIC

36◦29

S144◦36

E26/05/11

29/11/11

2.1

5539

307

LK

BL

3L

ake

Bola

cV

IC37◦58

S142◦90

E17/05/11

06/12/11

0.9

6554

267

MN

GN

6M

ingen

ewW

A29◦19

S115◦16

E14/05/11

24/10/11

1.9

1469

396

PT

LI5

Pt

Lin

coln

SA

34◦43

S135◦51

E13/05/11

12/11/11

0.6

5565

362

SS

TL

6S

thS

tirling

Ran

ges

WA

34◦57

S118◦27

E11/05/11

30/11/11

0.8

6484

325

WA

GG

2W

agga

WA

34◦93

S147◦35

E12/05/11

23/11/11

0.9

5664

319

YO

RK

6Y

ork

WA

31◦53

S116◦46

E09/05/11

15/11/11

1.3

2460

395

108


Tab

le8.2

:N

um

ber

ofen

trie

s(u

pp

ertr

iangl

e)an

dp

are

nts

(low

ertr

ian

gle

)in

com

mon

bet

wee

np

air

sof

tria

lsin

the

2011

CB

WA

ME

Td

ata

set.

AL

BR

2B

UN

T6

CC

RK

2E

LM

R3

LK

BL

3M

NG

N6

PT

LI5

SS

TL

6W

AG

G2

YO

RK

6A

LB

R2

167

228

223

123

171

123

222

212

426

BU

NT

6156

178

163

158

231

177

166

163

321

CC

RK

2160

164

124

216

178

218

211

123

423

EL

MR

3165

159

152

221

171

220

219

124

424

LK

BL

3143

155

158

160

167

122

223

220

417

MN

GN

6160

155

160

167

161

180

116

172

335

PT

LI5

143

157

164

161

144

156

212

228

423

SS

TL

6160

157

155

160

151

141

162

227

425

WA

GG

2159

151

143

144

156

158

157

159

423

YO

RK

6196

194

196

197

189

196

195

194

192

109


8.2.2 Statistical models

The analysis for a series of t trials, is an extension of the single trial model in Equation

7.3, Chapter 7. This MET model was initially proposed by Smith et al. (2001b), with

an extension for pedigree information developed by Oakey et al. (2007). The MET

model with pedigree information can be written as,

y = Xτ +Zv(ua + ui) +Zpup + e (8.1)

where y = (yT1 ,yT2 , ...,y

Tt )T , that is now a concatenated vector of data of individual

plot yields in t/ha combined across trials. X and Zv, are design matrices for fixed

effects and random genetic effects respectively. τ = (τT1 , τ

T2 , ...., τ

Tt )T is the p × 1

vector of fixed effects, and includes effects such as linear regression on rows associated

with spatial trend; ua = (uaT1 ,ua

T2 , ....,ua

Tt )T is the mt× 1 vector of random additive

genetic effects; ui = (uiT1 ,ui

T2 , ....,ui

Tt )T is the mt× 1 vector of random non-additive

genetic effects; up = (upT1 ,up

T2 , ....,up

Tt )T is the q × 1 vector of random peripheral

effects and e = (eT1 , eT2 , ...., e

Tt )T is the vector of residuals ordered as per the data

vector. Note that the vector of random peripheral effects (up) comprises blocking

effects for each trial and effects such as random row to accommodate spatial trend and

has the associated design matrix Zp. In this data set t = 10 and m = 1544.

The variance model for additive and non-additive genetic effects are:

var (ua) = Gea ⊗A

var (ui) = Gei ⊗ Im

where Ges, s = a, i, is the genetic variance matrix of dimensions t× t, for environments

e, with the main diagonal being the genetic variance for each trial and the off-diagonals

the genetic covariances between pairs of trials. A is as previously defined, the additive

relationship matrix and Im is an identity matrix of order m.

The variance model for errors is the same as in Chapter 7 Equation 7.2, however

the extension across trials using the approach of Smith et al. (2001b) is,

var (e) = R = ⊕tj=1Rj

110


where Rj is the variance matrix for the errors for the jth trial (j = 1, .., t). This

extension, enables a separate spatial covariance structure for error in each trial.

The variance model for the random peripheral effects are:

var (up) = Gp = ⊕tj=1 ⊕

bjl=1 σ

2pjlIqjl

where each trial has a maximum of bj random peripheral terms and the lth term (l =

1, ..., bj) has qjl effects and an associated variance component of σ2pjl

.

The first step of the analysis is to check for outliers and the adequacy of the spatial

model for each trial. As this was already undertaken in Chapter 7, the process was

only re-examined for the MET data set. This was achieved using a diagonal model for

the genetic variance matrix (Ges, s = a, i), so that,

Gea = diag {σ2aj}

Gei = diag {σ2ij}

where σ2aj is the additive genetic variance and σ2

ij is the non-additive genetic variance

of each trial, j for j = 1, ..., t. The diagonal model is the equivalent of running t = 10

individual trial analyses as undertaken in the previous chapter. The spatial models

were adequate and the same terms for global trend and extraneous variation for each

trial in Chapter 7 Table 7.3, were included in this MET analysis.

As in Chapter 7, AOMM statistics (Smith et al. unpublished paper) were used

to identify outliers in the 2011 trial data. An absolute Scres value of 3 was used

to diagnose outliers. Identified outliers were then examined in the datafile, with the

criterion for omission based on the Scres of other replicates within the trial and if

these were not present, the Scres of sister-lines with common parentage. For example

consider the trial MNGN6, the entry CBD1310 at column 9 row 24, was dropped after

identifying the Scres of its sister lines (i.e; they have the same mother). The Scres for

the sisterlines, were 3.63 and 0.86 respectively (see Table 8.3) compared to -5.67 for

the identified outlier. In concurrence with this outlier diagnostic, the detected outliers

were also confirmed with the plant breeder as erroneous, and then set to a missing

value delimiter in the data file.

111


A FA variance structure of Smith et al. (2001b) was then used to model GxE effects.

This was applied using the extension to including pedigree information as described by

Oakey et al. (2007) for MET data sets. The FA model for Ges, s = a, i is,

Ges = (ΛesΛTes +ψes)

where Λes is the t×k matrix of factor loadings (for k factors) and ψes is a t×t diagonal

matrix of trial specific variances. The resulting variance assumptions for additive and

non-additive genetic effects are,

var (ua) = (ΛeaΛTea +ψea)⊗A

var (ui) = (ΛeiΛTei +ψei)⊗ Im

Note that the spatial models identified earlier under the diagonal model were retained.

8.2.3 Model fitting and examination of GxE

The process for estimation and fitting of mixed models are as described in Chapter

3 with the additions for pedigree as described in Chapter 8. However the focus of

this chapter is the influence of environments on entry performance, and this is best

observed through the visual tools for exploring GxE developed by Cullis et al. (2010).

These tools include heatmap representations (R Development Core Team, 2012) of

REML estimations of genetic correlation matrices (Ces, where s = a, i) between trials,

with trial ordering within the heatmap based on an agglomerative (nested) hierarchical

cluster algorithm obtained from the ‘agnes’ package in R (R Development Core Team,

2012). This package produced both dendrogram and heatmap outputs.

Within an FA(k) framework of analysis, the correlation matrix for genetic effects

Ces, for s = a, i is,

Ces = DesGesDes

= Des(ΛesΛTes +ψes)Des

here, Des is a diagonal matrix with the diagonal elements given by dsii = 1/√gsii,

where gsii is the ith diagonal element of Ges. The terms Λes and ψes are as described

above in the Section 8.2.2.

112

8.3 Results

Also of interest in this chapter, is the investigation of the correlation matrix for total

genetic effects (Ceg). Ceg involves elements of the additive genetic relationship matrix

(A). Consider that the total genetic variance for an entry r, r = 1, ....,m is,

var (uvr) = arrGea +Gei (8.2)

where arr is the rth diagonal ofA. Hence this shows that the total genetic variance for a

trial and correlations between trials will differ depending on the inbreeding coefficient

of entries. This will be discussed in relation to the specific entry types, hybrid and

non-hybrid, for this data set in Section 8.3.6.

Table 8.3: Outliers detected from the AOMM statistic at the MNGN6 site.

Entry Column Row Replicate Block Yield Mum Dad Scresvalue

CBD1310 9 24 1 2 0.27 CBD0003 CBCV004 -5.67CBD1310 4 8 1 1 2.07 CBD0003 CBCV004 3.63CBD1308 7 17 1 2 2.11 CBD0003 CBCV004 0.86

8.3 Results

8.3.1 N-gen variance modeling

The peripheral effects and spatial models were described in detail in Chapter 7 and are

only summarized here. Global trend and extraneous variation components were needed

for 4 out of the 10 trials in the data set (Table 8.4). Three of the trials were observed to

have extraneous variation in the column dimension and only the YORK6 trial needed

terms for both column and row dimension. REML estimates of error variance ranged

from 0.003 (PTLI5) to 0.084 (CCRK2). Block variance components were zero or close

to zero across all trials in the data set. Autocorrelation values for row and column

dimensions were relatively small, the largest being 0.53 for the column dimension at

BUNT6.

113


Tab

le8.4

:S

patia

lm

od

eling

forth

e20

11M

ET

.R

EM

Lestim

ates

of

error,

blo

ckan

dau

tocorrelation

param

etersfor

rows

and

colu

mn

s.T

erms

ad

ded

forglob

altren

dor

extran

eous

varia

tion

an

dth

enu

mb

erof

outliers

removed

.

Erro

rB

lock

Au

tocorrelation

Glob

altren

d&

Nu

mb

erof

Colu

mn

Row

extran

eous

variationterm

sou

tliers

AL

BR

20.0

60.00

0.130.26

1B

UN

T6

0.0

30.00

0.100.22

rd(R

)#

2C

CR

K2

0.080.00

0.200.16

EL

MR

30.08

0.000.34

0.262

LK

BL

30.0

80.00

0.110.59

MN

GN

60.0

70.01

0.080.32

1P

TL

I50.0

00.00

0.040.33

1S

ST

L6

0.030.00

0.410.08

rd(C

)#

1W

AG

G2

0.020.00

-0.040.24

rd(C

)2

YO

RK

60.0

20.00

0.000.67

lin(R

)+

&rd

(C)

1+

lin(R

)in

dica

tesa

fixed

linea

rreg

ression

on

rownum

ber;

#rd

(R)

and

rd(C

)in

dica

tesra

ndom

rowand

random

colu

mn

effects

respectiv

ely.

114

8.3 Results

8.3.2 Outliers

All trials except CCRK2 and LKBL3 had at least one outlier removed (Table 8.4).

A majority of plots in column 1 of the CCRK2 trial were identified as outliers. This

was confirmed with the plant breeder as resulting from an issue of uneven germination

in this column (C Beeck, pers.comm.). As a result, this first column of the trial was

dropped; that is set to a missing value delimiter for this trial.

8.3.3 FA Model

Commencing with the base line diagonal model for Ges, an FA(1) followed by an FA(2)

structure were then fitted. There was an increase in log-likelihood (lr) from the diagonal

model up to the FA(2) (Table 8.6). The FA(2) model provided a superior fit (P<0.001)

and accounted for 79.88% of trial additive genetic variance and 76.25% of trial non-

additive genetic variance (Table 8.6). Due to computational limitations impacting on

time to analysis, the FA(2) was the last model fitted. Note however, that on average

a large percentage of both additive and non-additive genetic variance is explained by

the FA(2).

REML estimates of the percent variance accounted by the two factors in the FA(2)

model for additive genetic effects at each trial are summarized in Table 8.5. A large

proportion of the additive trial variance was accounted for by the first factor at the

trials ALBR2, BUNT6, CCRK2, ELMR3 and YORK6, which all had greater than 70%

explained. The percent variance accounted for by the first factor was poor for the trials

PTLI5 and SSTL6 (Table 8.5). The remaining trials, LKBL3, MNGN6 and WAGG2

had greater than 50% variance accounted for by the first factor. The second factor

however accounted for a large amount of variance for the trials BUNT6 and SSTL6,

22.06 and 34.87 respectively. Total percent variance under the FA(2) model for additive

effects was 100% for ALBR2, BUNT6 and CCRK2. At all other trials, the total percent

variance explained by the FA(2) for additive trial effects was greater than 70% except

for MNGN6 (68.18%) and PTLI5 (12.62%).

REML estimates of percent variance accounted by the two factors in the FA(2) model

for non-additive genetic effects at each trial are summarized in Table 8.5. The percent

115


variance accounted for by the first factor was large (greater than 70%) for the trials

ELMR3, MNGN6 and WAGG2. The trials CCRK2, SSTL6 and YORK6 had the lowest

percent variance accounted under the first factor. The percent variance accounted for by

the second factor however, was almost 0% at the trials PTLI5 and SSTL6. In contrast

the second factor explained the largest proportion for the trials ALBR2 (36.90%) and

BUNT6 (43.48%). The total variance explained by both factors for trial non-additive

effects was almost 100% at BUNT6, EMLR3, MNGN6 and WAGG2.

Table 8.5: REML estimates of percent of variance accounted by the FA(2) model foradditive and non-additive genetic effects.

Trial Additive genetic effects (%) Non-additive genetic effects (%)factor 1 factor 2 Total factor 1 factor 2 Total

ALBR2 89 11 100 54 37 91BUNT6 78 22 100 57 43 100CCRK2 82 18 100 32 8 40ELMR3 72 2 74 81 19 100LKBL3 58 14 72 66 7 73

MNGN6 56 13 68 76 24 100PTLI5 11 2 13 68 0 68SSTL6 47 35 82 48 1 49

WAGG2 68 3 71 71 29 100YORK6 72 1 73 13 28 42

Table 8.6: Models fitted for the genetic variance matrix Ges, s = a, i. REML log-likelihood (lr), REMLRT and estimates for percent of variance accounted by the FA-kmodel for additive and non-additive components.

Model for Log-Likelihood REMLRT (P value) Variance accounted (%)Ges ratio test (lr) Additive Non-additive

diagonal 4540.98FA(1) 5084.74 71.40 70.26FA(2) 5122.50 75.52 (P<0.001) 79.88 76.25

8.3.4 GxE for additive effects

The dendrogram of the dissimilarity matrix (It−Cea) of additive effects for yield (Fig.

8.1) suggests three clusters for additive effects. The first cluster includes the trials

ALBR2, ELMR3, WAGG2, LKBL3 and SSTL6. The second cluster includes BUNT6,

CCRK2 and MNGN6 and YORK6. The third cluster is PTLI5, on its own which

116

8.3 Results

appears to be unrelated to these two major clusters, indicating poor agreement for

additive effects.

The heatmap for additive correlations (Fig. 8.2) confirms high correlations within

clusters and moderate to weak correlations between clusters. PTLI5 had correlations

of less than 0.35 with all other trials. SSTL6 showed poor agreement (less than 0.40

correlation) with 4 trials and greater than 0.50 correlations with the remainder trials.

The presence of the two main clusters with low to moderate correlations between them

indicates the presence of GxE within this single year MET.

