Conservation by Consensus: Reducing Uncertainty from Methodological Choices in Conservation-Based Models

by

Mark S. Poos

A thesis submitted in conformity with the requirements for the degree of Doctorate of Philosophy

Department of Ecology and Evolutionary Biology University of Toronto

© Copyright by Mark S. Poos 2010

ii

Conservation by Consensus: Reducing Uncertainty from

Methodological Choices in Conservation-Based Models

Mark S. Poos

Doctorate of Philosophy

Department of Ecology and Evolutionary Biology University of Toronto

2010

Abstract

Modeling species of conservation concern, such as those that are rare, declining, or have a

conservation designation (e.g. endangered or threatened), remains an activity filled with

uncertainty. Species that are of conservation concern often are found infrequently, in small

sample sizes and spatially fragmented distributions, thereby making accurate enumeration

difficult and traditional statistical approaches often invalid. For example, there are numerous

debates in the ecological literature regarding methodological choices in conservation-based

models, such as how to measure functional traits to account for ecosystem function, the impact of

including rare species in biological assessments and whether species-specific dispersal can be

measured using distance based functions. This thesis attempts to address issues in

methodological choices in conservation-based models in two ways. In the first section of the

thesis, the impacts of methodological choices on conservation-based models are examined across

a broad selection of available approaches, from: measuring functional diversity; to conducting

bio-assessments in community ecology; to assessing dispersal in metapopulation analyses. It is

the goal of this section to establish the potential for methodological choices to impact

conservation-based models, regardless of the scale, study-system or species involved. In the

second section of this thesis, the use of consensus methods is developed as a potential tool for

reducing uncertainty with methodological choices in conservation-based models. Two separate

applications of consensus methods are highlighted, including how consensus methods can reduce

uncertainty from choosing a modeling type or to identify when methodological choices may be a

problem.

iii

Acknowledgments

No accomplishment is ever singular, and my doctoral work is no different. For me, I have

had the great fortune of a loving family, a group of wonderful friends, supportive colleagues, a

thorough and considerate academic committee, and a brilliant advisor. I think Marston Bates was

right when she said “Research is the process of going up alleys to see if they are blind.” In my

personal journey through this doctoral work, I have gone through many blind alleys. Without the

help and support of all the people in my life, this thesis would not have been possible.

First, I am grateful for the opportunity to have conducted this research under the supervision

of Dr. Don Jackson. I have learned a great deal about a great many things from Don, including:

multivariate statistics, sampling aquatic systems and natural ecology. Don was always willing to

share his thoughts and ideas, leave his door open door for questions, and let me steal a cup of

coffee; you couldn`t ask for a better combination. I am also very thankful for the mentorship of

Dr. Harold Harvey. Harold is my academic grandfather, a mentor and I hope to say friend. I am

thankful for Harold’s advice, his stories, and providing laughter and support. I will greatly miss

our daily conversations (and the recession cookies). I hope one day I am half as wise as Harold

is. I am deeply indebted to Dr. Nicholas Mandrak for all his help and advice. Not only was Nick

my M.Sc. supervisor, he was a member of my Ph.D. academic committee, and he was integral

part from the start of my thesis. Nick’s insights into biology of fishes are second to none, and I

owe a great deal of gratitude for all his mentorship throughout the years. Nick was also

instrumental in helping to obtain funds (IRF #1410) to keep aspects of this project going; which

saved the project. I am also greatly indebted to my academic committee members: Dr. Marie-

Josee Fortin, Dr. Brian Shuter and Dr. Keith Somers, whose expertise and knowledge greatly

improved this thesis and my own research. All of my academic committee members were

wonderful in guiding me through their areas of expertise, from spatial ecology (Thanks Marie-

Josee), to fisheries techniques (Thanks Brian), to multivariate statistics and bioassessments

(Thanks Keith). Finally, I am thankful to Dr. Bill Matthews for acting as my external examiner.

Bill has always been one of my academic idols, and it was an honor and pleasure to have him as

an external. I will never forget it. From my advisor, to my academic committee, to those

involved on the appraisal/defense; this thesis was clearly built on the shoulders of giants.

They say it takes a village to raise a child, for this thesis, it took a small army. None of the

field work would have been possible without the dedication and support of many people. In

iv

particular I wish to thank Andrew Drake (aka D.A.R. Rubington III) and Cavan Harpur. So

frequently I’d found myself in dire straits, without field crew or transportation, a rag tag

assemblage of gear, missing GPS coordinates, lack of fish (or fill in any number of other

circumstances); yet Andrew and Cavan were always there to help out and make it work. Their

perseverance and friendship turned what should have been an unquestionable failure into what I

hope is a great success. I can’t thank them enough.

An enormous thank you goes to the Toronto Region Conservation Authority, and in

particular the Aquatics group: Christine Tu, Dave Lawrie, Trevor Parker, and Tim Rance.

