chris brien 1 , bronwyn harch 2 , ray correll 2 & rosemary bailey 3

Principles in the design of multiphase experiments with a later laboratory phase: orthogonal designs

Chris Brien1, Bronwyn Harch2, Ray Correll2 & Rosemary Bailey3

1University of South Australia, 2CSIRO Mathematics, Informatics & Statistics, 3Queen Mary University of London

[email protected]

http://chris.brien.name/multitier


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Outline1. Primary experimental design principles.

2. Factor-allocation description for standard designs.

3. Single set description.

4. Principles for simple multiphase experiments.

5. Principles leading to complications, even with orthogonality.

6. Summary.

1) Primary experimental design principles Principle 1 (Evaluate designs with skeleton ANOVA

tables) Use whether or not data to be analyzed by ANOVA.

Principle 2 (Fundamentals): Use randomization, replication and blocking (local control).

Principle 3 (Minimize variance): Block entities to form new entities, within new entities being more homogeneous; assign treatments to least variable entity-type.

Principle 4 (Split units): confound some treatment sources with more variable sources if some treatment factors:i. require larger units than others,

ii. are expected to have a larger effect, or

iii. are of less interest than others.3

A standard athlete training example 9 training conditions — combinations of 3 surfaces and 3

intensities of training — to be investigated. Assume the prime interest is in surface differences

intensities are only included to observe the surfaces over a range of intensities.

Testing is to be conducted over 4 Months: In each month, 3 endurance athletes are to be recruited. Each athlete will undergo 3 tests, separated by 7 days, under 3

different training conditions. On completion of each test, the heart rate of the athlete

will be measured. Randomize 3 intensities to 3 athletes in a month and

3 surfaces to 3 tests in an athlete. A split-unit design, employing Principles 2, 3 and 4(iii).

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Peeling et al. (2009)

2) Factor-allocation description for standard designs

Standard designs involve a single allocation in which a set of treatments is assigned to a set of units: treatments are whatever are allocated; units are what treatments are allocated to; treatments and units each referred to as a set of objects;

Often do by randomization using a permutation of the units. More generally treatments are allocated to units e.g. using a spatial

design or systematically Each set of objects is indexed by a set of factors:

Unit or unallocated factors (indexing units); Treatment or allocated factors (indexing treatments).

Represent the allocation using factor-allocation diagrams that have a panel for each set of objects with: a list of the factors; their numbers of levels; their nesting

relationships. 5

(Nelder, 1965; Brien, 1983; Brien & Bailey, 2006)

Factor-allocation diagram for the standard athlete training experiment

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One allocation (randomization): a set of training conditions to a set of tests.

3 Intensities3 Surfaces

9 training conditions

4 Months3 Athletes in M3 Tests in M, A

36 tests

The set of factors belonging to a set of objects forms a tier: they have the same status in the allocation (randomization): {Intensities, Surfaces} or {Months, Athletes, Tests} Textbook experiments are two-tiered.

A crucial feature is that diagram automatically shows EU and restrictions on randomization/allocation.

Some derived items

Sets of generalized factors (terms in the mixed model): Months, MonthsAthletes, MonthsAthletesTests; Intensities, Surfaces, IntensitiesSurfaces.

Corresponding types of entities (groupings of objects): month, athlete, test (last two are Eus); intensity, surface, training condition (intensity-surface combination).

Corresponding sources (in an ANOVA): Months, Athletes[M], Tests[MA]; Intensities, Surfaces, Intensities#Surfaces.

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36 tests

Skeleton ANOVA

Intensities is confounded with the more-variable Athletes[M] & Surfaces with Tests[M^A]. 8

training conditions tier

source df

Intensities 2

Residual 6

Surfaces 2

I#S 4

Residual 18

tests tier

source df

Months 3

Athletes[M] 8

Tests[M A] 24




36 tests

E[MSq]

2 2 2MAT MA M

1 3 9

I1 3 q

1 3

S1 q

IS1 q

1

Mixed model Take generalized factors derived from factor-allocation

diagram and assign to either fixed or random model:

Intensities + Surfaces + IntensitiesSurfaces | Months + MonthsAthletes + MonthsAthletesTests

Corresponds to the mixed model:Y = XIqI + XSqS + XISqIS + ZMuM+ ZMAuMA+ ZMATuMAT.where the Xs and Zs are indicator variable matrices for the generalized factors in its subscript, and

qs and us are fixed and random parameters, respectively, with

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2E and E .j j j j u 0 u u I

This is an ANOVA model, equivalent to the randomization model, and is also written:

Y = XIqI + XSqS + XISqIS + ZMuM+ ZMAuMA + e. To fit in SAS must set the DDFM option of MODEL to KENWARDROGER.