ALB

R2

ELM

R3

WA

GG

2

LKB

L3

SS

TL6

BU

NT

6

CC

RK

2

MN

GN

6

YO

RK

6

PT

LI5

0.0

0.2

0.4

0.6

Dendrogram of agnes(x = dis.mat, diss = T)

Agglomerative Coefficient = 0.7dis.mat

Hei

ght

Figure 8.1: Dendrogram of the dissimilarity matrix (It−Cea) of additive effectsfor yield. -

117


MNG

N6

PTLI5

YORK6

MNGN6

CCRK2

BUNT6

SSTL6

LKBL3

WAGG2

ELMR3

ALBR2

ALBR

2

ELM

R3

WAG

G2

L

KBL3

SS

TL6

BU

NT6

CC

RK2

YOR

K6

PTL

I5

−1.0

−0.5

0.0

0.5

1.0

C1

C2

C3

Figure 8.2: Heatmap of the REML estimate of the additive genetic correlationmatrix (Cea) - The trials are as ordered as in the dendrogram in Fig. 8.1 and the keyindicates the correlation scale.

118

8.3 Results

8.3.5 GxE for non-additive effects

Dendrogram of the dissimilarity matrix (It −Cei) of trial non-additive genetic effects

for yield can be seen in Fig. 8.3. The dendrogram suggests the presence of 2 clusters.

The first consists of the trials ALBR2, BUNT6, MNGN6, CCRK2, YORK6 and the

second cluster consists of the trials ELMR3, WAGG2, LKBL3, PTLI5 and SSTL6.

There appears to be good agreement of trials within a cluster.

The heatmap for non-additive correlations (Fig. 8.4) reflects the 2 clusters seen

in the dendrogram. A majority of the trials are highly correlated (greater than 0.70)

for non-additive effects. The exception to this is CCRK2 and YORK6 which had

correlations less than 0.54 for 3 trials and 2 trials respectively. Similar to additive

effects, the heatmap for non-additive effects indicates the presence of GxE by the

presence of two clusters which are not directly correlated with each other.

Both the additive and non-additive genetic correlations between trials are summa-

rized in Table 8.7. At a majority of trials the non-additive genetic correlations were

smaller than the corresponding trial additive genetic correlations. The only exceptions

to these were the trials MNGN6 and PTLI5, which appeared to have higher non-additive

genetic correlations than additive genetic correlations across trials.

119


ALB

R2

BU

NT

6

MN

GN

6

CC

RK

2

YO

RK

6

ELM

R3

WA

GG

2

LKB

L3

PT

LI5

SS

TL6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Dendrogram of agnes(x = dis.mat, diss = T)

Agglomerative Coefficient = 0.68dis.mat

Hei

ght

Figure 8.3: Dendrogram of the dissimilarity matrix (It − Cei) of trial non-additive genetic effects for yield -

120

8.3 Results

ALB

R2

B

UN

T6

SSTL6

PTLI5

LKBL3

WAGG2

ELMR3

YORK6

CCRK2

MNGN6

BUNT6

ALBR2

C

CR

K2

YO

RK6

E

LMR

3

WAG

G2

L

KBL3

PTLI

5

SST

L6

−1.0

−0.5

0.0

0.5

1.0

C1

C2

MNG

N6

Figure 8.4: Heatmap of the REML estimate of non-additive genetic correlationmatrix (Cei) - The trials are as ordered as in the dendrogram in Fig. 8.3 and the keyindicates the correlation scale.

121


Tab

le8.7

:R

EM

Lestim

ates

ofth

egen

eticcorrelatio

nm

atrix

for

ad

ditive

effects

(up

per

triangle)

and

non

-add

itiveeff

ectsb

etween

trials(low

ertrian

gle).

AL

BR

2B

UN

T6

CC

RK

2E

LM

R3

LK

BL

3M

NG

N6

PT

LI5

SS

TL

6W

AG

G2

YO

RK

6

AL

BR

20.68

0.720.85

0.840.59

0.350.84

0.840.77

BU

NT

60.9

50.99

0.680.50

0.820.22

0.330.64

0.80C

CR

K2

0.5

90.61

0.710.54

0.830.23

0.370.67

0.81E

LM

R3

0.3

90.38

0.380.70

0.580.30

0.670.72

0.71L

KB

L3

0.4

40.4

40.39

0.840.44

0.300.74

0.700.61

MN

GN

60.9

40.98

0.630.57

0.580.19

0.300.55

0.67P

TL

I50.6

10.6

20.47

0.740.67

0.720.31

0.290.26

SS

TL

60.57

0.580.42

0.580.54

0.650.57

0.670.52

WA

GG

20.2

90.28

0.320.99

0.820.47

0.700.53

0.68Y

OR

K6

0.590.6

30.36

0.090.16

0.580.31

0.300.02

122

8.3 Results

8.3.6 GxE for total genetic effects

Recall that the correlation matrix for total genetic effects, Ceg, is dependent on values

of the additive relationship matrix, arr, see Equation 8.2. In this section three different

cases of arr values are considered: based on all entries (which is the standard case),

hybrid entries only and non-hybrid entries only. This approach is warranted based on

the unique make up of the motivational data set, as 555 (51%) of entries were hybrids

(0 inbreeding) and the remainder entries had varying levels of inbreeding, reflected by

their Fgen value and corresponding Fj (Table 8.8).

Table 8.8: Levels of inbreeding (Fgen) for entries in the 2011 MET. Specific Fgen valueswere previously explained in Table 6.7, Chapter 6.

Fgen Number of entries

0.00 1970.74 10.90 110.94 12.00 13.00 354.00 2827.00 110.00 555

8.3.6.1 Total genetic effects: all entries

In this case, Ceg is based on the average inbreeding coefficient of all entries in the

pedigree. Hence arr was evaluated at a = 1.82. This was the method used by Cullis

et al. (2010), Oakey et al. (2007) and Crossa et al. (2006).

The heatmap of total genetic correlations across all entries, suggests the presence of

3 clusters (Fig. 8.5). The first cluster consists of the trials ALBR2, SSTL6, ELMR3,

WAGG2 and LKBL3. The second cluster consists of the trials BUNT6, CCRK6,

MNGN6 and YORK6, which are all located in W.A except for CCRK2. PTLI5 appears

to have its own cluster, indicating poor agreement for total genetic correlations with

the other trial trials. There is close agreement of trials within clusters and moderate

agreement between clusters. The exception is the trial PTLI5, which appears to be

123


weakly correlated with a majority of other trials (correlations are less than 0.35 across

all trials).

BU

NT6

C

CR

K2

P

TLI5

E

LMR

3

S

STL6

ALB

R2

PTLI5

YORK6

MNGN6

CCRK2

BUNT6

LKBL3

WAGG2

ELMR3

SSTL6

ALBR2

LK

BL3

M

NG

N6

Y

ORK6

−1.0

−0.5

0.0

0.5

1.0

C1

C2

C3

WAG

G2

Figure 8.5: Heatmap of the REML estimate of the total genetic correlationmatrix (Ceg, where a = 1.82) - The key indicates the correlation scale.

8.3.6.2 Total genetic effects: hybrid entries & non-hybrid entries

In this section, Ceg is based on the average inbreeding of hybrid entries only and then

non-hybrid entries only in the pedigree. For hybrid entries, arr is evaluated at ah = 1.12

and for non-hybrid entries arr is evaluated at anh = 1.97.

Total genetic correlations across trials for hybrid and non-hybrid entries were sum-

marised in Table 8.9. There were no large differences for total genetic correlations across

124

8.3 Results

trials for hybrid entries (lower triangle) and non-hybrid entries (upper triangle). The

clustering for hybrids and non-hybrids were the same as that obtained for total genetic

effects across all entries. That is 3 clusters were suggested, the first cluster consists of

the trials ALBR2, SSTL6, ELMR3, WAGG2 and LKBL3. The second cluster consists

of the trials BUNT6, CCRK6, MNGN6 and YORK6. The third cluster consists only

of PTLI5.

125


Tab

le8.9

:R

EM

Lestim

atesof

the

total

genetic

correla

tion

matrix

for

the

2011

ME

T.

Th

elow

ertrian

glecon

sistsof

totalgen

eticco

rrelation

sb

etween

trialsfo

rhyb

riden

triesan

dth

eu

pp

ertria

ngle

con

sistsof

tota

lgen

eticcorrelation

sb

etween

trialsfor

non

-hyb

riden

tries.

AL

BR

2B

UN

T6

CC

RK

2E

LM

R3

LK

BL

3M

NG

N6

PT

LI5

SS

TL

6W

AG

G2

YO

RK

6

AL

BR

20.7

40.72

0.740.69

0.690.44

0.590.71

0.65B

UN

T6

0.6

70.89

0.710.53

0.810.29

0.470.67

0.79C

CR

K2

0.630.84

0.690.51

0.790.28

0.460.65

0.77E

LM

R3

0.730.68

0.640.66

0.660.42

0.570.68

0.63L

KB

L3

0.7

00.49

0.450.70

0.510.49

0.600.64

0.47M

NG

N6

0.6

50.80

0.760.65

0.500.29

0.450.62

0.70P

TL

I50.48

0.2

90.26

0.480.55

0.310.41

0.410.26

SS

TL

60.61

0.450.42

0.610.64

0.450.46

0.550.42

WA

GG

20.7

00.6

40.60

0.700.68

0.620.47

0.590.59

YO

RK

60.59

0.750.72

0.590.44

0.680.26

0.400.56

126

8.3 Results

8.3.7 Selection

The dual aims of a plant breeding program are parental selection and entry promotion.

In this section, it is outlined how the predictions from the MET/FA analysis with

pedigree information can be used for the basis of such selection decisions. The relevant

predictions are the so-called regression BLUPs (referred to as Reg-BLUPs) (Cullis

et al., 2010). Reg-BLUPs were obtained for each entry across each trial in the MET

and averaged across all trials belonging to the cluster groups suggested for additive

(see, Section 8.3.4) and total genetic effects (see, Section 8.3.6). As Reg-BLUPS were

averaged for a cluster, they will be referred to as C-BLUPs. The actual interpretation

of what these cluster groups represent will be considered in detail in the discussion

section.

8.3.7.1 Commercial selection

Commercial selection is based on the total genetic effect for two potential market seg-

ments: hybrid entries and non-hybrid entries. Given a selection target of the top 10

entries (this is an arbitrary number and can vary with breeding program strategy) this

following section considers comparisons between entries within an entry segment

Considering only the hybrid entries, C-BLUPs for total genetic effect were plotted

for the two main clusters, that is excluding the singleton cluster at PTLI5, see Fig.

8.6. The vertical and horizontal lines indicate the top 10 entries that would be selected

for each cluster. Entries on the right hand side of the vertical line indicate the top

10 entries with high total genetic C-BLUPs in Cluster 1 and the entries above the

horizontal line indicate the top 10 entries with high total genetic C-BLUPs for Cluster

2. Of these top 10 entries for each cluster group 6 entries in the top right hand corner

had the highest total genetic C-BLUPs for both cluster groups. This indicates that

there are 4 entries that would be selected for one cluster and not the other. There

was also a large amount of GxE present, indicated by the lack agreement of C-BLUPs

between cluster groups.

127


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00.

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40.

6

Cluster 1

Clu

ster

2

Figure 8.6: Total genetic C-BLUPs for hybrid entries from Cluster 2 plottedagainst Cluster 1 - The axes units indicates yield in t/ha, horizontal and vertical linesindicate the top 10 performing entries.

128

8.3 Results

For the non-hybrid entries, C-BLUPs for total genetic effect were plotted for the

two main clusters, again excluding the singleton cluster at PTLI5, see Fig. 8.7. Only

2 entries were located in the top right hand corner indicating the highest C-BLUPs

across both cluster groups. Entries on the right hand side of the vertical line and above

the horizontal line indicate the top 10 entries with the highest C-BLUPs for Cluster

1 and Cluster 2 respectively. Considering the top 10 entries selected from this plot, 8

entries would be selected for one cluster and not the other.

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Cluster 1

Clu

ster

2

Figure 8.7: Total genetic C-BLUPs for non-hybrid entries from Cluster 2 plot-ted against Cluster 1 - The axes units indicates yield in t/ha, horizontal and verticallines indicate the top 10 performing entries.

129


8.3.7.2 Selection for parents

The selection of parents for the next cycle of breeding is based on the additive genetic

effect, also known as the breeding value. Only non-hybrid entries can be used for par-

ents for the next cycle of breeding, so additive genetic effects are only considered for

this subset of entries. Selection for parents should be limited to the trials where the

MET/FA model explains a high proportion of the additive genetic effects i.e. BUNT6,

CCRK2, ELMR3, MNGN6, WAGG2 and YORK6 (Table 8.10). However, in this anal-

ysis this is not a requirement as C-BLUPs were averaged across all the trials within a

cluster grouping.

Table 8.10: REML estimates of proportion of additive (%), non-additive (%) and totalgenetic variance from the FA(2) model. Note that total genetic variance is evaluated usingthe average inbreeding coefficient of all entries (a = 1.82).

Trial Additive Non-additive Total

ALBR2 46.61 53.39 0.0506BUNT6 80.72 19.28 0.0508CCRK2 73.53 26.47 0.0325ELMR3 69.49 30.51 0.1319LKBL3 38.13 61.87 0.0573

MNGN6 82.26 17.74 0.0589PTLI5 33.25 66.75 0.0038SSTL6 46.60 53.40 0.0826

WAGG2 64.84 35.16 0.0291YORK6 81.75 18.25 0.0562

C-BLUPs for additive genetic effect were plotted for the two main clusters, again

excluding the singleton cluster with PTLI5, see Fig. 8.8. Entries in the top right

hand corner indicate the top 2 entries that have the highest C-BLUPs for both cluster

groups. Entries on the right hand side of the vertical line indicate the top 10 entries

with higher C-BLUPs for Cluster 1 trials and entries above the horizontal line indicate

the top 10 entries with higher additive genetic C-BLUPs for Cluster 2. Considering the

top 10 entries selected from this plot, 8 entries would be selected for one cluster and

not the other. There was evidence of GxE indicated by the lack of agreement of entry

C-BLUPs between cluster groups.

130

8.3 Results

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Figure 8.8: Additive genetic C-BLUPs for non-hybrid entries from Cluster 2plotted against Cluster 1 - The axes units indicates yield in t/ha, horizontal and verticallines indicate the top 6 performing entries.

131


8.4 Discussion

MET/FA analysis is routinely used in plant breeding programs and entry evaluation for

commercial release and farmer recommendations (Kelly et al., 2007, Smith et al., 2005).

However what is not as common is the integration of pedigree information within this

framework of analysis. In the literature review on pedigree information (Chapter 5)

two reasons were forwarded for this, the first being the lack of worked examples and

the second being the complexity in the analysis. The aim in this chapter is to address

both these concerns by using a plant breeding data set to demonstrate the application

of such an analysis framework but also to provide a discussion on the interpretation of

results from this analysis, within the context of a commercial plant-breeding program.

One of the main aims of MET analysis is to obtain predictions of entry performance

across locations and thereby an estimate of the impact of GxE (Smith et al., 2001b).

MET/FA models are the preferred framework of analysis for such data, as it allows for

heterogeneity of GxE variance, and correlations among GxE interactions (Smith et al.,

2001b). In terms of errors, it is flexible in accounting for spatial variation at a trial

level and heterogeneity of error variance between environments (Smith et al., 2001b,

Stefanova and Buirchell, 2010). Overall, this enables flexibility in the modeling frame-

work for accounting for GxE, which is known to be a large factor in target environments

in Western Australian (Gilmour et al., 1996) as well as greater canola production zones

in Australian (Beeck et al., 2010, Cullis et al., 2010).

The MET/FA models provided a good fit for the data set, with the FA(2) accounting

for 79.88% of trial additive effects and 76.25% of trial non-additive effects. The additive

variance component across trials was high, averaging 61.72% of total genetic variance,

but ranged from 33.25% to 82.26%. Similar to the study of Beeck et al. (2010), it was

observed that the additive variance heterogeneity and proportion of additive genetic

variance is dependent on the environment the entries were grown in.