Through a simple handshake, we became partners, and their help and support ensured that much

of this research would get done. When no-one was there to help, TRCA was always ready to

come to the rescue. Regardless of the obstacles, their support never faltered and I am so happy

that we worked together. Dave Lawrie deserves special recognition for being the absolute best

colleague you could ask for. Dave was always willing to give you the shirt off his back, I know

of few people with his dedication and passion for preserving aquatic species.

Too all the volunteers who helped sample and collect data; thank you. Of course, sampling

endangered species comes with its own set of surprises. The fire ants were my favorite. My guess

is that Dave was right when he observed a negative correlation between returning volunteers and

fire ants; yet so many of you returned for multiple feedings. So thank you all the dedicated

volunteers from- the University of Toronto - Andrew Drake, Cavan Harpur, Brie Edwards, Meg

St. John, Maggie Neff, Paul Venturelli, Alex Manning, Monica Granados, Jonathan Ruppert,

Sapna Sharma, Nicole Puckett, Cristal Hart, Chris Howard, Moe Luksenberg, John Brett, Don

Jackson, Steve Walker – and the Toronto Region Conservation Authority - Dave Lawrie,

Christine Tu, Trevor Parker, Tim Rance, Brennan Paul, Elyssa Elton, Brad Stephens, Cristal

Hart, Laura DelGiudice, Brian Moyle, and Maria Parish – And: Peter Ng, Michell Wong Ken,

Connie Zehr, Bev Edwards, Yuko Nozoe, Derek Trim, Kenny Lee, Kari Jean (ABCA), Davin

Heinbuck (ABCA), Doug Forder (Ontario Streams), Daniel Morodvanschi, and Kenny Lee.

I also wish to thank the Jackson lab past and present (Sapna Sharma, Steve Walker, Maggie

Neff, Meg St.John, Brie Edwards, Monica Granados, Riku Pavola, Lifei Wang, Jean Bernard

Caron, Angela Strecker, Karen Wilson, and Theo Willis), and Harvey labs (Andrew Drake,

Cavan Harpur) for their enormous support of my project. Most (if not all) members of these labs

participated in some form or another in my project, and if not, their influence from discussions

can be found throughout the following pages. In particular thanks to Steve Walker for his help in

v

developing tests for measuring sensitivity in functional diversity. I never thought it would take as

long or work out nearly as well as it did; my sincere thanks to Steve.

I thank the following agencies and individuals for provided data for portions of this research:

Royal Ontario Museum (ROM), Fisheries and Oceans (DFO), Ontario Ministry of Natural

Resources (OMNR), Toronto Region Conservation Authority (TRCA), Conservation Halton

(CH), Lower Lake Simcoe Conservation Authority (LLSCA), Credit Valley Conservation, Dr.

Nicholas E. Mandrak (DFO), Doug Forder (Ontario Streams), Jeff J. Anderson (LLSCA), Les

Stanfield (OMNR), Erling Holm (ROM), David Lawrie (TRCA), Scott Jarvie (TRCA), Sherwin

Watson-Leung (CH), Dr. Scott Reid (OMNR), John Pisapio (OMNR), and Mark Heaton

(OMNR).

To my extended academic family, the department of Ecology and Evolutionary Biology,

thank you. Dr. Lisa Manne was very kind in providing desk space at UTSc in the later stages of

my thesis for writing. I am ever so grateful as it saved me many hours of commuting and kept me

close to my family (who also thank you). I also really enjoyed coming in and talking ecology

with her and her lab (Caroline Tucker, Christopher Grouios, and all the undergraduate students).

Dr. Helen Rodd has also been a great sounding board for ideas and has been wonderfully

supportive. Dr. Locke Rowe and Dr. Spencer Barrett provided laboratory space for imaging and

genetics work, which didn’t make its way into the thesis, but will hopefully be out in publication

shortly and I thank them nonetheless. Bill Cole and Jen Perry were also wonderful in helping me

navigate through their worlds of genetics and imaging.

To my friends and family ... thanks. Mom, dad, Nicole, Dave, Mau, Evan, Hayley, Shannon,

my extended family, and friends; thanks for keeping me sane and always bringing a smile to my

face. Last but not least, none of this would be possible without the constant love and

encouragement from my wife Jessica and son Jacob. My absolute favourite part of everyday is

coming home to see you. Thank you just isn’t enough. Thanks for accepting the fish smell in the

car (you just can’t get that stink off), for missing your own events, or getting dropped off

early/late to accommodate mine, for the missed evenings and weekends, the added stress, the

crappy pay, the long hours, and everything else. To Jacob, I hope one day you can take some

inspiration from these pages. Not for its content (I would never do that to you), but from reading

in-between these lines and noting a worthy lesson: that even when something is difficult,

perseverance and hard work can help you find your way through any blind alley.

vi

Table of Contents Acknowledgments .......................................................................................................................... iii 

Table of Contents ........................................................................................................................... vi 

List of Tables .................................................................................................................................. x 

List of Figures ............................................................................................................................... xii 

List of Appendices ....................................................................................................................... xvi 

Chapter 1 General Introduction ...................................................................................................... 1 

I) Impacts of Methodological Choices in Conservation-Based Models .................................... 3 

Functional Diversity ............................................................................................................ 4 