Brien & Demétrio (2009).

3) Single-set description Single set of factors that uniquely indexes observations:

{Months, Intensities, Surfaces} (Athletes and Tests omitted). What are the EUs in the single-set approach?

A set of units that are indexed by Months-Intensities combinations and another set by the Months-Intensities-Surfaces combinations.

Of course, Months-Intensities(-Surfaces) are not actual EUs, as Intensities (Surfaces) are not randomized to those combinations.

They act as a proxy for the unnamed units. Mixed model is: I + S + IS | M + MI + MIS.

Previous model: I + S + IS | M + MA + MAT. Former more economical as A and T not needed. In SAS, default DDFM option of MODEL (CONTAIN) works.

However, MA and MI are different sources of variability: inherent variability vs block-treatment interaction.

This "trick" is confusing, false economy and not always possible.

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e.g. Searle, Casella & McCulloch (1992); Littel et al. (2006).

Single-set description ANOVA Single set of factors that uniquely indexes observations:

{Months, Intensities, Surfaces}

Use factors to derive skeleton ANOVA

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source df

Months 3

Intensities 2

M#I 6

Surfaces 2

I#S 4

Residual 18

E[MSq]

2 2 2MI M

1 3 9

I1 3 q

1 3

S1 q

IS1 q

1

Confounding not exhibited, and need E[MSq] to see that M#I is denominator for Intensities.

Summary of factor-allocation versus single-set description

Factor-allocation description is based on the tiers and so is multi-set.

It has a specific factor for the EUs and so their identity not obscured.

Single-set description factors are a subset of those identified in the factor-allocation description and so more economical.

Skeleton ANOVA from factor-allocation description shows: The confounding resulting from the allocation; The origin of the sources of variation more accurately

(Athletes[Months] versus Months#Intensities).

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4) Principles for simple multiphase experiments

Suppose in the athlete training experiment: in addition to heart rate taken immediately upon completion of a

test, the free haemoglobin is to be measured using blood specimens

taken from the athletes after each test, and the specimens are transported to the laboratory for analysis.

The experiment is two phase: testing and laboratory phases. The outcome of the testing phase is heart rate and a blood

specimen. The outcome of the laboratory phase is the free haemoglobin.

How to process the specimens from the first phase in the laboratory phase?

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Some principles

Principle 5 (Simplicity desirable): assign first-phase units to laboratory units so that each first-phase source is confounded with a single laboratory source. Use composed randomizations with an orthogonal design.

Principle 6 (Preplan all): if possible. Principle 7 (Allocate all and randomize in laboratory):

always allocate all treatment and unit factors and randomize first-phase units and lab treatments.

Principle 8 (Big with big): Confound big first-phase sources with big laboratory sources, provided no confounding of treatment with first-phase sources.

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A simple two-phase athlete training experiment Simplest is to randomize specimens from a test to

locations (in time or space) during the laboratory phase.

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36 tests

36 Locations

36 locations


source df

Intensities 2

Residual 6

Surfaces 2

I#S 4

Residual 18

tests tier

source df

Months 3

Athletes[M] 8

Tests[M A] 24

locations tier

source df

Locations 35

E[MSq]

2 2 2 2L MAT MA M

1 3 91

I1 3 q1

1 31

S1 q1

IS1 q1

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Composed randomizations

A simple two-phase athlete training experiment (cont’d)

No. tests = no. locations = 36 and so tests sources exhaust the locations source.

Cannot separately estimate locations and tests variability, but can estimate their sum.

But do not want to hold blood specimens for 4 months.16


source df

Intensities 2

Residual 6

Surfaces 2

I#S 4

Residual 18

tests tier

source df

Months 3

Athletes[M] 8

Tests[M A] 24

locations tier

source df

Locations 35

E[MSq]

2 2 2 2L MAT MA M

1 1 3 9

I1 1 3 q

1 1 3

S1 1 q

IS1 1 q

1 1

A simple two-phase athlete training experiment (cont’d) Simplest is to align lab-phase and first-phase blocking.