Pedigree information in the MET/FA mixed model framework is important as it

enables individual estimation of additive and non-additive genetic effects as demon-

strated by Oakey et al. (2006, 2007) thereby enabling the joint aims of selecting the

best entries for promotion/commercialization and the selection of parents for crossing

132

8.4 Discussion

in the next cycle of breeding. This not only increases the efficiency of the breeding

program, but also enables effective selection of parents for target environments due to

the analysis enabling identification of GxE for additive genetic effects.

An important component of this chapter, which differentiates this study from pre-

vious studies utilising this MET/FA framework with pedigree information, is the par-

titioning of total genetic variance for hybrid and non-hybrid entries at a trial. Previous

studies are based on total genetic variance obtained from the average of all the diagonal

elements of the relationship matrix (A) (Beeck et al., 2010, Crossa et al., 2006, Oakey

et al., 2006, 2007). This was relevant for the majority of these respective studies, given

that their data sets were composed of a single ‘type’ of entry, that is inbreds or hybrids,

rather than a combination of the two. For example Oakey et al. (2006) worked on fully

inbred wheat lines grown across southern Australia. The only exception to this was the

study by Beeck et al. (2010), who analyzed oil and yield traits for inbred canola entries

across southern Australia, with a data set consisting of 578 entries with only 55 hybrid

entries. However in this current CBWA data set this would not have been accurate

given the make up of the entry types that compose the breeding program, 51% were hy-

brids and 49% of non-hybrids. Such a data set is especially relevant as canola-breeding

programs increasingly market hybrids alongside open-pollinated entries.

Open pollinated entries (non-hybrids) are produced/bred from multiple generations

of selfing; hybrid entries on the other hand are produced from the cross of two inbred

parents. The inbreeding coefficients of these entries as a result are vastly different,

hybrids should have little or no inbreeding and non-hybrids (SSD and DH entries) have

close to the maximum inbreeding coefficient of 1. This was observed in our data set, with

the inbreeding coefficient estimated at 0.12 and 0.97 for hybrid and non-hybrid entries

respectively. In addition hybrid and non-hybrid entries within a breeding population do

not comprise one homogeneous population. As a result there is a need within MET/FA

framework to separate total genetic variance for hybrid and non-hybrid entries.

Of most importance in the differentiation of total genetic variance for hybrids and

non-hybrids is the study of GxE on these entry types. Previous studies have demon-

strated that heterosis may be effective in some environments and not in others due to

impact of GxE (Xu and Zhu, 1999) and that GxE differs for additive and non-additive

133


genetic components (Beeck et al., 2010, Cullis et al., 2010). In terms of selection, it

would be beneficial to determine the impact of environment on entry type performance.

The results of this chapter indicate that there was very little heterogeneity of total

genetic correlations between trials for the two entry types (Table 8.11). The cluster

groupings of the total genetic correlations for each entry type also resulted in the same

cluster groupings as those obtained across all entry types. Regardless, due to the

makeup of the motivational data set and the differing levels of inbreeding, entry type

should be differentiated for total genetic variance. Further, in other plant breeding

data sets, the novelty of splitting total genetic effects may enable the study of different

impacts of GxE for total genetic effects in hybrid and non-hybrid segments.

MET trials also enable an understanding for the nature and patterns of GxE in

target environments (Cooper and DeLacy, 1994). An understanding of GxE thereby

enables the exploiting of specific and general adaptation or identifies target environment

clusters (Bernardo, 2002). Cullis et al. (2010) developed and demonstrated tools for

exploring the impacts of GxE using heatmaps of the genetic correlation matrix, with

trial ordering within the heatmap subject to clustering. An association of environmental

factors such as rainfall, frost and drought as well as biotic factors such as blackleg

incidence can then be used to give insight into possible impacts of these on drivers

of GxE for additive, non-additive and total genetic cluster groupings. While cluster

groupings are not perfect, they enable a starting point for an interpretation of target

environment groupings (Cullis et al., 2010).

There appeared to be three clusters for additive genetic effects, with a majority

of trials in two main clusters, see Table 8.11. The trials ALBR2, ELMR3, WAGG2,

LKBL2 and SSTL6 were in cluster one and reflected trials that had higher annual

rainfall and longer growing season. This cluster consisted predominantly of Eastern

Australian trials, with the exception of SSTL6, which is in a high rainfall area of SW

Western Australia. Cluster two on the other hand referred to trials that had a lower

annual rainfall and shorter growing season. The majority of trials in this cluster were

W.A. wheatbelt trials with the exception of CCRK2. CCRK2 while a high rainfall trial

in northern NSW was impacted by frost in the middle of the growing season, which

could explain why it was clustered in the second cluster. While PTLI6 was classed as

134

8.4 Discussion

a high rainfall trial it was not within the first cluster, and instead formed a cluster on

its own. A possible explanation to this was that the trial was very low yielding, which

was the result of severe end of season drought and a delayed harvest which led to pod

shattering (D. Tabah pers.comm.).

The two main clusters of trials for additive genetic effects reflects the adaptation of

entries to either short or long growing season environments. Entry adaptation to these

particular environment clusters could be based on maturity type, as maturity genes are

known to be additive in nature (Brandle and McVetty, 1989). This fits with breeding

objectives as later flowering entries are targeted for high rainfall environments with

longer growing seasons and early maturity entries are often targeted for low rainfall

zones, which are characterized by drought and higher temperatures (Si et al., 2003).

Interestingly, the average yield for cluster one was 1.28 t/ha less than that of cluster

two at 1.69 t/ha. This could reflect the achievement of one of CBWA’s breeding objects

which is producing entries adapted to low rainfall environments, as the second cluster

corresponded with environments characterized by low rainfall. In contrast, it could

also mean that the current resistance of entries to blackleg is not as adequate in high

rainfall environment/sites conducive to blackleg disease. This ultimately limits the

yield potential as observed by the lower yield of cluster one trials. These two broad

clusters have implications for breeding program objectives. The two clusters represent

broad adaptation environments, which the breeder can then take advantage of for the

selection of parents on the basis of regional adaptation.

In terms of selecting parents (only non-hybrid entries), it is evident from the plot of

C-BLUPs for additive genetic effects (Fig. 8.8) that there are entries adapted to high

annual rainfall/long growing season (Cluster 1) or low annual rainfall/short growing

season (Cluster 2). This enables breeding/selection specifically for regional adaptation.

However, there are also 2 entries that have high C-BLUPs across both cluster groups,

which indicates overall adaption to the two environment groupings. In addition, the

C-BLUP plot also indicates the presence of GxE across cluster groups, indicated by the

lack of agreement of entry rankings between cluster groups (Fig. 8.8.)

For non-additive effects, the trials were all clustered within two main groups, see

Table 8.11. The first cluster included ALBR2, BUNT6, MNGN6, CCRK2 and YORK6,

135


and the second cluster consisted of the trials, ELMR3, WAGG2, LKBL3, PTLI5, and

SSTL6. For non-additive effects it appeared that cluster one consisted of trials that

did not have blackleg disease present and cluster two consisted of trials that had been

impacted by blackleg and in some cases severely (LKBL3, PTLI5, and SSTL6).

The basis of clustering for non-additive effects could be explained by the pres-

ence/absence of blackleg disease. Thus it implies the adaptation of some entries to

blackleg disease environments through non-heritable combinations of alleles. As this

clustering is based on non-additive effects it is reflective of resistance based on gene

complexes (inferring polygenic resistance) that has resulted from the chance crossing of

parents. Polygenic resistance is also known to be a variable mechanism strongly affected

by environmental conditions (Balesdent et al., 2001, Delourme et al., 2008, Fitt et al.,

2006). In terms of outcomes for the breeding program the clustering indicates that the

entries in the MET have two main environments which indicate different adaptation

of germplasm. However it poses a complexity, as such effects are non-heritable in the

progeny resulting from crosses of parents within the cluster.

The clustering observed for total genetic effects as well as total genetic effects dif-

ferentiated for hybrid entries and non-hybrid entries all indicated 3 possible clusters

(Table 8.11). The trials within these clusters were the same as those obtained for

additive genetic effects and interpreted similarly as well.

Entry selection for promotion/commercialization is based on two market segments,

hybrid and non-hybrid. Total genetic C-BLUPs were plotted for hybrid and non-

hybrid entries (Fig. 8.6 and Fig. 8.7) for the two main cluster groups interpreted

for total genetic effect in Table 8.11. As a result, selection of C-BLUPs for Cluster

1 corresponds to environments with high annual rainfall, longer growing seasons and

blackleg incidence, and selection of C-BLUPs for Cluster 2 corresponds to environments

with lower annual rainfall and shorter growing season with some trials affected by

drought or frost events. There is also the possibility of selecting for overall adaptation;

six hybrid entries in the top right hand corner of Fig. 8.6 or two non-hybrid entries in

the top right hand corner of Fig. 8.7. Both plots also indicate the presence of GxE by

the lack of agreement of entries between the two cluster groups.

136

8.4 Discussion

The target environments in this data set were highly representative of the canola

cropping zones within Australia, that is latitudes below 32◦ (Kirkegaard et al., 2011).

Hence it is a representative group of environments to test the magnitude of GxE inter-

actions. The clustering analysis of trials revealed that groupings were based on weather

(rainfall drought and frost) and biotic factors such as blackleg incidence. The clustering

analysis also indicated that within this breeding program there are two main types of

environments that are bred for and there is a clear adaptation pattern for entries to

these. The first environment includes dryland agricultural zones, predominantly W.A.

wheatbelt trials which are characterized by winter dominated rainfall and northern

sand plain agriculture and the second environment consists of trials characterized by

long season, equi-seasonal rainfall which is predominantly cropped on clay loamy soils

from the eastern states of Australia. Such an understanding of the grouping of target

environments could result in strategies to exploit such adaptations for GxE.

The MET/FA analysis from data in a single season did not indicate a large amount

of GxE. This is expected, as growing seasons tend to be more variable than trials

within a year for additive, non-additive and total genetic effects. Variability has been

previously reported as the result of seasonal conditions in Australia, with the paper

by Cullis et al. (2010) stating that in their data set, the common causes of GxE being

related to yearly change from sowing date, blackleg pressure, and rainfall distribution.

Previous plant breeding studies often use more than a single growing season data, for

example see Beeck et al. (2010), Cullis et al. (2010) which used 2 years worth of MET

data and higher order FA(k) models. As a result these papers had a greater structure

obtained from a 2-year analysis hence, much more GxE was seen. Data from a single

growing season were sufficient for the purpose of illustrating the method, and at the

same time highlighting the issue outlined in the next chapter, that is the computational

limitations from MET with pedigree analysis, resulting in extensive time to analysis

completion. The analysis with FA(1) took 568.9 seconds per iteration and the analysis

for FA(2) took 2288.7 seconds per iteration.

137


8.5 Summary

The objective of this chapter was to demonstrate the application of FA to the genetic

modeling of MET data sets, and to provide a discussion on the interpretation of the

results of such an analysis. This chapter has demonstrated the importance of envi-

ronment and pedigree information in improving the efficiency of selection in a plant

breeding program. The MET/FA approach in this chapter not only enables the estima-

tion of additive and non-additive genetic effect of entries, but also the impact of GxE

on these genetic effects. In this chapter this was extended by deriving total genetic

variance for hybrid and non-hybrid entries, to observe the impact of GxE on these

entry types. The clustering analysis resulting from MET/FA analysis did not indicate

large differences in GxE on trial groupings for hybrid and non-hybrid entry segments

in the CBWA motivational data set. However, the method outlined is a more accurate

selection tool given the differences in inbreeding levels between entry types. In other

plant breeding datasets that jointly trial hybrid and non-hybrid entries it may indicate

broad insights into the basis of possible sources of GxE on trial groupings.

138

8.5 Summary

Tab

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139


140

Chapter 9

Analysis completion times: MET

analysis with pedigree

information

9.1 Introduction

Plant breeding data sets are often large, comprising of replicated entry performance

trials across locations and years to sample a large population of target environments.

The basis of this is to observe the magnitude of GxE, which in Australian agriculture is

known to be highly variable. Chapter 8 demonstrated that even within a single growing

season there is GxE present, however across seasons Beeck et al. (2010) and Cullis et al.

(2010) demonstrate that GxE can be substantial. Thus in the annual MET analysis

for selection decisions it is important for plant breeding programs to include data from

as many relevant years and locations as possible.

Recent studies have utilized a mixed model framework with pedigree information to

estimate additive (and sometimes dominance) values in plant breeding METs (Oakey

et al., 2006, 2007). However their use in the routine analysis of plant breeding programs

is limited (Beeck et al., 2010). The reasons for this could be the limited access to

electronically stored pedigree information or the lack of published reports, which outline

the process and benefits of such an analysis. In addition to this it was found that

141

9. ANALYSIS COMPLETION TIMES: MET ANALYSIS WITHPEDIGREE INFORMATION

increased adoption of this analysis framework is impeded by the ever increasing data

set size, which is prohibitive on the time to analysis completion.

A literature search of recent studies using MET/FA analysis with pedigree informa-

tion, indicates data sets range in sizes. The study of Mathews et al. (2007) analyzed

a set of 106 environments, with 41 varieties, Crossa et al. (2006) analyzed two data

sets, the first had 47 varieties across 10 sites and the second 49 varieties and 15 sites.

Beeck et al. (2010) and Cullis et al. (2010) analyzed a 19 environment data set with

332 varieties. In contrast, current practice in CBWA’s breeding program is to use a

MET analysis with pedigree spanning three years of data. A three year MET is a

compromise for the annual analysis as a previous attempt of a 4 year MET comprising,

13 locations, 48 trials and 2624 entries with pedigree information took approximately 4

weeks (and longer) to run to completion (Dr. David Tabah pers.comm.). It is obvious

that there is a huge computational burden given the size of the data set. Such lengthy

analysis completion times is a hindrance to a breeding program, because it impedes the

efficiency of the program, as selection decisions could not be made in time for the next

seasons trial planning before the analysis is completed.

Variance parameter estimation in mixed models is via REML (Patterson and Thomp-

son, 1971). This is achieved in ASReml-R through a computing strategy termed the

average information (AI) algorithm (Gilmour et al., 1995). The mixed model approach

of Smith et al. (2001b) for MET data sets requires the estimation of large numbers of

variance parameters, associated with spatial models and error variance heterogeneity

for each trial and those for the FA model for GxE. The inclusion of pedigree information

results in an even more complex model (Kelly et al., 2009). This estimation of variance

parameters requires the inversion of large matrices and even with current sparse matrix

methods and the AI algorithm, the large size of plant breeding data sets and inclusion

of extensive pedigree information is limiting the speed of the analysis.

A previous study by Atkin et al. (2009) aimed to reduce this computational burden

by examining how computational time may be decreased by reducing the size of the data

set and pedigree information. However, this is not possible for the motivational data

obtained from CBWA, where the records from the newer generations (generations 2 to

7) have a larger number of records (i.e number of new parents) than later generations

142

9.2 Computation background

(generations 8 to 13) (see Table 6.8, Chapter 6). This appears to be common for self-

pollinated plant breeding data sets. Hence, the “trimming” of pedigree has no impact

on significantly reducing the data set size. Instead, this chapter aims to investigate the

algorithm used for analysis in ASReml-R and to quantify the actual times taken for

different variance models for a series of CBWA MET data sets.