Bioassessments ................................................................................................................... 5 

Metapopulation Viability Models ....................................................................................... 6 

II) Using Consensus Methods to Reduce Uncertainty from Methodological Choices .............. 7 

Scope .......................................................................................................................................... 8 

Statement of Contribution .......................................................................................................... 9 

Publication of Thesis Material ................................................................................................... 9 

References ................................................................................................................................ 10 

Section I: ...................................................................................................................................... 18 

The importance of methodological choices in ecological models ................................................ 18 

Chapter 2: The importance of methodological choices in influencing the measure of functional diversity across ecological communities ..................................................................... 19 

Abstract .................................................................................................................................... 19 

Introduction .............................................................................................................................. 20 

Methods .................................................................................................................................... 22 

Results ...................................................................................................................................... 26 

Discussion ................................................................................................................................ 29 

vii

Acknowledgements .................................................................................................................. 32 

References ................................................................................................................................ 33 

Appendices ............................................................................................................................... 36 

Chapter 3: Addressing the removal of rare species in bioassessments with other choices in multivariate analyses ..................................................................................................................... 43 

Abstract .................................................................................................................................... 43 

Introduction .............................................................................................................................. 44 

Methods .................................................................................................................................... 47 

Evaluating Decisions in Multivariate Bioassessments ..................................................... 48 

Assessing the Statistical Argument ................................................................................... 51 

Assessing the Biological Argument .................................................................................. 52 

Results ...................................................................................................................................... 54 

Discussion ................................................................................................................................ 58 

Acknowledgements .................................................................................................................. 61 

References ................................................................................................................................ 61 

Appendices ............................................................................................................................... 71 

Chapter 4: Contrasting direct versus indirect dispersal in metapopulation viability analyses ..... 73 

Abstract .................................................................................................................................... 73 

Introduction .............................................................................................................................. 74 

Methods .................................................................................................................................... 77 

Determining Metapopulation Viability ............................................................................. 78 

Colonization Rate: ............................................................................................................ 78 

Extinction Rate: ................................................................................................................ 79 

Incorporating Dispersal Directly into the Metapopulation Model: ................................... 80 

Regional Stochasticity ...................................................................................................... 81 

Comparing Viability of Metapopulations Using Direct versus Indirect Parameterization ................................................................................................... 81 

viii

Results ...................................................................................................................................... 82 

Metapopulation Dynamics ................................................................................................ 82 

Metapopulation and Patch Viability ................................................................................. 83 

Discussion ................................................................................................................................ 87 

Acknowledgements .................................................................................................................. 92 

References ................................................................................................................................ 92 

Appendices ............................................................................................................................. 100 

Section II: ................................................................................................................................... 102 

Reducing uncertainty from methodological choices using consensus methods ......................... 102 

Chapter 5: Reducing uncertainties in modeling the distribution of endangered species using habitat-based ensemble models ................................................................................................... 103 

Abstract .................................................................................................................................. 103 

Introduction ............................................................................................................................ 104 

Methods .................................................................................................................................. 106 

Study Area and Species .................................................................................................. 106 

Building Individual Models ............................................................................................ 109 

Evaluating Individual Models ......................................................................................... 113 

Building Ensemble Models ............................................................................................. 113 

Evaluating Ensemble Models ......................................................................................... 114 

Results .................................................................................................................................... 115 

I) Individual Models ....................................................................................................... 115 

II) Ensemble Models ....................................................................................................... 117 

Discussion .............................................................................................................................. 120 

Conclusion .............................................................................................................................. 123 

Acknowledgements ................................................................................................................ 123 

ix

References .............................................................................................................................. 124 

Appendices ............................................................................................................................. 131 

Chapter 6: Using consensus methods to identify (and reduce) sensitivity from methodological choices when measuring functional diversity ............................................................................. 136 

Abstract .................................................................................................................................. 136 

Introduction ............................................................................................................................ 137 

Methods .................................................................................................................................. 139 

Using Consensus Methods to Identify Uncertainty when Measuring FD ...................... 140 

Results .................................................................................................................................... 141 

The Relationship between FD, Distance Measure & Clustering Algorithm .................. 141 

Identifying Sensitivity in FD Using Consensus Methods ............................................... 144 

Discussion .............................................................................................................................. 144 

Acknowledgements ................................................................................................................ 147 

References .............................................................................................................................. 147 

Appendices ............................................................................................................................. 152 

Chapter 7: General Conclusions ................................................................................................. 153 

A) Conservation-based Models in General ............................................................................ 153 

B) Functional Diversity .......................................................................................................... 154 

Recommendations .................................................................................................................. 157 

References .............................................................................................................................. 158 

x

List of Tables Table 2.1 – The maximum probability of FD sensitivity for five communities previously used to

examine FD (Petchey and Gaston 2007; Podani and Schmera 2006). The number of species and

the number of assemblage pairs tested are also shown. Data sources are as follows: A) Holmes

et al. (1979), B) Munoz and Ojeda (1997), C) Jaksic and Medel (1990), D) Golluscio and Sala