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36 tests

4 Batches

9 Locations in B

36 locations


source df

Intensities 2

Residual 6

Surfaces 2

I#S 4

Residual 18

tests tier

source df

Months 3

Athletes[M] 8

Tests[M A] 24

locations tier

source df

Batches 3

Locations[B] 32

E[MSq]

2 2 2 2 2BL B MAT MA M

1 1 3 99

I1 1 3 q

1 1 3

S1 1 q

IS1 1 q

1 1

Note Months confounded with Batches (i.e. Big with Big).

Composed randomizations

A simple two-phase athlete training experiment – mixed model

Form generalized factors and assign to fixed or random: I + S + IS | M + MA + MAT + B + BL.

ANOVA shows us i) there will be aliasing and ii) model without lab terms will fit and be sufficient – a “model of convenience”.

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36 tests

4 Batches

9 Locations in B

36 locations


source df

Intensities 2

Residual 6

Surfaces 2

I#S 4

Residual 18

tests tier

source df

Months 3

Athletes[M] 8

Tests[M A] 24

locations tier

source df

Batches 3

Locations[B] 32

E[MSq]


1 9 1 3 9

I1 1 3 q

1 1 3

S1 1 q

IS1 1 q

1 1

The multiphase law DF for sources from a previous phase can never be

increased as a result of the laboratory-phase design. However, it is possible that first-phase sources are split

into two or more sources, each with fewer degrees of freedom than the original source.

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source df

Intensities 2

Residual 6

Surfaces 2

I#S 4

Residual 18

tests tier

source df

Months 3

Athletes[M] 8

Tests[M A] 24

locations tier

source df

Batches 3

Locations[B] 32

E[MSq]


1 9 1 3 9

I1 1 3 q

1 1 3

S1 1 q

IS1 1 q

1 1

DF for first phase sources unaffected.

Factor-allocation in multiphase experiments While multiphase experiments will often involve multiple

allocations, not always: A two-phase experiment will not if the first phase involves a survey

i.e. no allocation e.g. tissues sampled from animals of different sexes.

A two-phase experiment may include more than two allocations: e.g. a grazing trial in the first phase that involves two composed randomizations.

Factor-allocation description is particularly helpful in understanding multiphase experiments with multiple allocations.

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5) Principles leading to complications, even with orthogonality

Principle 9 (Use pseudofactors): An elegant way to split sources (as opposed to introducing

grouping factors unconnected to real sources of variability). Principle 10 (Compensating across phases):

Sometimes, if something is confounded with more variable first-phase source, can confound with less variable lab source.

Principle 11 (Laboratory replication): Replicate laboratory analysis of first-phase units if lab variability

much greater than 1st-phase variation; Often involves splitting product from the first phase into portions

(e.g. batches of harvested crop, wines, blood specimens into aliquots, drops, lots, samples and fractions).

Principle 12 (Laboratory treatments): Sometimes treatments are introduced in the laboratory phase and

this involves extra randomization. 21

A replicated two-phase athlete training experiment Suppose duplicates of free haemoglobin to be done:

2 fractions taken from each specimen; One fraction taken from 9 specimens for the month and analyzed; Then, the other fraction from 9 specimens analyzed.

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4 Months2 Fractions in M, A, T3 Athletes in M3 Tests in M, A

72 fractions

4 Batches2 Rounds in B

9 Locations in B, R

72 locations

2 F1 in M

Problem: 18 fractions in a month to assign to 2 rounds in a batch: Use F1 to group the 9 fractions to be analyzed in the same round. An alternative is to introduce the grouping factor FGroups,

but, while in the analysis, not an anticipated variability source.

locations tier

source df

Batches 3

Rounds[B] 4

Locations[B R] 64

A replicated two-phase athlete training experiment (cont’d)

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source df

Intensities 2

Residual 6

Surfaces 2

I#S 4

Residual 18

fractions tier

source df

Months 3

Fractions[M A T]1 4

Athletes[M] 8

Tests[M A] 24

Fractions[M A T] 32




72 fractions

2 F1 in M

E[MSq] 2 2 2 2 2 2 2BRL BR B MATF MAT MA M

1 2 118 2 6 18

1 2 1

I1 q1 2 6

1 1 2 6

S1 q1 2

IS1 q1 2

1 21

1 1

Note split source


9 Locations in B, R

72 locations

A replicated two-phase athlete training experiment (cont’d)