In this first section of this chapter, the algorithm for fitting FA models in ASReml-R

is examined in detail. Three formulations of the model are investigated. The first

two were described by Thompson et al. (2003) as the “Independent formulation” and

the “Dependent formulation”, the third will be referred to as the reduced rank (RR)

version of the Dependent formulation.

9.2.1 Independent formulation

In this formulation as well as the following formulations, the spatial mixed model

approach of Smith et al. (2001b) is used. This approach assumes that entries are

independent, that is there are no relationships between entries in a trial. For the

independent formulation, the analysis for a series of t trials and m entries can be

written as,

y = Xτ +Zgug + e (9.1)

where y = (yT1 ,yT2 , ...,y

Tt )T , that is a concatenated vector of data of individual plots

combined across trials. X and Zg, are design matrices for fixed effects and random

genetic effects respectively. τ = (τT1 , τ

T2 , ...., τ

Tt )T is the p × 1 vector of fixed effects;

ug = (ugT1 ,ug

T2 , ....,ug

Tt )T is the mt × 1 vector of random genetic effects and e =

(eT1 , eT2 , ...., e

Tt )T is the vector of residuals ordered as per the data vector. Note that

peripheral effects have been omitted from this section.

143


Under the FA model of Smith et al. (2001b), entry effects for each environment are

written as:

ug = (Λ⊗ Im)f + δ

where Λ = [λ1....λt] is the (t × k) matrix of factor loadings (for k factors) at t trials;

Im is an identity matrix of order m; f = (fT1 ,f

T2 , ...,f

Tk )T is a mk × 1 vector of entry

scores and δ is an mt× 1 vector of residual genetic effects.

In terms of variance assumptions, f , δ and e are assumed to have a multivariate

normal distribution with a zero mean vector and variance matrix:Ik ⊗ Im 0 0

0 ψ ⊗ Im 0

0 0 R

where ψ = diag {ψj}, that is a (t× t) diagonal matrix, where the diagonals correspond

to the trial specific variances. Hence the variance matrix for entry effects in each

environment is,

var (ug) = Ge ⊗ Im = (ΛΛT +ψ)⊗ Im

The mixed model equations (MME) for the model in Equation 9.1 are,[XTR−1X XTR−1Zg

ZgTR−1X Zg

TR−1Zg +Ge−1 ⊗ Im

][τ

ug

]=

[XTR−1y

ZgTR−1y

](9.2)

For illustrative purposes in this section, the coefficient matrix in Equation 9.2 is split

into two parts; data and variance:

C =

[XTR−1X XTR−1Zg

ZgTR−1X Zg

TR−1Zg

]+

[0 0

0 (ΛΛT +ψ)−1 ⊗ Im

](9.3)

9.2.1.1 Toy example

To illustrate the independent formulation, a toy data set consisting of two entries,

replicated twice across four sites was created. This data set was then analyzed using

144


the mixed model of the form in Equation 9.1 with k = 1 factors. The model included a

fixed effect for each site (no overall mean was fitted) and random entry effects for each

site. For ease of illustration and interpretation the latter were ordered as sites within

entries and it was assumed that R = I.

The coefficient matrix, (C) of the MME was then derived using Equation 9.3, see

Fig. 9.1. The pattern of C was observed by replacing actual values with colour to

indicate cells which consisted of data (red), variance (blue) and data plus variance

estimates (purple). The labels of rows and columns in Fig. 9.1 are, “site1, .... , site4”

- indicating site fixed effects, “site1:entry1, .... , site4:entry1” - indicating the random

effects for entry 1 at each site and “site1:entry2, .... , site4:entry2” - indicating the

random effects for entry 2 at each site.

145


site4:entry2

site3:entry2

site2:entry2

site1:entry2

site4:entry1

site3:entry1

site2:entry1

site1:entry1

site4

site3

site2

site1

site1

site2

site3

site4

site1

:ent

ry1

site2

:ent

ry1

site3

:ent

ry1

site4

:ent

ry1

site1

:ent

ry2

site2

:ent

ry2

site3

:ent

ry2

site4

:ent

ry2

Figure 9.1: Toy example of the independent formulation - Indicates the patternof the coefficient matrix for the independent formulation of the mixed model in Equation9.1.

146


9.2.2 Dependent formulation

Thompson et al. (2003) provided a different formulation to reduce computational loads.

In their formulation of the model in Equation 9.1, they first consider the partitioning

of the vector ug as ug = (ug1,ug2) which leads to,[ug1

ug2

]=

[Λ1 ⊗ ImΛ2 ⊗ Im

]f +

[δ1

δ2

](9.4)

where the elements of δ1 are all non-zero, but all the elements of δ2 are zero. Λ1 and

Λ2 are t1 × k and t2 × k matrices of loadings partitioned within ug. Also note that

t1 + t2 = t. This main model gives rise to two simplified forms, which enables some or

all of the specific variances to be zero:

1. t1 = t and t2 = 0, which assumes that all specific variance are non-zero

2. t1 = 0 and t2 = t, which assumes that all specific variance are zero

The model that Thompson et al. (2003) consider for a series of t trials and m entries

is of the form,

y = Xτ +Zcuc + e (9.5)

where y, τ andX are as previously stated; Zc = [Zf2, Zg1], whereZf2 = Zg2(Λ⊗Im);

uc = (f , ug1). The dependent formulation considered in this section, is the case where

there are no zero ψ, which results in: uc = (f , ug) and Zc = [0, Zg].

The variance assumptions are as previously stated, so that

var (uc) = Gc ⊗ Im =

[Ik ΛT

Λ ΛΛT +ψ

]⊗ Im

The MME for the model in Equation 9.5 are,[XTR−1X XTR−1Zc

ZcTR−1X Zc

TR−1Zc +Gc−1 ⊗ Im

][τ

uc

]=

[XTR−1y

ZcTR−1y

](9.6)

147


where Gc−1 = [

Ik + ΛTψ−1Λ −ΛTψ−1

−ψ−1Λ ψ−1

]

Similar to the independent formulation the coefficient matrix in Equation 9.6 is split

into two parts, data and variance: XTR−1X 0 XTR−1Zg

0 0 0

ZgTR−1X 0 Zg

TR−1Zg

+ 0 0 0

0 (Ik + ΛTψ−1Λ)⊗ Im −ΛTψ−1 ⊗ Im0 −ψ−1Λ⊗ Im ψ−1 ⊗ Im

9.2.2.1 Toy Example

The same toy data set described in the previous section, was analysed using the de-

pendent formulation in Equation 9.5. The model included a fixed effect for each site

(no overall mean), random entry effects for each site and a random factor score for

each entry. The coefficient matrix, C of the MME was then derived using Equation

9.6, see Fig. 9.2. Similarly to the previous section, the pattern of C was observed by

replacing actual values with colour. The labels of rows and columns in Fig. 9.2 are

“site1, .... , site4” - indicating site fixed effects, “fac:entry1” - random factor score

for entry 1,“site1:entry1, .... , site4:entry1” - indicating the random effects for entry

1 at each site, “fac:entry2” - random factor score for entry 2, and “site1:entry2, .... ,

site4:entry2” - indicating the random effects for entry 2 at each site.

148


site4:entry2

site3:entry2

site2:entry2

site1:entry2

fac:entry2

site4:entry1

site3:entry1

site2:entry1

site1:entry1

fac:entry1

site4

site3

site2

site1

site1

site2

site3

site4

fac:e

ntry

1

site1

:ent

ry1

site2

:ent

ry1

site3

:ent

ry1

site4

:ent

ry1

fac:e

ntry

2

site1

:ent

ry2

site2

:ent

ry2

site3

:ent

ry2

site4

:ent

ry2

Figure 9.2: Toy example of dependent formulation - Indicates the pattern of thecoefficient matrix for the dependent formulation of the model in Equation 9.5.

149


9.2.3 Reduced rank version - dependent formulation

The fully reduced rank (RR) formulation of Thompson et al. (2003) with an additional

explicit term to accommodate specific variances is considered. This will be referred to

as the RR+diag model. The RR+diag model is written as,

y = Xτ +Zcuc +Zgδ + e (9.7)

where uc = f and Zc = Zf = Zg(Λ ⊗ Im). Note that this is the case of t1 = 0 and

t2 = t in Section 9.2.2.

The MME for the model in Equation 9.7 are, XTR−1X XTR−1Zf XTR−1Zg

ZfTR−1X Zf

TR−1Zf + Ik ⊗ Im ZfTR−1Zg

ZgTR−1X Zg

TR−1Zf ZgTR−1Zg +ψ−1 ⊗ Im

τfδ

=

XTR−1y

ZfTR−1y

ZgTR−1y

(9.8)

Similarly to the independent formulation the coefficient matrix in Equation 9.8 is split

into two parts, data and variance: XTR−1X XTR−1Zf XTR−1Zg

ZfTR−1X Zf

TR−1Zf ZfTR−1Zg

ZgTR−1X Zg

TR−1Zf ZgTR−1Zg

+ 0 0 0

0 Ik ⊗ Im 0

0 0 ψ−1 ⊗ Im

9.2.3.1 Toy Example

The same toy data set, and process was utilized for the RR+diag version of the de-

pendent formulation in Equation 9.7. The model included a fixed effect for each site

(no overall mean), a random factor score for each entry and random entry effect for

each site. Note that in contrast to Sections 9.2.2.1 and 9.2.1.1 the latter represent the

150


“residual” entry effects at each site, ie. δ rather than the total entry effect ie. ug.

The coefficient matrix, C of the MME was then derived using Equation 9.8, see Fig.

9.3. The pattern of C was observed by replacing actual values with colour. The labels

of rows and columns in Fig. 9.3 are “site1, .... , site4” - indicating site fixed effects,

“fac:entry1” - random factor score for entry 1,“site1:entry1, .... , site4:entry1” - indi-

cating the random effects for entry 1 at each site, “fac:entry2” - random factor score for

entry 2 and “site1:entry2, .... , site4:entry2” - indicating the random effects for entry 2

at each site.

151


site4:entry2

site3:entry2

site2:entry2

site1:entry2

fac:entry2

site4:entry1

site3:entry1

site2:entry1

site1:entry1

fac:entry1

site4

site3

site2

site1

site1

site2

site3

site4

fac:e

ntry

1

site1

:ent

ry1

site2

:ent

ry1

site3

:ent

ry1

site4

:ent

ry1

fac:e

ntry

2

site1

:ent

ry2

site2

:ent

ry2

site3

:ent

ry2

site4

:ent

ry2

Figure 9.3: Toy example of RR version of dependent formulation - Indicates thesparsity pattern of the coefficient matrix for the RR+diag formulation of the mixed modelin Equation 9.7.

152


9.2.4 Absorption

Solving the MME requires the inversion of the coefficient matrix (C). This is achieved

using the process of absorption (or Gaussian elimination) and back-substitution. In

this section the process of absorption is detailed.

After the ordering of the coefficient matrix, C, ASReml-R undertakes absorption

sequentially, that is a single row at a time beginning from the bottom line, and con-

tinuing upwards. Given a set of MME, the coefficient matrix can be written and then

subsequently partitioned as:

C =

C11 C12 C13 · · · C1,(N−1) C1N

C21 C21 C23 · · · C2,(N−1)

......

......

... · · ·...

C(N−1),1

......

.... . . C(N−1),N

CN1 · · · · · · · · · · · · CNN

=

[C11 c1N

cT1N cNN

]

where N is the order of the coefficient matrix C. C11 is the top portion of C, a matrix

of dimensions (N − 1) × (N − 1); c1N and cT1N are the far right column and bottom

row of the matrix C respectively. c1N is a vector of length (N − 1). cNN is a scalar,

and is often referred to as the pivot (Gilmour et al., 1995).

Given C above, the process of absorption involves forming the updated matrix,

C? = C11 − c1NcT1N/cNN (9.9)

where C? is an (N − 1) × (N − 1) matrix. The absorption process is then applied to

C? to form an updated matrix of dimension (N − 2)× (N − 2) and so on.

9.2.5 Sparsity and ordering

The coefficient matrix, C of the MME is often sparse, that is, it contains many zero-

valued elements. The computational burden of the absorption process can be reduced

if this is taken into consideration (Gilmour et al., 1995, Thompson, 2009). Specifically,

153


computing time is reduced by eliminating operations on the zero elements. Since ab-

sorption is a sequential process, it is not only the sparsity of C but also the updated

matrices (of the form in Equation 9.9) that is important. The latter is often influ-

enced by the ordering of the equations. A good ordering is one that maintains sparsity,

that is, minimises fill-in during the absorption process (Meyer, 1989, Thompson, 2009).

Gilmour et al. (1995) summarises the point in their statement “Assuming we avoid

multiplication by zero and that cT1N (Equation 9.9) has ni non-zero values, the number

of multiplications is (ni +1)(ni +2)/2. Thus operations can be avoided by ordering the

equations to minimize ni at each stage” (pg 1449).

In terms of the fitting of FA models, the ordering of the equations in the MME

corresponding to the genetic effects is a key determinant of computing time. This is

most easily seen by considering the toy example. The sparsity pattern of C, for the

dependent formulation (Fig. 9.2) is such that the section corresponding to the genetic

effects (10 × 10 partition in the bottom right hand corner) is relatively sparse. Now

consider a single absorption using two different orderings:

• The C matrix is ordered as in Fig. 9.2

• The C matrix is ordered as above but fac:entry2 is moved to the last row and

column

For both these scenarios the sparsity pattern of the updated matrix (C?) after

absorption is obtained. In the first ordering scenario the absorption of site4:entry2

resulted in a sparse updated matrix (Fig. 9.4). However in the second scenario, the

absorption of a factor score, ie. fac:entry2 resulted in substantial fill-in (Fig. 9.5). As

a result, there will be a greater number of computations to be carried out for scenario

two, which would be computationally intensive in comparison to scenario one.

The sparsity pattern of C for the independent formulation (Fig. 9.1) is such that

the section corresponding to the genetic effects (8 × 8 partition in the bottom right

hand corner) contains two dense sub-blocks. Thus C itself is less sparse than in the

dependent formulation and no re-ordering can improve the absorption process. Note

that the independent formulation was the original formulation in ASReml-R. It was

replaced by the dependent (also called the “sparse”) formulation of Thompson et al.

(2003) after this was shown to result in substantial time savings. This is the current

154


site3:entry2

site2:entry2

site1:entry2

fac:entry2

site4:entry1

site3:entry1

site2:entry1

site1:entry1

fac:entry1

site4

site3

site2

site1

site1

site2

site3

site4

fac:e

ntry

1

site1

:ent

ry1

site2

:ent

ry1

site3

:ent

ry1

site4

:ent

ry1

fac:e

ntry

2

site1

:ent

ry2

site2

:ent

ry2

site3

:ent

ry2

Figure 9.4: Sparsity after absorption in a toy example of the dependent for-mulation with correct ordering - The updated coefficient matrix C?, indicating thefill in pattern resulting from the correct ordering of C. Black shading indicates non-zerocells and white indicates zero cells.

155


site4:entry2

site3:entry2

site2:entry2

site1:entry2

site4:entry1

site3:entry1

site2:entry1

site1:entry1

fac:entry1

site4

site3

site2

site1

site1

site2

site3

site4

fac:e

ntry

1

site1

:ent

ry1

site2

:ent

ry1

site3

:ent

ry1

site4

:ent

ry1

site1

:ent

ry2

site2

:ent

ry2

site3

:ent

ry2

site4

:ent

ry2

Figure 9.5: Sparsity after absorption in a toy example of the dependent formu-lation with incorrect ordering - The updated coefficient matrix C?, indicating the fillin pattern resulting from incorrect ordering of C. Black shading indicates non-zero cellsand white indicates zero cells.