(1993), and E) Chapin et al. (1996). ............................................................................................. 29 

Table 3.1 – Summary of ordination techniques, similarity coefficients and exclusion of rarely

sampled species being compared. Abbreviations are indicated in parentheses and used in

subsequent figures and tables. All four approaches described in the “Exclusion of Rarely

Sampled Species” were used in each of the four “Similarity Coefficient” combinations with both

PCoA and NMDS. As Correspondence Analysis has the implicit chi-squared distance measure,

only the four approaches used in the “Exclusion of Rarely Sampled Species” were included in

that set of analyses. ....................................................................................................................... 49 

Table 3.2 –Partitioning of variation in sum of squared deviations of Procrustes analyses (m2

statistic) across various choices in multivariate analyses, including: i) removal of rare species; ii)

ordination technique; and, iii) choice of similarity measure. Abbreviations are those noted in

Table 1. ......................................................................................................................................... 56 

Table 4.1 – Summary of mark-recapture information of the endangered fish the reside dace

(Clinostomus elongatus) used to directly parameterize stochastic patch occupancy

metapopulation models. Shown are two locations in the Greater Toronto Area, Ontario, Canada:

A) Leslie Tributary, and B) Berczy Creek. Note: items denoted with a single asterisk (*)

represent a significant difference between populations (Mann-Whitney U for average dispersal,

G-test for stationary tags, p<0.05). ............................................................................................... 82 

Table 4.2 – Intrinsic mean time (in years) to extinction (Grimm and Wissel 2004) of two

metapopulations of the endangered fish the redside dace (Clinostomus elongatus) for different

levels of regional stochasticity. ..................................................................................................... 84 

Table 5.1 – Summary of the hierarchical habitat-based model used for predicting the presence of

endangered minnow, redside dace (Clinostomus elongatus). Seven predictive models were used

xi

including: LR (logistic regression), CT (classification trees), MARS (multivariate adaptative

regression splines), ANN (artificial neural networks), DA (discriminant analysis), RF (random

forest), and BR (boosted regression trees). Variables were derived using forward-selection

procedures on five independent datasets and are shown as a percentage of datasets where each

variable was selected (parentheses indicate negative associations). Variables selected from only

one dataset were omitted. A prioi predictions (ap) based on habitat predictors thought to

influence the decline of the species are shown for reference, where + indicates a positive

correlation, - negative correlation, 0 none. ................................................................................. 108 

Table 5.2 – A comparison of model sensitivity (correct classification of species presence),

specificity (correct classification of species absence), and overall classification (correct

classification of both species presence and absence) for redside dace (Clinostomus elongatus).

Single models are: LR (logistic regression); CT (classification trees); BR (boosted regression

trees); MARS (multivariate adaptive regression splines); ANN (artificial neural networks); DA

(discriminant analysis); and, RF (random forest). Ensemble forecasts are: consensus model

(CM); principal component analysis (PCA); weighted average using overall classification (WA);

mean (Mn); and, median (Md). ................................................................................................... 116 

Table 6.1 – Consensus measures of dendrogram group fidelity across distance measures

(Euclidean and cosine) for each clustering algorithm: single linkage, unweighted pair group

method with arithmetic means (UPGMA), and complete linkage. Group fidelity was determined

by majority rules consensus tress using CI(C) consensus index. ................................................ 144 

xii

List of Figures Figure 2.1 –Measuring the sensitivity of FD in a hypothetical eleven species community. The

procedure consists of randomly dividing the community into two assemblages, noting how FD

orders the two assemblages and assessing how sensitivity effects this order (i.e. did assemblage 1

always have higher FD given differences in methodology?). Each species is represented as a

letter and the assemblages are represented as overlapping rectangles that contain the letters

associated with their component species. The first set of rectangles represents one random

division of the community into two assemblages. A new random division can be obtained by

leaving the overlapping rectangles fixed and simply permuting the order of the species. The

second set of overlapping rectangles gives an example of such a permutation. ........................... 24 

Figure 2.2 - The effect of alpha and beta diversity on the probability of FD sensitivity for three

communities crossed with four sets of construction methods. Darker shading represents a higher

probability of sensitivity. The communities are A) Insectivorous birds (Holmes et al. 1979), B)

Intertidal fish (Munoz and Ojeda 1997) and C) Predatory vertebrates (Jaksic and Medel 1990).

Each column is for a different set of construction methods. For the first and second columns,

overall PS and PG, all nine methods of dendrogram construction (three distance measures times

three clustering algorithms) were used with the PS and PG methods respectively. For the third

column, all three clustering algorithms were used with Gower’s distance and the PS method. For

the fourth column, all three distance measures were used with UPGMA and the PS method. .... 27 

Figure 2.3 –The effect of alpha and beta diversity on the probability of FD sensitivity for two

communities crossed with two methods of FD calculation. Darker shading represents a higher

probability of sensitivity. The communities are D) Patagonian forbs (Golluscio and Sala 1993)

and E) Artic vegetation (Chapin et al. 1996). For these communities, only Gower’s distance

could be calculated and so only three construction methods could be compared, corresponding to

the three clustering algorithms. Each column is for a different method of FD calculation. The

first is for the PS method and the second is for the PG method. .................................................. 28 

Figure 3.1 – Example of rank-ordered, site-level vector residuals of Procrustean multivariate

comparison. The length of a vector residual indicates an overall lack of fit for a site between two

multivariate analyses. Shown is a comparison of full dataset of Principal Coordinates with

xiii

Jacaard’s distance and the same dataset where species occurring at 5% of sites were removed.