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Last line measures laboratory variation. Again, no. fractions = no. locations = 72 and so fractions

sources exhaust locations sources. Consequently, not all terms could be included in a mixed model. Pseudofactors not needed in mixed model (cf. grouping factors).


source df

Intensities 2

Residual 6

Surfaces 2

I#S 4

Residual 18

fractions tier

source df

Months 3

Fractions[M A T]1 4

Athletes[M] 8

Tests[M A] 24

Fractions[M A T] 32


1 2 18 1 2 6 18

1 2 1

I1 1 2 6 q

1 1 2 6

S1 1 2 q

IS1 1 2 q

1 1 2

1 1

locations tier

source df

Batches 3

Rounds[B] 4

Locations[B R] 64

A replicated two-phase athlete training experiment – mixed model

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Form generalized factors and assign to fixed or random: I + S + IS | M + MA + MAT + MATF + B + BR + BRL.

Removing aliased terms gives one “model of convenience”: I + S + IS | MA + MAT + B + BR + BRL (M, MATF omitted). Another model would result from removing B and BRL (need BR).


source df

Intensities 2

Residual 6

Surfaces 2

I#S 4

Residual 18

fractions tier

source df

Months 3

Fractions[M A T]1 4

Athletes[M] 8

Tests[M A] 24

Fractions[M A T] 32


1 2 18 1 2 6 18

1 2 1

I1 1 2 6 q

1 1 2 6

S1 1 2 q

IS1 1 2 q

1 1 2

1 1

locations tier

source df

Batches 3

Rounds[B] 4

Locations[B R] 64

Single-set description for example

Single set of factors that uniquely indexes observations: {4 Months x 2 Rounds x

3 Intensities x 3 Surfaces}.

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72 fractions

2 F1 in M

Source df

Months 3

Rounds[M] 4

Intensities 2

M#I 6

Surfaces 2

I#S 4

M#I[S] 18

Residual 32

E[MSq] 2 2 2 2 2

MR MIS MI M

1 2 2 6 18

1 2

I1 2 6 q

1 2 6

S1 2 q

IS1 2 q

1 2

1


9 Locations in B, R

72 locations

How do: the locations

sources; Fractions;

affect the response? Similar to model of convenience

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6) Summary

Factor-allocation, rather than single-set, description used, being more informative and particularly helpful in multiphase experiments.

Multiphase experiments usually have multiple allocations.

Have provided 4 standard principles and 8 principles specific to orthogonal, multiphase designs.

In practice, will be important to have some idea of likely sources of laboratory variation.

Are laboratory treatments to be incorporated?

Will laboratory replicates be necessary?

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References Brien, C. J. (1983). Analysis of variance tables based on experimental

structure. Biometrics, 39, 53-59. Brien, C.J., and Bailey, R.A. (2006) Multiple randomizations (with

discussion). J. Roy. Statist. Soc., Ser. B, 68, 571–609. Brien, C.J. and Demétrio, C.G.B. (2009) Formulating mixed models for

experiments, including longitudinal experiments. J. Agr. Biol. Env. Stat., 14, 253-80.

Brien, C.J., Harch, B.D., Correll, R.L. and Bailey, R.A. (2010) Multiphase experiments with laboratory phases subsequent to the initial phase. I. Orthogonal designs. Journal of Agricultural, Biological and Environmental Statistics, submitted.

Littell, R. C., G. A. Milliken, et al. (2006). SAS for Mixed Models. Cary, N.C., SAS Press.

Nelder, J. A. (1965). The analysis of randomized experiments with orthogonal block structure. Proceedings of the Royal Society of London, Series A, 283(1393), 147-162, 163-178.

Peeling, P., B. Dawson, et al. (2009). Training Surface and Intensity: Inflammation, Hemolysis, and Hepcidin Expression. Medicine & Science in Sports & Exercise, 41, 1138-1145.

Searle, S. R., G. Casella, et al. (1992). Variance components. New York, Wiley.

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Web address for link to Multitiered experiments site



chris brien 1 , bronwyn harch 2 , ray correll 2 & rosemary bailey 3

Documents

set of treatments

set of units

set of factors

split units

splitunit design

single allocation

factorallocation diagrams

single set description