156


formulation implemented in ASRreml-R, and has been reported as resulting in a savings

of 50% of computational time when t = 17 and k = 3 and up to 90% when t = 62 and

k = 3 (Thompson et al., 2003, Thompson, 2009). A summary of the iteration times for

two data sets analyzed under independent and dependent formulations by Thompson

et al. (2003) were reproduced in Table 9.1.

Table 9.1: Time taken (in seconds) for completion of an iteration for independent anddependent formulations. This table has been reproduced from (Thompson et al., 2003)

AlgorithmDataset Model Dependent Independent (Smith et al., 2001b)

Lupins, p = 17 FA(1) 1.2 5.1Lupins, p = 17 FA(2) 2.2 5.5Lupins, p = 17 FA(3) 2.6 5.9Barley, p = 62 FA(1) 30 786Barley, p = 62 FA(2) 50 833Barley, p = 62 FA(3) 101 940

As discussed above it is vital with the dependent formulation that all site:entry

effects are absorbed prior to the factor scores. This means that the MME must be

re-ordered to allow this. Thompson et al. (2003) suggested that “A simple algorithm

would be to (i) count the non-zero elements in each row, (ii) absorb the row with

the least number of non-zero elements and update C, then repeat the process on the

updated C matrix” (pg 402). In terms of the genetic effects in the toy example (Fig.

9.4) this would result in the absorption of the rows associated with the site:entry effects

(each of the associated rows has only 3 non-zero elements) prior to the factor scores

(each of the associated rows has 5 non-zero elements). However, in real examples,

the non-genetic models are much more complex so that the ordering may not be as

clear-cut. The problem is exacerbated with the inclusion of pedigree information which

creates a far more dense set of equations for the site:entry effects. With these models it

is important to not only consider absorption of site:entry effects prior to factor scores

but also to order so that offspring are absorbed prior to parents. It is difficult to find

an ordering algorithm that can deal with such complexities and it is hypothesized that

currently in ASReml-R the inclusion of pedigree information in a MET analysis leads

to an inefficient ordering, resulting in factor scores being absorbed prior to site:entry

effects.

In an attempt to overcome this, the third formulation of the FA model, namely the

157


RR+diag formulation is considered. The sparsity pattern ofC for the toy example (Fig.

9.3) is slightly more dense than for the dependent formulation (Fig. 9.2). However in

this formulation the factor scores are “loaded” with data (i.e. there are more red cells

and no blue cells in Fig. 9.3 compared with Fig. 9.2) with the end result that the

number of non-zero elements in the rows corresponding to factor scores far exceeds

that for the rows corresponding to site:entry effects (9 compared with 3). Thus for

more complex models (including METs with pedigree information) it is, arguably, more

likely that the factor scores will be absorbed after the site:entry effects and hence the

computational efficiencies associated with sparsity will be exploited. This hypothesis

is examined empirically in the next section.

9.3 Example: Analysis completion times

9.3.1 The data set

In this section, the time required for the completion of a MET/FA analysis with and

without pedigree information is quantified. The complete CBWA canola MET data

set as described in Chapter 6 was used for the analyses conducted. From the CBWA

data set, four sub data sets of varying length (years) were created with corresponding

pedigree files, see summary in Table 9.2. Pedigree files were trimmed to contain only

entries (and parents of entries) present in the corresponding data file using the package

“Pedicure” (Butler, 2012) in R (R Development Core Team, 2012). Data subsets were

based on one to four years, and included the most recent growing season, 2011.

Table 9.2: Summary information on CBWA data subsets analyzed.

Data subsets Years Trials Number of records Entries Mums Dads

2011 1 10 5940 1084 235 3132011-2010 2 22 12444 1952 274 5662011-2009 3 35 17832 2533 302 7792011-2008 4 47 20928 2624 307 784

158


9.3.2 Analysis

The spatial model for each trial was determined using the techniques outlined in Chap-

ter 7. These spatial models were retained for all subsequent analyses. Three stages

of models were fitted, the first assumed independence of entries, the second fitted ad-

ditive genetic effects only and the third fitted both additive and non-additive genetic

effects. For each stage of models, different forms of the genetic variance matrix were

considered, beginning with the diagonal model, followed by a factor analytic model

with k = 1 factors, fitted using the formulation described in Section 9.2.2 (denoted

FA(1)) and then the formulation described in Section 9.2.3 (denoted RR(1)+diag). In

doing so a step wise procedure was utilized to understand the cause of the extensive

time to completion observed for mixed model analysis with pedigree information.

9.3.3 Computation

Each of the following models were run in ASReml-R 3.0− 1 (library 3.0hj) on a Apple

Macintosh core 2.66GHz Intel Core i7 processor with 8GB RAM. The workspace was

set at a standard of 100e7 (or 800, 000, 000 bytes) for each of the models fitted. The

time (in seconds) required to complete the second iteration was obtained from the

difference between the times required for the second iteration and first iteration. The

time taken to complete the second iteration was used as an accurate representation for

time to analysis, as it excludes the initial setting up of design matrices that correspond

to the time taken for the completion of the first iteration.

9.3.4 Results & Discussion

The number of trials in the data sets increased from 10 in the one year data set to

47 in the four year data set (Table 9.3). There is almost a linear increase in the

number of entries for these respective data sets, increasing from 5940 to 20928. This

is approximately an increase of 3.5 times. There was a small increase in comparison

for the number of mums and dads in the pedigree files, between the one and four year

data sets (Table 9.3).

159


Table 9.3: Sequence of models fitted for genetic variance structures.

Model number Model summary Additive Non-additive Total

1 ND none none diag2 NF none none FA(1)3 NR none none RR(1) + diag

4 DN diag none5 FN FA(1) none6 RN RR(1) + diag none7 DD diag diag8 FF FA(1) FA(1)9 RR RR(1) + diag RR(1) + diag

Model acronyms: diag = diagonal model, FA(1) = factor analytic model of order 1 fitted using the

formulation in Section 9.2.2 and RR(1)+diag = factor analytic model of order 1 fitted using the

formulation in Section 9.2.3

Considering the models without pedigree information (ND, NF and NR in Fig. 9.6),

these all took less than 1 minute for the completion of the second iteration for all data

sets. The inclusion of pedigree information and the modeling of additive genetic effects

(i.e models DN, FN, RN) resulted in substantial differences between models in terms

of the time taken for the completion of the second iteration. The time required for the

FN model ranged from 125.9 seconds for the one year data set, to 1522.1 seconds for

the four year data set. In comparison to RN models, the FN models were 4.9, 7.7, 10.3

and 12.9 times larger for the one to four year data sets respectively. For the models

that included additive and non-additive genetic effects (models DD, FF, RR) there

was a large increase in completion of second iteration times for FF and RR models.

Comparing the FN and FF models, there was an increase in magnitude of computation

of 4.8, 3.3, 2.6, 2.1 for the one to four year data sets. In comparison to the RR models,

the FF model was 2.4, 2.9, 3.1 and 3.3 times slower for the one to four year data sets.

160


Genetic Model

Tim

e fo

r se

cond

iter

atio

n (s

econ

ds)

010

020

030

040

050

060

0

ND NF NR DN FN RN DD FF RR

1

050

010

0015

00


2

050

010

0015

0020

0025

00


3

010

0020

0030

00


4

Figure 9.6: Second iteration completion times - The time (in seconds) taken forsecond iteration completion for different genetic models (see acronyms in Table 9.3) andvarying length of data sets, 1 to 4 years indicated by the panel title.

161


It is evident that the inclusion of pedigree information in MET/FA analysis increases

the length of computation times for both FA(1) and RR(1)+diag models and this

increases with data set/pedigree length. However, the results indicate that the re-

parameterized RR(1)+diag model takes a third of the time as the FA(1) to complete

an iteration when pedigree information is included. This could be explained for the

toy example in Section 9.2.5, as the RR(1)+diag formulation loads up the factor rows

in the coefficient matrix, C, which prevents the absorption of the factor first and thus

avoids the resulting fill in effect.

9.3.5 Summary

During the course of this thesis, it was observed that the time for completion of

MET/FA models with pedigree information can be costly. This was investigated in this

chapter by firstly examining the possible model formulations available in ASRreml-R

followed by quantifying the actual times required for the completion of this analysis for

a real plant breeding data set. It is possible that long computational times, could be

attributed to issues with ordering when pedigree information is included in the model.

The formulation proposed in Section 9.2.3 appears to offer a quicker alternative in terms

of computational time.

162

Chapter 10

General Discussion

10.1 Introduction

The aim of a plant breeding program is to produce new varieties that are superior to

those already in the market, in terms of traits of economic importance such as yield

and quality etc. As a result, the process of plant breeding involves the manipulation

of complex traits in unpredictable environments (Hammer et al., 2006). Testing plant

breeding entries for many traits across a range of environments is costly, and the results

can be problematic if there are large errors or entries rank differently across environ-

ments; hence, the success of a plant-breeding program is linked to the efficiency of

selection methods. The topics of correlated traits, ancestry and environments were

researched in this thesis with the aim of demonstrating how the utilization of this ad-

ditional information can be used to improve the efficiency of selection within a plant

breeding program. This chapter brings together the main aims and findings across

all the chapters and provides a discussion on how the results from such analysis can

be interpreted and utilized with respect to a breeding program aims/objectives and

outcomes.

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10. GENERAL DISCUSSION

10.2 Correlated traits

The first half of the thesis focused on correlated traits. The motivation behind this

study was the fact that while selection is usually undertaken on several traits within

a breeding program, plant breeding programs rarely use multivariate methods or an

index of selection, which is commonplace in animal breeding programs (Comstock et al.,

1996, Piepho et al., 2008).

Using a plant survival data set for blackleg disease of canola, a bivariate linear mixed

model approach was proposed in which the two variables are the initial and final plant

survival counts. The literature review in Chapter 2 discussed how such counts can be

subject to different biological, environmental and genetic factors, and showed how the

bivariate framework is statistically more accurate in accommodating this. The value of

the bivariate method (Chapter 3) was the modeling of spatial variation for each trait,

trait based outlier detection, and estimation of correlations between genetic effects and

errors between the two traits. This method clearly demonstrated that each set of plant

counts should be treated as separate traits as is the case under a bivariate analysis.

In terms of efficiency of selection, the bivariate approach not only provided a more

detailed picture for selection for disease resistance but also a more accurate assessment

of the impact of disease resistance compared with the historical analysis of percentage

survival data. The use of correlated traits in a bivariate framework of analysis enabled

a two dimensional view of selection, and three sources of information for selection

namely emergence counts, maturity counts and percentage survival values. Thus under

a bivariate analysis it was possible to obtain insights into the outcome of the plant-

pathogen interaction from the beginning to the end of the season, that would otherwise

not be observed under the historical analysis.

Efficiency of selection for the bivariate approach was evaluated by determining the

improvements in accuracy of prediction afforded over the univariate approaches for the

plant survival data. Modest improvements were achieved in prediction accuracies in

the bivariate model for the (log) emergence and (log) maturity traits and also for the

difference (log maturity − log emergence). Further it was also demonstrated that if

selection was preferred on the “difference” trait (log maturity− log emergence), this still

164

10.2 Correlated traits

should be done with reference to emergence. Thus the bivariate analysis was preferred

over the univariate analyses for the traits emergence, maturity and the difference.

The bivariate analysis allowed for a correlation between entry effects between emer-

gence and maturity to be estimated. This enabled an insight into plant/pathogen

relationships on particularly how blackleg disease could impact on entry emergence,

which was ignored under the historical univariate analysis. Studies on blackleg disease

have long reported that the type of genetic resistance against blackleg disease infection

is based on the stage of plant growth (Ballinger and Salisbury, 1996, Light et al., 2011,

Rempel and Hall, 1996, Roy, 1984). This would have an impact on entry selection,

as selection for resistance against blackleg disease could be undertaken based on plant

survival counts at emergence and/or maturity plant stages of growth.

While plant breeding experiments are multivariate in nature, there are few stud-

ies addressing issues of multivariate analysis in plant breeding. Chapter 4 addresses

a subset of studies on multivariate selection in plant breeding trials based on covari-

ance analysis, which is the most common approach in the plant breeding literature to

analyze for a single trait while “adjusting” for the presence of another trait. This is

effectively the historical approach to analysis of blackleg plant survival values (Chapter

3). Chapter 4 also discussed the potential application of a bivariate approach to other

plant selection experiments, which preferentially use covariance analysis to adjust for

one trait in the presence of another. These experiments included disease resistance

and grain yield and a subset of QTL studies in the areas of disease resistance, grain

yield and protein content. The covariance approach was contrasted with the bivari-

ate approach and it was discussed why the bivariate method would be preferred over

covariance analysis methods.

Covariance analysis has several problems, as highlighted in the subset of studies

reviewed in Chapter 4: (i) the covariate is often a trait that is also of interest for

selection, which implies that it has its own genetic variance (ii) covariance adjustment

to a common level, generates unrealistic varieties that do not exist in the experiment or

the breeding population, and (iii) the assumption that the covariate should be measured

without any error, is nearly impossible in plant breeding experiments. For points (i)

and (ii), above, the bivariate analysis enables a two-dimensional view for selection

165


based on the two traits, without the need for an adjustment. Thereby it is a more

flexible framework of analysis as both traits can be incorporated into the selection

process. For point (iii), however, the bivariate framework enables the modeling of

spatial error for each trait, which is effectively ignored under covariance analysis. In

terms of efficiency, the bivariate framework should improve the efficiency of selection

in plant breeding programs, as there is no need to adjust one trait for another. While

multivariate analysis is common animal breeding (Falconer, 1981), the application of

multivariate analysis to plant breeding is complicated by the nature of plant breeding

trials. Bivariate analysis of field trials, as described in Chapter 3, is a novel method for

the achievement for multiple trait selection objectives in plant breeding with widespread

applications in the areas of disease resistance, grain yield, and protein content, as well

as QTL studies Chapter 4.

10.3 Ancestry & Environments

The second half of the thesis focuses on the impact of environment on selection and

the combined use of ancestry and environments to improve efficiency of selection.

The literature review presented in Chapter 5 found that the inclusion of pedigree

information in plant breeding METs resulted in major improvements in BLUP-based

prediction methods from mixed model analysis. Yet, few studies in the current scientific

literature are based on applied plant breeding programs (the exceptions are (Beeck

et al., 2010) and (Cullis et al., 2010)). This could be the result of complexity in the

fitting of these models, issues with the interpretation of results within the context of

plant breeding objectives and also limited worked examples. The processes described

in Chapters 7 (spatial analysis (N-gen modeling) of trials with pedigree information)

and 8 (MET/FA analysis), addressed these issues. The motivating data set for these

chapters was obtained from Canola Breeders Western Australia Pty Ltd (see description

in Chapter 6), a breeding program which utilize METs and mixed model analysis with

pedigree information in their canola breeding program.

Of the subset of studies reviewed in Chapter 5 that included pedigree information

in mixed model analysis, the studies by Crossa et al. (2006) and Burgueno et al. (2007)

166


omitted the use of spatial models. Chapter 7 as result, focused on demonstrating why

spatial modeling of non-genetic variance is important, especially in terms of efficiency

within a plant breeding context. Chapter 7 illustrated on an individual sites basis

the spatial modeling process and demonstrated the importance of pedigree information

in the spatial modeling of trials. With respect to (p-rep) trials, the results indicated

that the inclusion of pedigree information in spatial analysis is especially important.