Vectors shown in grey indicate those sites where at least one species was removed, whereas

vectors in black indicate sites where no species were removed. The ratio of mean vector

residuals between sites where species were removed versus those sites that did not have species

removed indicates the distribution of impacts of the removal of rare species across multivariate

analyses. Where most vector residuals for sites having species removed are largest, they indicate

that these observations (sites) have been changed the most in their position between two

ordinations. .................................................................................................................................... 53 

Figure 3.2 –Principal Coordinates Analysis (PCoA) of the sum of squares deviations (m2

statistic) comparing the concordance between solutions based on different ordination techniques,

similarity coefficients and treatments of excluding rarely sampled species. A minimum-spanning

tree was overlaid on Axes 1 and 2 to highlight connections between groups of points. Dashed

lines indicate deviations from group membership in cases where clear groupings do not exist

(e.g. M10 for Axes 2 and 3). Short forms are continued from Table 3.2. .................................... 55 

Figure 3.3 – Site-level impact of the removal of rare species. Shown are box and whisker plots of

the ratios of Procrustes vector residuals between sites for which rare species were removed and

those sites that did not have any species removed. All comparisons were done by comparing

site-level Procrustes vector residuals from the full datasets and with the removal of rare species

across all similarity coefficients and ordination methods. ............................................................ 57 

Figure 4.1 – Study sites on Rouge River, Ontario where redside dace (Clinostomus elongatus)

dispersal and patch dynamics were monitored. Study locations: A) Leslie Tributary, and B)

Berczy Creek, were sub-divided into extensive sites (black), where redside dace were tagged

with a color-coded visual implant elastomer tag, and extended sites (grey), which were

monitored for tag movement. ........................................................................................................ 76 

Figure 4.2 – Metapopulation viability of the endangered species the redside dace (Clinostomus

elongatus) in two stream metapopulations: A) Leslie Tributary, and B) Berczy Creek. Shown are

the probabilities of extinction (y-axis) in years (x-axis) of a stochastic patch-based

metapopulation model. Models were parameterized using: I) indirect parameterization of

colonization and dispersal via patch distance, and; II) direct parameterization of colonization and

dispersal using empirical estimates from a mark-recapture study. Legend: Vertical hashes

xiv

represent a time interval of 100 years, solid lines indicate population trajectories where regional

stochasticity was set to 0, dashed lines set to 0.1 and dotted lines set at 0.2. ............................... 85 

Figure 4.3 – Differences in patch viability parameterized using indirect (y-axis) and direct (x-

axis) patch dynamics of the endangered species the redside dace (Clinostomus elongatus) in two

stream metapopulations: A) Leslie Tributary (L6-L15), and B) Berczy Creek (B6-B18). Shown

are the mean probabilities of persistence of a given patch across 10,000 simulations. To

demonstrate the variability in patch viability, 25% quantiles are overlaid as the negative of both

the vertical and horizontal axes, while 75% quantiles are overlaid as the positive vertical and

horizontal axes. The dashed line is a 1:1 line. .............................................................................. 86 

Figure 5.1 – Distribution of sampling locations between 1997-2007. Closed circles indicate

redside dace occurrences, whereas, open circles indicate where redside dace were absent. ...... 107 

Figure 5.2 .................................................................................................................................... 110 

Figure 5.3- Cluster analysis showing the relationship with the observed distribution (Obs.) of

redside dace (Clinostomus elongatus) and: A) the seven individual modeling approaches alone,

and B) with ensemble forecasts included. Model short forms are carried over from Table 5.2. 117 

Figure 5.4 – Box and whisker plots showing the variability in consensus ensemble forecasts for

predicting the presence of an endangered redside dace. Consensus ensemble models were

compared across all combinations of one (n =7), three (n =35), five (n =21) and seven (n=1)

input models (x-axis). Boxes are 25th and 75th percentiles, horizontal lines indicate the median,

vertical lines indicate the upper and lower values, diamonds indicate the mean and are connected

by dashed lines. Modeling metrics were: A) model sensitivity (i.e. correct classification of

species presence); B) model specificity (i.e. correct classification of species absence); and, C)

overall classification. Dashed lines indicate 95% confidence intervals. .................................... 119 

Figure 6.1 - The relationship between distance measure (Euclidean or cosine) and clustering

algorithm (SL = single linkage / nearest neighbor, UPGMA = unweighted pair group method

with arithmetic mean, CL = complete linkage / farthest neighbor) with FD using five community

data sets: A) Arctic vegetation (Chapin et al. 1996); B) Insectivorous birds (Holmes et al. 1979);

C) Patagonian forbs (Golluscio and Sala 1993); D) Intertidal fish (Munoz and Ojeda 1997); and,

E) Predatory vertebrates (Jaksic and Medel 1990). FD values were re-scaled relative to the

xv

Arctic vegetation data, which has the highest FD values. This standardization leads to the

appearance of a constant outcome for the Arctic dataset, but this consistency is solely an artifact

of using it as the reference point rather than the outcomes not differing depending on the

resemblance measure or clustering algorithm. ............................................................................ 142 

Figure 6.2 - The relationship between distance measure (solid lines = Euclidean distance, dashed

lines = cosine distance) and building a dendrogram using a clustering algorithm (1 = complete

linkage / farthest neighbor, 2 = unweighted pair group method with arithmetic mean / UPGMA,

3 = single linkage / nearest neighbor) where species are individually removed when calculating

FD. Five community data sets are shown: A) Arctic vegetation (Chapin et al. 1996), B)

Insectivorous birds (Holmes et al. 1979), C) Patagonian forbs (Golluscio and Sala 1993), D)

Intertidal fishes (Munoz and Ojeda 1997), and E) Predatory vertebrates (Jaksic and Medel 1990).

Shown inset are 50% majority rule consensus trees demonstrating lack of between group fidelity

of species where calculating functional diversity using different distance measures, but the same

clustering approach. .................................................................................................................... 143 

xvi

List of Appendices Appendix 2.1 – Derivation of Equation 1. .................................................................................... 36 

Appendix 2.2 – MatLAB Code for testing sensitivity of FD ........................................................ 38 

Appendix 2.3 – MatLAB Code for calculating FD via Podani and Schmera ............................... 41 

Appendix 2.4 – MatLAB Code for calculating probabilities of sensitivity .................................. 42 

Appendix 3.1 – Site-level effects of methodological choices in bioassessments. Shown are the

ratios between mean site-level vector residuals from Procrustes analyses of sites having species

removed and those sites having no species removed. Mean site-level vector residual values were

separated by sites which had rare species removed (M1: n=2; M5: n=19; and M10: n=63); and

compared with those sites that not. ............................................................................................... 71 

Appendix 3.2 – Summary of three-dimensional ordination results. Shown are eigenvalues for

Principal Coordinates Analyses (PCA) and Correspondence Analyses (CA), with percent

variance explained shown in parentheses. Stress values are shown for Non-Metric

Multidimensional Scaling (NMDS). ............................................................................................. 72 

Appendix 4.1 –The endangered redside dace (Clinostomus elongatus). Photo credit: Mark Poos.

..................................................................................................................................................... 100 

Appendix 4.2 –Visual implant elastomer (VIE) tag inserted subcutaneously on the ventral surface

of the endangered redside dace (Clinostomus elongatus). Photo credit: Mark Poos. ................. 100 

Appendix 4.3 – Model parameters used in the stochastic patch-occupancy models. Shown are

the number of emigrants (mi), the mean probability of detection (PD), the number of emigrants

adjusted for potentially missed tags (Mi), the number of immigrants needed to start a new sub-

population, and the rate of extinction (Ei). Other parameters include: the incidence function for

Leslie Tributary (dI = 210, x = 0.4926, e = 3.685, y = 6.12), and Berczy Creek (dI = 150, x =

0.5652, e = 4.187, y = 7.01). Not shown: dij given it is a pairwise estimate rather than unique for

each pool. .................................................................................................................................... 101 

xvii

Appendix 5.1 – Model metrics for all combinations of consensus ensemble models. Models are:

LR (logistic regression), CT (classification trees), MARS (multivariate adaptive regression

splines), RF (random forest), ANN (artificial neural networks), BR (boosted regression trees, and

DA (discriminant analysis). ........................................................................................................ 131 

Appendix 5.2 – R Code for testing configurations of 1,3, 5 and 7 prediction consensus models

..................................................................................................................................................... 133 

Appendix 6.1 – MatLAB Code for calculating total branch lengths of dendrograms from various

species combinations .................................................................................................................. 152 

1

Chapter 1 General Introduction

“All models are wrong, some models are useful” George Box, Statistics Professor

The overall aim of ecology is to broaden the understanding of the relationship between species

and their environment (Krebs1998). This objective can be achieved through a myriad of ways

from broadening the understanding of the structural components of an organism (e.g. molecular

or cellular biology, physiology, evolutionary ecology), to understanding specific aspects of the

organism itself (e.g. behavioral or evolutionary ecology, genetics), to understanding the interface

between the organism and its biotic and abiotic environment (e.g. community or population

ecology). Due to the complexity involved in each of the disciplines, and regardless of how

ecologists view the world, simplification of natural processes are needed and a statistical model

is often required.

Statistical models provide the framework for an interpretation of nature. A statistical model is,

by definition, merely a mathematical representation of some aspect of nature (Quinn and Keough

2002) and can take many forms. Models can be predictive in that they can be used to forecast,

among other things, species distributions (e.g. Araújo and Guisan 2006; Elith et al. 2006), habitat

importance (e.g. Olden and Jackson 2001; Guissan and Thuiller 2005), or potential changes in

the environment and species response (e.g. climate change; Thuiller 2004; Araújo et al. 2005).