Pedigree relationships in plant breeding trials can be used to borrow information from

relatives, which aids in the explanation of entry performance. The results of Chapter 7

also indicated that base-line non-genetic modelling should always include pedigree infor-

mation for the determination of site-specific spatial models, as there may be differences

between spatial models and outliers identified as a result of pedigree information.

There is a high level of adoption of p-rep trial designs, especially for the testing of

early generation entries (Cullis et al., 2006). Such trials attempt to provide accurate

selection of entries, with limited replication, for commercialization and parents for

the next cycle of breeding. As a result, it is important to minimize error, such as

environmental heterogeneity. Environmental effects are common in all designed field

trials, and if not accounted for can lead to biased estimates of treatment effects (Basford

and Cooper, 1998). Hence spatial analysis (N-gen modeling) should be undertaken

as standard for plant breeding trial analysis. However, published research has not

addressed the benefits of including pedigree information in finding optimum spatial

models in p-rep trials. The estimation of genetic merit in breeding trials is critical

for the efficiency of plant breeding programs, and spatial mixed model analysis with

pedigree information provides an improvement in efficiency.

Chapter 8 completed this process of model fitting by demonstrating the MET/FA

genetic modeling of the trials in Chapter 7 as well as providing an interpretation of the

results.

A key motivation for the use of a MET/FA model with pedigree information is

the independent estimation of additive and non-additive genetic effects as well as an

estimate of GxE on these components. This study extended this area of research by

considering different summaries of total genetic variance based on inbreeding coefficients

of entry types. Total genetic variance was summarized in three ways: for the entire

167


data set, hybrid entries only and non-hybrid entries only. Previous studies are based

on total genetic variance evaluated from the average of all the diagonal elements of the

relationship matrix (Beeck et al., 2010, Oakey et al., 2006, 2007, Crossa et al., 2006).

This was relevant for the majority of these studies, given that their data sets were

composed of a single ‘type’ of entry; that is either hybrid or non-hybrid, rather than a

combination of the two. The motivational data set from CBWA however, represents a

current trend in plant breeding programs producing entries for both hybrid and open

pollinated market segments - our data set indicates a 50:50 split of entries for these

market segments.

MET/FA analysis provides an estimate of the presence and magnitude of GxE.

The findings of Chapter 8 indicate that within a single year (growing season) there is

substantial GxE. GxE was shown to be a large factor for Crop Variety Testing trials

in Western Australian (Gilmour et al., 1996) and for canola across production zones

in southern Australia (Beeck et al., 2010, Cullis et al., 2010). While cross over GxE

has the potential to complicate varietal selection (Chapters 5 and 8), an understand-

ing of genotypic adaptation to heterogeneous environments (Fukai and Cooper, 1995)

(Chapter 5) would enable broad or specific adaption patterns to be exploited for selec-

tion. Hence the interpretation of entry performance across environments is critical in a

plant breeding context as it maintains genetic gain in selection and this translates into

breeding program efficiency.

The results of cluster analysis for additive, non-additive and total genetic corre-

lations between trials indicated strong association of GxE with environmental factors

such as rainfall, drought, and frost as well as the biotic factor blackleg disease incidence.

While the target environments in CBAs breeding program are diverse (Fig. 6.1), the

clustering analysis suggested two main types of environments and different adaptation

of entries to these environments. The first cluster group of environments comprised

dryland agricultural zones, predominantly Western Australian wheatbelt sites, which

were characterized by short growing seasons with winter dominated rainfall and sandy

soils, and the second cluster group consisted of long season, equi-seasonal rainfall on

clay loamy soils in the eastern states of Australia.

Of most importance in this chapter is the differentiation of total genetic variance for

168


hybrids as well as non-hybrids, and the study of the impact of GxE on these entry types.

The resulting clustering analysis for hybrid and non-hybrid entry types indicated similar

patterns of GxE to that of total genetic variance across all entry types. The clustering

for hybrid and non-hybrid entries indicated adaptation patterns associated with annual

rainfall and length of growing season. Regardless, this differentiation enabled an insight

to the impacts of GxE on the different entry segments. For example, the C-BLUPs of

hybrid entries indicated disagreement between the two main cluster groups (Fig. 8.6)

which may indicate that heterosis is effective in some environments and not others (Xu

and Zhu, 1999). This would not have been otherwise observed by taking the average

of the inbreeding coefficients of all entries within the data set. Hence it highlights

the impact of environment on entry performance for these two entry types and would

impact on selection decisions. This highlights the fact that environment may influence

variety performance differentially for hybrids and non-hybrids, which (if not recognised)

may have a significant effect on selection decisions.

In terms of selection decisions, this chapter also demonstrated how predictions can

be used alongside the interpretation of cluster groups (Table 8.11). For additive genetic

effects (non-hybrid entries only) and total genetic effects (hybrid and non-hybrid en-

tries) it was possible to then identify entries with regional adaptation, that is to either

a particular cluster group, or even overall adaptation to both cluster groups. Efficiency

in selection would result from the ability to tailor selection to target environments en-

countered within the breeding program. This chapter in particular demonstrates how

genetic gain can be maintained and even improved in terms of selection by account-

ing for GxE. GxE would otherwise complicate selection and even mask genetic gain

impacting negatively on selection efficiency.

It is interesting to note that the cluster analysis for non-additive effects could be

explained by the presence or absence of blackleg disease at the trial sites. The challenge

to plant breeders is to exploit these non-heritable effects, as some may be due to epistasis

which can be fixed in inbred lines through crossing and recombination.

During the study of the MET/FA framework with pedigree information in Chapter

8, it was found that the time to complete these analyses was a potential limitation to

practical use in a commercial breeding program. As a result Chapter 9 researched the

169


causes of these extensive times for analysis completion, namely the algorithm used for

analysis in ASReml-R and quantifying the actual times taken for analysis of data set(s)

with different genetic models.

The results from this chapter indicate that analysis times did increase substantially

for FA models when pedigree information was included. However, the RR+diag for-

mulation of the FA model reduced second iteration completion times to one third of

the time than that under the standard formulation of the FA model. The results also

indicate that with the inclusion of pedigree information, complications in the ordering

of equations in the coefficient matrix (C) could be the cause of the long computational

times. Given that as the size of plant breeding datasets increases, and molecular marker

data are integrated within mixed model analyses, these timing problems will only be

exacerbated, hence further research is needed in this area of mixed model computation.

10.4 Future directions of research: correlated traits, an-

cestry and environments

It is demonstrated in this thesis that the analysis of two correlated traits should be

managed within a bivariate framework rather than through covariance analysis. Further

research in the area should aim to extend the bivariate method to a MET analysis from

the single trial analysis in Chapter 3. Substantial benefits would arise from a MET

analysis of the blackleg disease resistance ratings in Chapter 3, as GxE for resistance

is likely to be identified and should be used in decisions in plant breeding and variety

recommendation.

Pedigree information provides an estimate of genetic relationship based on Mendelian

expectations of relatives in the pedigree. Another option is to estimate relationship

through molecular marker information (Chapter 5). However the results in this the-

sis indicate that the use of pedigree data enables gains in the efficiency of selection.

Pedigree data from crossing records may have missing values, but unlike molecular

marker data, missing values in pedigree data do not violate the requirement of the A

matrix to be at least positive semi-definite. There are few reports in the literature on

170

10.5 Conclusion

the practical application of molecular marker data for estimation of genetic relation-

ships. Pedigree-based information can be as efficient as molecular marker information

to predict genetic effects (Mrode and Thompson, 2005, Maenhout et al., 2009). Further

there could be issues with the cost of molecular markers. Pedigree records are available

within plant breeding data bases, though it needs to be electronically accessible and

well managed in a database. Nevertheless the benefits of such information for a plant

breeding program makes this worth pursuing as it can result in efficiency of selection

within a plant breeding program, for very little added cost.

10.5 Conclusion

Statistical models in plant breeding programs aim to model the gene to phenotype

relationship (Cooper et al., 2005). This thesis examines how this can be improved with

the inclusion of information from correlated traits, ancestry and environments. This

thesis dealt with the efficiency of selection for the traits disease resistance and grain

yield. However it can be extended to a majority of crop production traits. It is hoped

that the efficiency gains demonstrated in this thesis will lead to greater adoption of

these methodologies in plant breeding programs.

171


172

Appendices

173

Appendix A

Published paper based on

Chapter 3

175

A bivariate mixed model approach for the analysis of plantsurvival data

Aanandini Ganesalingam • Alison B. Smith •

Cameron P. Beeck • Wallace A. Cowling •

Robin Thompson • Brian R. Cullis

Received: 23 September 2011 / Accepted: 21 August 2012 / Published online: 30 August 2012

� Springer Science+Business Media B.V. 2012

Abstract Disease resistance is often measured as

plant survival, which involves taking multiple counts

of plants before and after disease incidence. Often,

survival data are analyzed by forming a single derived

variable, namely final counts expressed as a percent-

age of initial counts. In this study we propose a

bivariate linear mixed model approach in which the

two variables are the initial and final counts. This

approach is demonstrated using data from nine

blackleg disease nurseries in the 2009 growing season

in Australia. Replicated experiments were grown at

each nursery with a mixture of commercial Australian

canola cultivars and breeding lines (collectively called

‘entries’) being tested. Plant survival was determined

by counting all the seedlings at emergence and then

recounting the number surviving at maturity in each

plot. The counts were considered as two ‘traits’, which

were log transformed prior to a bivariate linear mixed

model analysis. Each trait had different error vari-

ances, spatial components (both local and global) and

outliers. The variance of entry effects was non-zero for

both traits at all locations. The correlation of entry

effects between the traits ranged from 0.218 to 0.935

across locations. Best Linear Unbiased Predictors

(BLUPs) of entry effects at both sampling times

provided three possible indices for selection: (log)

counts at emergence, (log) counts at maturity and the

difference between these two which could be expon-

entiated to provide percentage survival values. Thus

the bivariate mixed model approach for the analysis of

A. Ganesalingam (&) � A. B. Smith � C. P. Beeck

School of Plant Biology M084, The University of Western

Australia, 35 Stirling Highway, Crawley, WA 6009,

Australia

e-mail: [email protected]

A. B. Smith � B. R. Cullis

School of Mathematics and Applied Statistics, Faculty

of Informatics, University of Wollongong, Wollongong,

NSW, Australia

C. P. Beeck � W. A. Cowling

Canola Breeders Western Australia Pty Ltd, South Perth,

WA, Australia

W. A. Cowling

The UWA Institute of Agriculture, The University

of Western Australia, Crawley, WA, Australia

R. Thompson

Rothamsted Research, Harpenden, UK

R. Thompson

Queen Mary University of London, London, UK

B. R. Cullis

Division of Mathematics, Informatics and Statistics,

CSIRO, Canberra, ACT, Australia

123

Euphytica (2013) 190:371–383

DOI 10.1007/s10681-012-0791-0

plant survival data provided a more detailed picture of

the impact of disease resistance compared with the

univariate analysis of percentage survival data. Addi-

tionally the predicted entry effects for survival were

more accurate in the bivariate analysis.

Keywords Plant survival � Blackleg disease �Bivariate mixed models

Introduction

Blackleg disease of canola (Brassica napus L.) is

caused by the fungal pathogen Leptospheria maculans

(Punithalingam and Holliday 1972). This disease is

one of the most economically devastating diseases of

canola in Australia (Sivasithamparam et al. 2005),

North America and Europe (Fitt et al. 2006; West

et al. 2001). Disease infection is often associated with

yield losses ranging from 10 to 50 % (West et al.

2001). In Western Australia alone, the losses associ-

ated with blackleg disease during the 1998 and 1999

growing seasons, were $20M and $50M respectively

(Khangura and Barbetti 2001). Disease resistance in

Australian commercial cultivars is one of the main

methods of controlling disease incidence.

The National Blackleg Resistance Ratings for new

canola varieties are published annually (http://www.

australianoilseeds.com/commodity_groups/canola_

association_of_australia/pests__and__disease). These rat-

ings are based on plant survival data from blackleg

disease nursery trials conducted across southern Aus-

tralia over several years. Disease nurseries are located in

medium to high rainfall areas of Western Australia,

South Australia, Victoria and New South Wales

(Fig. 1). Disease nurseries are run by commercial

canola breeding companies and publicly funded

research groups, and are coordinated by the National

Blackleg Group (NBG). Plant survival is calculated by

dividing the number of plants at maturity by the number

of plants at emergence in each plot, expressed as a

percentage. Percent survival data are then subjected to

an analysis across sites after an appropriate transfor-

mation. We will refer to this as the historical approach.

The historical approach involves the univariate

analysis of a variable derived from two observed sets

of measurements. In this paper we propose a bivariate

mixed model analysis in which the two original variables

are maintained. This allows for the estimation of genetic

effects for each trait (initial and final counts) which may

reveal greater insight into variety performance including

establishment and disease resistance. Additionally the

data (for both the bivariate and historical approaches)

typically require a transformation to better meet the

assumptions of the analyses. If a logarithmic transfor-

mation is used the differences between the estimated

genetic effects for (log) final counts and (log) initial

counts in the bivariate analysis are analogous to the

estimated genetic effects for (log) survival in the

historical (univariate) analysis. Importantly however

the predicted entry effects for (log) survival from the

bivariate analysis are likely to be more accurate than

those from the univariate analysis (Thompson and

Meyer 1986; Mrode and Thompson 2005).

In this paper, the bivariate mixed model approach is

developed for blackleg survival data. The analysis is

developed by first considering the univariate analysis

of each trait then the bivariate analysis of both traits

together. This is fully described for one disease

nursery site (York), and results summarised across

nine disease nursery sites. We conclude with a

discussion on the value of the bivariate mixed model

approach for blackleg disease counts.

Materials and methods

Description of data

Nine blackleg disease nursery locations in 2009 are

summarised in Table 1. Each of these nurseries was

managed by a different private breeding company. As

such there were some differences in experimental

design, subject to some basic protocols as set out by

the NBG (Marcroft 2009). The most obvious differ-

ence is that at some locations entries were divided on

the basis of herbicide tolerance and then a separate

experiment conducted for each, whereas at others all

entries (irrespective of herbicide group) were com-

bined into a single experiment. The former approach

was adopted by some companies as they felt it enabled

more efficient management practices to be applied. A

key point is that resistance ratings are presented across

all herbicide groups so that it is important that all

groups tested at a nursery are analysed together as a

single set of entries. This is also a requirement for any

372 Euphytica (2013) 190:371–383

123

multi-environment trial (MET) analysis as it ensures

that entry concurrence (commonality) is maximised

across locations within a year.