Models can also be used to test hypotheses, e.g. is environment A better than B for a given

species (e.g. Matthews et al. 1992; Grossman et al. 2002; Skyfield and Grossman 2008)? Lastly,

models can be strictly informative, such as what is the current condition of environment A or

species B (e.g. Grossman 1984; Mathews et al. 1982; 1986)? In all cases, models are assumed as

useful if they can provide a realistic representation of the underlying natural world (Hilborn and

Mangel 1997). However, many factors may influence models and understanding methodological

choices may provide improved understanding of how ecologists interpret nature.

The history of statistical methods has demonstrated that understanding the impact of

methodological choices in today’s modeling approaches is sorely needed. Prior to the

development of modern computers, ecologists opted for models where the structure was

arbitrarily simplified (Quinn and Keough 2002). As such, there were relatively few

2

methodological choices, as data were often ‘massaged’ to fit the statistical approach or,

alternatively, practitioners used their favorite method ad-hoc (Jackson 1993). Now, with

complex statistical software and powerful computers readily available, ecologists have a plethora

of approaches to choose from and many methodological choices to make.

Understanding how methodological choices may impact results is an area of active and ongoing

research (Jackson et al. 1989; Hilborn and Mangel 1997; Grossman et al. 2006). Over the last

decade alone, methods for the analysis of species distribution have diversified and, at this point,

dozens of different approaches are available (Guisan and Zimmermann 2000; Guisan and

Thuiller 2005; Elith et al. 2006). Previous studies have demonstrated that methodological

choices made at one level of analysis may have cascading impacts on subsequent choices

(Jackson et al. 1989; Grossman et al. 1991; Dormann et al. 2008). For example, Dormann et al.

(2008) demonstrated that changes in both data quality and choice of model type had large

impacts for modeling the distribution of the critically endangered bird the Great Grey Shrike

(Lanius excubitor). Further, in a review of modeling species distributions with presence only

data, Elith et al. (2006) showed large differences in predictive performance among modeling

methods at both regional and species levels. These examples highlight that, despite substantial

effort in improving statistical methods for conservation-based species, there remains great

uncertainty regarding the impacts of methodological choices.

There are numerous debates in ecological literature regarding methodological choices in diverse

areas such as how to measure functional traits to account for ecosystem function (Petchey and

Gaston 2002; Podani and Schmera 2006), the impact of including rare species in biological

assessments (Cao et al. 1998; Marchant 1999), and whether species-specific dispersal can be

measured using distance-based functions (Heinz et al 2005; 2006). Given the choices that

ecologists must make, knowing the pitfalls and assumptions of conservation-based models

become increasingly important. For example, ecologists should know the impact of

methodological choices given the sampling design and kind of data that they wish to collect

(Quinn and Keough 2002).

My thesis attempts to address issues in methodological choices in conservation-based models in

two ways. In the first section of the thesis, the impacts of methodological choices inherent in

conservation-based models will be examined across a broad selection of available approaches

3

from measuring functional diversity (Chapter 2), to developing bio-assessments in community

ecology (Chapter 3), to assessing dispersal in metapopulation analyses (Chapter 4). It is the goal

of this section to illustrate the potential for methodological choices to impact conservation-based

models, regardless of the scale, study system or species involved.

For clarity, a conservation-based model refers to a model used to interpret/advance knowledge of

species with conservation concern (e.g. conservation designation, rare, or declining). These

models can include population viability analyses, models of habitat suitability or models of

ecological function. A methodological impact refers to a difference in conclusions (e.g.

population viability, habitat suitability, and ecological function) derived from methodological

choices that alter ecological interpretation and/or lead to differences in recovery/management.

Species with conservation concern refers to those which occur infrequently, have a conservation

status (e.g. endangered or threatened) or are declining (i.e. undergoing reductions in population

size but do not have a conservation designation). As there are many forms of rarity (Rabinowitz

et al. 1986), the definition of conservation-based models and species with conservation concern

used in this thesis is purposefully inclusive to allow for the greatest possible impact on

conservation applications. In the second section of this thesis, the use of consensus methods will

be demonstrated as a potential tool for reducing uncertainty with methodological choices in

conservation-based models. Two separate applications of consensus methods will be

highlighted, including how consensus methods can reduce uncertainty from choosing a modeling

type (Chapter 5) and can be used to identify when methodological choices may be a problem

(Chapter 6).

I) Impacts of Methodological Choices in Conservation-Based Models

To some researchers, there is perhaps no greater goal of ecology than to preserve the biological

diversity of species (Ehlrich and Wilson 1991). The loss of biological diversity through human

impact is of concern for the structure of ecological communities and its affects may, in turn,

affect the structure of ecosystems (e.g. keystone species). Current rates of decline of species

worldwide are thought to be hundreds to thousands of times higher than pre-human levels (Jelks

et al. 2008). In particular, freshwater systems may have rates of decline many times higher than

terrestrial systems (Ricciardi and Rasmussen 1999). With the rate of species decline increasing,

the need for conservation-based models that reflect biological phenomena is becoming timely.