High disease levels at each nursery were promoted

by growing entries on or with blackleg-infested

stubble from the previous season. The stubble source

at each disease nursery is given in Table 2. Disease

nurseries were only included in the data set if the

susceptible control entry, Karoo had less than 30 %

survival (Marcroft 2009). Standard management prac-

tices were followed across all disease nurseries and are

set out in the protocols determined by the NBG

(Marcroft 2009). It is current NBG protocol to omit

plots with less than 20 emergence counts due to poor

Fig. 1 Geographic

locations of nine blackleg

(Leptospheria maculans)

disease nurseries across

southern Australia during

the 2009 growing season

Table 1 Description of

blackleg disease nursery

experiments during the 2009

growing season

Each location is composed of

one or more experiments based

on herbicide type. The number

of entries, columns, rows and

blocks are listed for each

experiment in this data set

Conv conventional, Clclearfield�, TT triazine

tolerant, RR round up ready�

Location Experiment Herbicide type Entries Columns Rows Blocks

Bakers Hill BH Conv, Cl, TT 57 3 57 3

Bordertown BT1 Cl 13 3 13 3

BT2 TT 28 6 14 3

BT3 Conv 24 6 12 3

BT4 Conv, Cl, TT 33 3 33 3

Clear Lake CL Conv, Cl, TT 18 4 20 4

Lake Bolac LB1 Conv, TT 24 12 8 3

LB2 RR 31 12 10 3

LB3 Conv, Cl, TT, RR 107 12 30 3

Nurcoung NU1 RR 24 12 8 3

NU2 Conv, Cl, TT, RR 107 12 30 3

Shenton Park SP Conv, Cl, TT 65 22 9 4

Wagga Wagga WA Conv, Cl, TT 74 15 16 3

Wonwondah WO Conv, TT 31 12 10 3

York YK Conv, TT 78 3 79 3

Euphytica (2013) 190:371–383 373

123

germination, and to truncate plots with greater than

100 % survival to 100 %. For the bivariate analysis

however, the data for all plots were retained.

All experiments in this data set were designed as

randomized complete block designs (RCB) with either

3 or 4 replicates and extra plots of controls. All

experiments were laid out as a rectangular array

indexed by rows and columns (Table 1). Sufficient

seed was sown by hand or machine to target 100

established plants per plot. The number of emerged

seedlings per plot was counted at 4–6 weeks after

sowing, and the number of surviving plants per plot

was counted when plants were mature, at the wind-

rowing stage.

Statistical methods

Univariate analysis

The first step in the bivariate analysis is to identify

appropriate spatial models for each trait, following the

spatial mixed model approach of Gilmour et al.

(1997). This is most readily achieved by conducting

a separate univariate analysis for each trait. The

univariate models are then incorporated into the

bivariate analysis.

We describe the approach for one disease nursery

(York) that comprises a single experiment (Table 1).

Let r and c be the number of rows and columns

respectively so that the total number of plots is given

by n = rc, m is the number of entries and b the number

of blocks in the RCB design. At York, n = 237 plots,

r = 79 rows, c = 3 columns, b = 3 blocks and

m = 78 entries (Table 1). Note, that there were extra

plots of the variety GT61 sown, as there was not

enough seed for the variety CBTM Mallee HTTM. The

data are ordered as rows within columns. The base line

univariate mixed model is developed for the data at

sampling time j, (j = 1, 2), where j corresponds to 1

for emergence counts and 2 for maturity counts, and is

given by:

yj ¼ Xsj þ Zvuvj þ Zbubj þ ej ð1Þ

where yj is the n 9 1 vector of data; sj is the vector of

fixed effects (in this case the overall site mean) with

associated design matrix X; uvj is the m 9 1 vector of

random entry effects with associated design matrix

Zv; ubj is the b 9 1 vector of random block effects

with associated design matrix Zb, and ej is the vector of

residuals ordered as per the data vector. There are no

sub-scripts associated with the design matrices since,

for the base-line model, they are the same for both

sampling times.

Entry effects are assumed to be independent with

variance r2v j and the block effects to be independent

with variance r2bj. This is written as var ðuvjÞ ¼ r2

v jIm

and var ðubjÞ ¼ r2bjIb where Im and Ib are identity

matrices of dimensions m 9 m and b 9 b respectively.

In terms of the errors, a separable autoregressive

process of order one (AR1) as proposed by (Gilmour

et al. 1997) was assumed. Thus we write var ðejÞ ¼Rj ¼ r2

j Rcj � Rrj; where rj2 is the error variance at

sampling time j;Rcj is a c 9 c correlation matrix for

trend in the column dimension and Rrj a r 9 r

correlation matrix for trend in the row dimension.

Each matrix is a function of a single autocorrelation

parameter qcj and qrj for the column and row

Table 2 Location based

details: state, stubble type,

number of experiments,

number of entries and average

plant counts at emergence

(eme) and maturity (mat) for

each of the 2009 blackleg

disease nurseries

Location State Stubble type Experiments Entries Average

Eme Mat

Bakers Hill WA Bravo TT 1 57 60 35

Bordertown SA Mixture 4 74 38 13

Clear Lake VIC 45Y77 1 18 50 33

Lake Bolac VIC ATR-Marlin 3 148 28 7

Nurcoung VIC ATR-Cobbler 2 128 50 15

Shenton Park WA CB Telfer 1 65 75 57

Wagga Wagga NSW Bravo TT 1 74 34 10

Wonwondah VIC AV-Garnet 1 31 37 14

York WA ATR-Cobbler 1 78 59 13

374 Euphytica (2013) 190:371–383

123

dimensions respectively. Experiments with four or

less columns, were assumed to have independence for

errors in the column dimension, so that Rcj ¼ Ic:

After the base line mixed model was fitted,

diagnostics were used to assess the adequacy of the

spatial models. These included plots of residuals

against row number for each column and a three

dimensional display of the sample variogram

(Gilmour et al. 1997). These diagnostics were used

to identify outliers and determine if additional fixed

and/or random terms were required in the model.

Nurseries with several experiments required a more

complex model. Here, the data vector included the

individual plot data combined across experiments,

ordered as rows within columns within experiments.

Thus the total number of plots is n ¼Ps

i¼1 ni where s

is the number of experiments and ni is the number of

plots in experiment i, i.e. (i ¼ 1; . . .; s). The base line

mixed model for sampling time j is given by:

yj ¼ Xsj þ Zvuvj þ Zeuej þ Zbubj þ ej ð2Þ

where uv is as previously defined (and m is the total

number of entries for the disease nursery location); ub

is now the b 9 1 vector of random block effects for

each experiment (so that b ¼Ps

i¼1 bi); ue is the s 9 1

vector of random experiment effects. The variance

assumptions for the random effects are now varðuvjÞ ¼

r2v j Im; varðuej

Þ ¼ r2e jIs and varðubj

Þ ¼ diagðrbji2IbiÞ:

Note, that the variance matrix for ubj is block diagonal

and implies a separate block variance for each

experiment. A separate spatial model for the errors

was allowed for each experiment, so: varðejÞ ¼ Rj ¼diagðRjiÞ where Rji ¼ ðr2

jiRcji � RrjiÞ: The use of

distinct spatial models and block variances for each

experiment was due to the fact that the experiments at

each nursery are physically separated from each other,

that is not adjoining.

The count data were first log-transformed before

analysis to approximate a Gaussian distribution with a

constant variance. Further, this transformation ensured

that the predicted counts were non-negative, which is

of biological significance to this analysis.

All models in this paper were fitted in ASReml-R

(Gilmour et al. 2009). This provided residual maxi-

mum likelihood (REML) estimates of the variance

parameters, (empirical) Best Linear Unbiased Esti-

mates (BLUEs) of the fixed effects and (empirical)

Best Linear Unbiased Predictions (BLUPs) of the

random effects.

Bivariate analysis

Having identified the appropriate spatial models from

the univariate analyses for the emergence and maturity

data, the bivariate analysis is then conducted. Again,

consider the simplest case of a nursery comprising a

single experiment. Let y ¼ ðy01; y02Þ0, be the combined

vector of data across sampling times. The mixed

model for the bivariate analysis is given by

y ¼ X�sþ Z�vuv þ Z�bub þ Z�ouo þ e ð3Þ

where uv ¼ ðu0v1u0v2Þ0

is the 2m 9 1 vector of random

entry effects and Z�v ¼ I2 � Zv is the associated design

matrix; ub ¼ ðu0b1; u0b2Þ0

is the 2b 9 1 vector of

random block effects and Zb� ¼ I2 � Zb is the asso-

ciated design matrix; e ¼ ðe01; e02Þ0

is the vector of

errors ordered as for the data vector. The vector of

fixed effects, s, includes an overall mean for each

sampling time and any other fixed effects as identified

in the spatial modelling (e.g. linear regression on

rows) from the univariate analyses. Any random

effects identified in the univariate analyses are

included in the vector uo.

The variance assumptions for the random entry

effects are,

varðuvÞ ¼ varuv1

uv2

� �

¼ r2v1

rv12 r2v2

� �

� Im ð4Þ

where rvj2 (j = 1, 2) are as previously defined (i.e. the

the variance of entry effects for each sampling time)

and rv12 is the covariance between the entry effects at

emergence and maturity. For ease of interpretation, we

converted the covariance between entry effects to a

correlation, i.e.

qv12 ¼rv12ffiffiffiffiffiffiffiffiffiffiffiffiffir2

v1r2v2

p ð5Þ

The variance assumptions for the block effects were

similar except the covariance between sampling times

was omitted if one or both block variances were small.

The variance assumptions for the vector uo were

chosen appropriately for the terms involved. For the

Euphytica (2013) 190:371–383 375

123

errors a separable spatial correlation model was

assumed, namely

varðeÞ ¼ vare1

e2

� �

¼ r21

r12 r22

� �

� Rc � Rr ð6Þ

where rj2 (j = 1, 2) is as previously defined (i.e. the

error variances for each sampling time) and r12 is the

covariance between the errors at emergence and

maturity. The latter accommodates the repeated mea-

sures nature of the data with two sampling times for

each plot. The error covariance was converted to a

correlation between the traits, as in Eq. 5. The (spatial)

correlation matrices Rc and Rr correspond to autore-

gressive processes of order one, that is, functions of

single parameters qc and qr respectively. The separa-

bility assumption implies that the same spatial corre-

lation parameters are applicable for both sampling

times. It may be desirable to allow different param-

eters, but such models are not yet available and are the

subject of current research.

After fitting this model the predicted entry means

(i.e. BLUPs of entry means) were obtained for each

sampling time. Let pjk denote the predicted entry mean

for entry k at sampling time j. These may be back-

transformed to the original scale as exp(pjk). The back-

transformed difference between the predicted means

for maturity and emergence is given by

expðp2k � p1kÞ ¼expðp2kÞexpðp1kÞ

ð7Þ

namely the ratio of maturity to emergence counts on

the back-transformed scale. This allows entries to be

assessed on the same basis as the historic approach,

namely in terms of percent survival.

Results

Univariate analysis

York disease nursery

Results of the univariate analyses are described in

detail for the York disease nursery in Western

Australia (Table 1; Fig. 1).

This experiment had three columns, so an AR1

spatial trend process was only modeled for the row

dimension in both trait models. After the model was

fitted, diagnostics including sample variograms and

residual plots were used to determine the adequacy of

spatial models. The plots for emergence (Fig. 2) showed

the existence of local spatial correlation in the row

direction (reflected in the smooth trend in the residual

plot), extraneous variation in the row direction (seen as

the up/down pattern in the variogram) and three outliers

(unusually low values on the residual plot). The outliers

were omitted from the subsequent analysis (i.e. the count

data were set as missing values). The extraneous

variation was accommodated by fitting random row

effects in the model, which improved the residual plot

and variogram (Fig. 3). The spatial correlation for trend

in the row direction was strong (0.72; see Table 3). In

contrast to the emergence analysis, there were no

extraneous effects in the maturity analysis. A single

outlier was detected and set to a missing value. The

spatial correlation in the row direction was much weaker

than for emergence (0.22; see Table 3).

The REML estimate of the variance of entry effects

for the emergence trait (0.400) was almost as large as

that for the maturity trait (0.511) (Table 4). The

variance component for blocks was very close to zero

(0.041 and 0.066 for the emergence and maturity

models respectively). The error variance component

for the emergence mixed model (0.382) was larger

than the maturity model (0.299) (Table 3).

All disease nurseries

Across all the disease nursery locations, there were

more outliers removed from the univariate emergence

model than the maturity model (Table 3).

The terms fitted to the mixed models for non-

stationary trend and extraneous variation differed for

each trait. There was more extraneous variation

present for emergence than maturity across all exper-

iments (Table 3). In 9 out of the total 15 experiments,

terms were required to encompass non-stationary

trend and extraneous variation for emergence and

maturity trait models.

In terms of stationary trend, there was variation

between trait models for column and row autocorre-

lation values (Table 3). Across all the emergence

mixed models, the largest column autocorrelation

value was 0.66 at BT2, and the largest row autocor-

relation value was 0.72 at YK (Table 3). For the

maturity mixed models, the largest column autocor-

relation value was 0.43 at NU1 and the largest row

376 Euphytica (2013) 190:371–383

123

autocorrelation value was 0.54 at NU1 (Table 3).

High autocorrelation values indicate the presence of

strong local trend in these experiments for the

respective trait model.

The REML estimates of variance of entry effects for

emergence and maturity were non-zero for all locations

(Table 4). Therefore, entries varied in emergence and

maturity traits at all disease nursery locations. Further-

more, the variance of entry effects for maturity was

substantially larger than that of emergence across all

experiments except Bakers Hill (Table 4).

Bivariate analysis

York disease nursery

For the bivariate analysis, the spatial terms from the

univariate trait mixed models were retained and the

bivariate model (Eq. 3) was fitted.

REML estimates of variance of entry effects for

emergence and maturity from the bivariate model

were close approximations of the variances of entry

effects obtained from the individual univariate mixed

models (Table 4). Similarly, the error variance com-

ponents from the bivariate model were close approx-

imations of the individual univariate analyses

(Table 3). The AR1 row correlation value under the

bivariate model was 0.362, which was close to the

average of the row correlations obtained under

the univariate trait analyses (Table 3).

The bivariate model included a correlation structure

for both the entry effects and the errors. The estimated

correlation of entry effects between the traits at York

was 0.71 and the correlation between trait errors was

0.59. The high correlation of entry effects demon-

strates an agreement between entry rankings for both

traits and the high error correlation reflects the impact

of the repeated measures nature of the data.

Fig. 2 Initial plot of residuals and sample variogram from the univariate emergence model at the York disease nursery

Euphytica (2013) 190:371–383 377

123

After fitting the bivariate model, BLUPs of entry

means at emergence and maturity for each experiment

were used to produce two plots. In the first, entry

means at maturity were plotted against entry means at

emergence (Fig. 4). In the second, the difference

between BLUPs of entry means for emergence and

maturity were plotted against the BLUPs of entry

means at emergence (Fig. 5). The difference between

the predicted entry means for emergence and maturity

corresponded to the percent survival scale of the

historical approach, when back-transformed (see

Eq. 7).

The maturity versus emergence plot showed large

variation in emergence of entries at York, from 10 to

100 plant counts (Fig. 4). The majority of entries

were clustered towards the centre of the graph, with

emergence counts between 20 and 50 and maturity

counts between 5 and 20. A regression line of

maturity against emergence, which is implicit in the

bivariate variance structure for the entry effects was

drawn in Fig. 4. This corresponds to the regression

of the true entry effects for maturity (i.e. uv2) on the

true entry effects for emergence (i.e. uv1). The slope is

given by

b ¼ qv12 �

ffiffiffiffiffiffiffir2

v2

r2v1

s

ð8Þ

The slope for York was 0.84, which indicates a strong

linear relationship between maturity and emergence

counts at this disease nursery location.

The percent survival versus emergence plot showed

that the control entry Surpass501TT had a very low

emergence count, with less than 20 plant counts, but an

average percentage survival value of 25 % (Fig. 5).

The highly resistant entry Hyola50 had average

emergence, but the highest percentage survival value

at 65 %. The entry 46Y20(J) had the highest plant

Fig. 3 Plot of residuals and sample variogram from the univariate emergence model at the York disease nursery after the addition of a

random row component and removal of outliers

378 Euphytica (2013) 190:371–383

123

emergence and maturity counts (Fig. 4) but average

percentage survival value of 25 % (Fig. 5).

All disease nurseries

In contrast to the York disease nursery, which had a

high correlation of entry effects between traits, the Lake

Bolac disease nursery had the lowest correlation of

entry effects between traits (Table 4). At Lake Bolac,

there was little agreement between entry rankings

across traits (Fig. 6). The slope of the regression line of

entry effects for maturity on emergence at this site was

0.49, which indicated weak linear relationship when

compared to York, which had a slope of 0.84.

Table 3 Spatial modelling in univariate analyses of emergence

(eme) and maturity (mat) trait data for each experiment: terms

added for global trend or extraneous variation, REML

estimates of error variance, REML estimates of autocorrelation

parameters (for columns and rows, where fitted) and number of

outliers removed

Expt Global trend & extraneous variation termsa Error variance Autocorrelation Number of outliers

Eme Mat Eme Mat Column Row Eme Mat

Eme Mat Eme Mat

BH rdRow 0.040 0.317 0.19 -0.03 1

BT1 0.102 0.150 -0.02 -0.33

BT2 0.062 0.410 0.66 0.26 0.03 0.35

BT3 0.265 0.261 0.6 0.42 -0.13 -0.1

BT4 0.136 0.191 -0.07 0.13 1 1

CL 0.017 0.064 0 0.26 1

LB1 rd(R) & rd(C) 0.078 0.26 -0.06 0.01 0.05 0.12 1

LB2 rd(R) & rd(C) 0.098 0.291 0.09 0.05 -0.07 0.04

LB3 rd(R) & rd(C) 0.160 0.283 0.01 0.04 0.01 0.04 1

NU1 0.169 0.089 0.23 0.43 0.39 0.54 1

NU2 rd(R) & rd(C) 0.183 0.111 0.09 0.35 0.12 0.16 3 4

SP lin(R) 0.015 0.054 0.02 -0.17 -0.05 0.17 2 2

WG rd(R) & rd(C) lin(C) 0.029 0.265 0.24 0.28 2

WO rd(R) & rd(C) 0.163 0.031 0.16 0.03 -0.13 -0.07

YK rd(R) 0.382 0.299 0.72 0.22 3 1

a lin(R) and lin(C) indicates a fixed linear regression on row or column number; rd(R) and rd(C) indicate random row and column

components

Table 4 REML estimates of

entry variance from univariate

trait model and bivariate model

at each disease nursery location

The correlation between entry

effects from the bivariate

model is also shown

Location Univariate Bivariate

Eme Mat Eme Mat Correlation

Bakers Hill 0.108 0.127 0.109 0.131 0.68

Bordertown 0.112 0.744 0.110 0.748 0.24

Clear Lake 0.042 0.232 0.047 0.259 0.68

Lake Bolac 0.124 0.622 0.126 0.629 0.22

Nurcoung 0.079 0.489 0.075 0.485 0.25

Shenton Park 0.191 0.657 0.194 0.636 0.94

Wagga Wagga 0.053 0.768 0.053 0.765 0.73

Wonwondah 0.033 0.687 0.034 0.691 0.73

York 0.400 0.511 0.354 0.493 0.71

Euphytica (2013) 190:371–383 379

123

The REML estimates of the variance of entry

effects for emergence and maturity at all sites were

similar to the approximations obtained from the

individual trait univariate analyses (Table 4). The

correlation of entry effects between the two traits

averaged 0.57 (range 0.22–0.94) across the nine

disease nursery locations.

Accuracy comparisons

As noted in the introduction one of the main advan-

tages in using a bivariate analysis is that it increases

the accuracy of predictions. The accuracy of a

prediction for a variety is defined here as the square

of the correlation between the true and predicted effect

for that variety. It can be computed using the estimated

genetic variance for the trait concerned and the

prediction error variance for the variety following

Mrode and Thompson (2005). In this paper the key

accuracy comparison is in terms of survival rates.

These accuracy values have been calculated for each

variety for each disease nursery from both the

bivariate analysis of (log) initial and (log) final counts

and the univariate analysis of the difference. To ensure

a fair comparison the estimates of genetic and

non-genetic variances (including block and error

variances) were held constant between the two

Emergence

Mat

urity

1

2

3

4

2.5 3.0 3.5 4.0 4.5 5.0


10 20 30 40 50 60 70 80

10

20

30

40

50

60

7080

Fig. 4 Predicted entry means at emergence plotted against

predicted entry means at maturity from the bivariate model for

the disease nursery at York. A regression line of maturity against

emergence was included, with the slope having a value of 0.84.

The axes are on a log scale (as for the analysis) with the back-

transformed scale (i.e. plant counts) shown inside each axis

Emergence

Mat

urity

− E

mer

genc

e

−2.5

−2.0

−1.5

−1.0

−0.5

2.5 3.0 3.5 4.0 4.5 5.0


10 20 30 40 50 60 70 80

10

20

30

40

50

60

70

Fig. 5 The difference between predicted entry means at

maturity and emergence (corresponds to percentage survival

when back transformed, these values are shown on the inside of

the y-axis) plotted against predicted entry means at emergence

from the bivariate model for the disease nursery at York

Emergence

Mat

urity

1

2

3

4

2.5 3.0 3.5


0504030201

10

20

30

40

50

60

7080

Fig. 6 Predicted entry means at emergence plotted against

predicted entry means at maturity from the bivariate model for

the disease nursery at Lake Bolac. A regression line of maturity

against emergence was included, with the slope having a value

of 0.49. The axes are on a log scale (as for the analysis) with the

back-transformed scale (i.e. plant counts) shown inside each

axis

380 Euphytica (2013) 190:371–383

123

approaches. Thus for Bakers Hill, for example, the

genetic variances for the bivariate analysis were

rv12 = 0.109 and rv2

2 = 0.131 for the emergence and

maturity traits respectively, with a genetic correlation

of qv12 = 0.68 (see Table 4) so that the genetic

variance for the univariate analysis of the difference

was constrained to be equal to rv12 ? rv2

2 - 2qv12 rv1

rv2 = 0.078. Non-genetic components were con-

strained in a similar manner. Additionally identical

sets of counts (namely the complete data after the

removal of the outliers described in Table 3) were

used to obtain both the bivariate and univariate

predictions. The accuracy of prediction from the

bivariate analysis was greater than that from the

univariate analysis for all varieties in all nurseries.

The average percentage gain for individual nurseries

ranged from 0.1 to 5.6 % with a mean of 1.1 % (see

Table 5). The gains were small for those nurseries

where the univariate accuracies were high (that is, near

the maximum possible value of 1.0), whereas more

substantial gains were observed for those nurseries

where the univariate accuracies were lower.

Discussion

In this article, a bivariate mixed model approach for

the analysis of plant survival data is described and

applied to data from nine Australian canola blackleg

disease nursery trials. The two traits (variables) in the

bivariate analysis are plant counts at emergence and

plant counts at maturity.

A valuable feature of the bivariate approach is the

ability to conduct spatial modelling separately for each

trait. The components of spatial variation (Gilmour

et al. 1997) often differed between emergence and

maturity counts. Global trend and extraneous variation

was found in many of the trials (see Table 3) and was

more prevalent for the emergence trait. Local station-

ary trend varied across experiments as seen by the

range in row and column autocorrelation parameters

(see Table 3). Autocorrelation values greater than 0.3

were observed in five experiments indicating the

existence of strong local spatial trend. In some cases

(for example at York) the trend differed between the

two traits. The number of outliers differed for each

trait, and emergence had more outliers than maturity.

The modelling of spatial trend is an important

component of the analysis of field experiments as it

has been shown to improve experiment precision

(Qiao et al. 2000) and leads to large reductions in

effective error variance (Smith et al. 2006). Our study

demonstrates the importance of trait based spatial

modelling, as there can be differences between the

traits within the model (Table 3). Under the historical

approach, this would not have been observed as the

derived variable (percentage survival) confounds the

errors associated with each trait.

Under the current Australian disease nursery anal-

ysis protocol, plots with less than 20 counts at

emergence are omitted and plots with greater than a

100 % survival are truncated to a 100 % (Marcroft

2009). Such values arise because emergence counts (in

addition to maturity counts) are subject to error. The

bivariate approach avoids such rules, since it accom-

modates error variation in emergence and maturity so

that all data points are retained for analysis. For

instance, under the historical protocol, 21 % of the

total number of plots at the York disease nursery were

removed, which is a substantial loss of data. In the

bivariate analysis, plots of BLUPs at maturity vs

emergence allow entry predictions to be discounted

where there is poor emergence, at the discretion of the

researcher. This is a more informed approach than

deletion of the raw data in the historical analysis.

The bivariate analysis allows us to examine the

entry effects for individual traits. In our analysis, the

entry variance components for both emergence and

maturity were non-zero for all nurseries (see Table 4).

The entry variance for maturity was greater than that

for emergence at all locations except Bakers Hill,

Table 5 The accuracy of prediction for the difference between

(log) initial and (log) final counts: absolute accuracy values

from univariate analyses and percentage gain in accuracy from

bivariate analyses compared with univariate

Location Difference

univariate

Percent improvement

difference

Bakers Hill 0.42 5.63

Bordertown 0.91 0.17

Clear Lake 0.87 0.14

Lake Bolac 0.86 0.40

Nurcoung 0.89 0.07

Shenton Park 0.91 0.79

Wagga Wagga 0.86 1.14

Wonwondah 0.85 0.89

York 0.73 0.93

Euphytica (2013) 190:371–383 381

123

where the two values were similar. Critically, the

bivariate approach demonstrated the existence of

variation among entries for emergence, which can

also be seen in the graphs of predicted entry means

(Figs. 4, 6). This information is lost in the historical

approach.

The variation in maturity counts between entries is

largely attributed to differential resistance to blackleg

disease since disease nursery management protocols

ensures that the effects of other pests and diseases are

minimised (Marcroft 2009). However the variation in

emergence counts between entries may arise either

from differential resistance to early blackleg infection

or seed source differences. Seedling emergence is

known to be affected by environmental factors such as

soil fertility, salinity, compaction, tillage and surface

residues (Forcella et al. 2000). It can also be affected

by seed lot factors such as age of seed (Finch-Savage

1986), the storage environment of the seed (Ellis and

Roberts 1980), and seed production environment

(Ellis et al. 1993). Seed source variation is a known

issue for Australian blackleg disease nurseries, how-

ever the impact of this variation has not been

previously quantified. Also, blackleg disease has the

potential to impact on seedling emergence (Li et al.

2007; Sosnowski et al. 2006). Li et al. (2007) dem-

onstrated that soil borne ascospores and pycnidiosp-

ores of L. maculans caused seedling death from early

infection, with seedling deaths as high as 59 % of

seedlings after sowing in infested soil. Differences in

entry emergence attributable to early infection would

constitute genetic effects of resistance.

The correlations between entry effects at emer-

gence and maturity were moderate to strong ([0.6) at

6 out of the 9 disease nursery locations (Table 4). This

highlights that even though the entry effects for

emergence and maturity may have different causes

they are still strongly correlated at most disease

nursery sites.

In terms of entry selection, the bivariate approach

provides a more detailed picture than using the

historical approach. The prediction of entry means at

emergence and maturity can be used to generate three

sources of information for selection: emergence

counts, maturity counts and percentage survival

values. Even if percentage survival values are

regarded as the most appropriate for selecting blackleg

entries, there are both biological and statistical reasons

why this should not be done without reference to entry

emergence. The biological issues have already been

discussed. A key statistical issue is that the accuracy of

prediction of variety survival is greater with the

bivariate approach. In our study the gains were modest

but importantly the accuracy of prediction from the

bivariate approach was greater than that from the

univariate analysis for all varieties for all data-sets.

The gains for any particular data-set are obviously

unknown prior to an analysis but may be larger than

reported here and are worth pursuing given that there

is little extra cost or difficulty involved in conducting

the bivariate analysis.

The existence of genetic variance for the emer-

gence counts raises an important issue regarding

another method of analysis that is widely used, namely

the analysis of covariance. In the application presented

in this paper this would involve the analysis of

maturity counts using the emergence counts as a

covariate. In such an analysis the entry means for

maturity counts would all be adjusted to correspond to

a single emergence value (typically the average value

across all entries). If differences in entry emergence

are linked to early blackleg infection and thence have a

genetic basis then from a biological point of view it is

inappropriate to adjust entries to a common emergence

value since this adjustment effectively creates varie-

ties that do not exist. This type of adjustment has long

been known to be dangerous (see Smith 1957;

Urquhart 1982). Smith (1957) considers an example

where the treatments are varieties of corn, the variable

under study is yield and the covariate is number of ears

at constant plant density. Smith (1957) says that ‘‘Ear

number, an innate variety characteristic, cannot be

altered at will. Comparison of yields adjusted to equal

ear number is therefore artificial …’’. Thus in the

application presented in this paper the use of analysis

of covariance is inappropriate.

The approach presented in this paper provides a

valid and informative statistical analysis for other

types of bivariate data that are often examined using

either univariate analyses of ratios of variables or

analysis of covariance. In the plant breeding context

an important example is varietal selection for quality

traits. These traits are often ‘adjusted for’ grain protein

using either of the methods just described. Typically,

however, there are genetic differences between vari-

eties in terms of protein so that a bivariate analysis as

presented in this paper would be the recommended

approach.

382 Euphytica (2013) 190:371–383

123

In conclusion, the bivariate approach is an improve-

ment on the method historically used, in which a

derived variable (counts at maturity expressed as

percentage of counts at emergence) is analyzed. The

modelling approach presented is for individual disease

nurseries, however it is noted that the current annual

blackleg disease resistance ratings are obtained from a

series of disease nurseries across years and sites,

known as METs. Future research will aim to extend

the bivariate mixed model approach for MET data.

Acknowledgments The authors would like to thank the

National Blackleg Committee for the use of the 2009

Australian National Blackleg Resistance Rating data and

Steve Marcroft and Chris Lisle for valued help. The authors

would also like thank the referees for helpful comments which

have greatly improved the manuscript. The authors gratefully

acknowledge the financial support of the Grains Research

and Development Corporation of Australia (GRDC) in various

aspects of this research. Aanandini Ganesalingam acknowl-

edges Bayer CropScience for a PhD scholarship.

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123

Appendix B

ASReml-R Code

B.1 ASReml-R Code for fitting the univariate trait mod-

els in Chapter 3

bh.asr <- asreml(yvar~1,random=~Entry+Block,

rcov=~id(Column):ar1(Row),data=bleg.dat)

For this call;

• yvar - the response variable, this is the trait analysed i.e plant survival counts at

emergence or maturity

• Entry - factor with 57 levels

• Block - a block term - factor with 3 levels

• Column - column term - factor with 3 levels

• Row - row term - factor with 57 rows

• AR1(Row) - AR1 structure fitted for rows

• id(Column) - identity structure fitted for columns

189

B. ASREML-R CODE

B.2 ASReml-R Code for fitting the bivariate trait models

in Chapter 3

bh.asr <-asreml(yvar~Sample,random=~corh(Sample):Entry+diag(Sample):Block

+at(Sample,’eme’):Row,

rcov=~corh(Sample):id(Column):ar1(Row),data= bleg.dat)

For this call;

• Sample - factor with two levels corresponding to the traits emergence and maturity

• Entry - factor with 57 levels

• Block - a block term - factor with 3 levels

• Column - column term - factor with 3 levels

• Row - row term factor with 57 rows

• AR1(Row) - AR1 structure fitted for rows

• id(Column) - identity structure fitted for columns

190

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