4

There are many methodological difficulties with developing conservation-based models and

understanding these challenges may help improve species management. First, species that are of

conservation concern suffer from low sample sizes and spatially fragmented distributions,

thereby making accurate enumeration difficult (Green and Young 1993) and traditional statistical

approaches often invalid (Ellison and Agrawal 2005). As a result, models fitted to a response of

species with conservation concern usually have low power or high uncertainty (Cunningham and

Lindenmayer 2005). Further, in many multivariate methodologies, researchers down weight or

exclude rare species altogether (Gauch 1982), thereby reducing or eliminating their usefulness

for conservation applications.

The role of methodological choices has been debated in several types of conservation-based

models. In the following section a few ecological examples will be highlighted for further

analysis in this thesis.

Functional Diversity

Given the current loss of biodiversity, it is important to be able to model how functional diversity

will respond to species loss. Quantification of the relationship between species loss and

functional diversity is necessary to highlight unique species traits that may be lost and their

implication to ecosystem function (Srivastava and Vellend 2005). Extinction should not affect

overall ecosystem function if all species have similar functions, but it will have a major effect if

each species has a different function (Fonseca and Ganade 2001). Simulations using natural

communities have shown that the loss of species reduces functional diversity disproportionately

relative to the number of species (Petchey and Gaston 2002; Larsen et al. 2005). These findings

are in agreement with empirical studies noting the loss of rare species disproportionately impacts

ecosystem function (Hooper et al. 2005).

One of the current metrics of functional diversity (sensu Petchey and Gaston 2002) uses the total

branch length of a dendrogram (tree) constructed from a matrix of ecological traits. This method

requires three steps before the total branch lengths can be calculated, including: 1) obtaining a

trait matrix; 2) converting the trait matrix into a distance matrix; and, 3) clustering of the

distance matrix to produce a dendrogram. Petchey and Gaston (2002) have suggested that the

particular analytical methods used to produce the functional dendrogram are of limited relevance

5

to the resultant metric as the relationship between functional diversity and species richness is

robust to changes in distance metric and clustering algorithm. However, previous research has

indicated that changes in distance metric have dramatic effects on clustering dendrograms

(Jackson et al. 1989; Jackson 1993) and, similarly, choice of clustering algorithm (Legendre and

Legendre 1998) can alter biological conclusions.

The role of methodological choices for altering functional diversity (FD; Petchey and Gaston

2002) has only been recently debated (Petchey and Gaston 2006, 2007; Podani and Schmera

2006, 2007; Poos et al. 2009). Functional diversity is an important component of conservation-

based models as researchers are often interested in the amount of functional variation with an

ecosystem (Loreau et al. 2001). Comparisons of the amount of functional variation that is lost

when species have been removed from an ecosystem has demonstrated that rare species play an

important role for maintaining ecosystem function (Fonseca and Ganade 2001; Petchey and

Gaston 2002b; Schmera et al. 2009). In Chapter 2, the importance of methodological choices on

estimating functional diversity is examined. In particular, Chapter 2 highlights whether how

functional diversity is measured can alter ecological insight.

Bioassessments

Similar to functional diversity, there has been a dramatic increase in the amount of

bioassessment literature in the past few decades (Bailey et al. 2004). Bioassessments have been

used by managers for decades to evaluate communities undergoing anthropogenic impact and

species decline (Green 1979; Barbour et al. 1999). Debates regarding the impact of

methodological choices on the assessment of biological communities (i.e. bioassessment) are

varied and include criticisms of multi-metric approaches (Hannaford and Resh 1995; Wallace et

al. 1996; Bowman and Somers 2006); the development of multivariate and predictive models

(Bailey et al. 2004), taxonomic resolution (Somers et al. 1998; Hewlett 2000), improvements to

rapid methods (Hannaford and Resh 1995; Somers et al. 1998), redundancy in metrics (Barbour

et al. 1992), and improvements to the study design of reference conditions (Norris and Thoms

1999; Bowman and Somers 2005; Bailey et al. 2009).

One ongoing debate in bioassessments is the importance of rare species (Cao et al. 1998; 2001;

Marchant 1999; Marchant 2002). Methods for identifying trends in multivariate analyses which

6

include rare species are often limited because they typically represent points of unusually high

leverage in the analysis. When rare species (i.e. species that occur infrequently) are included in

multivariate analyses, they often alter the analysis because, relative to more common species,

they over-fit an association of relatively few occurrences to a specific habitat type that, in turn,

unduly influences species-habitat differences (Legendre and Gallagher 2001). To compensate

researchers either down weight or exclude rare species from multivariate analyses without

consideration of the degree of influence they may have on the underlying biological data (Gauch

1982). Obviously, the benefits of this decision are negated when the species or relationship of

interest relates to a species that is rare, as is the case with many conservation applications.

Imperiled species (i.e. species with a conservation status), like rare species, are scarce and often

found in small numbers in specific habitat types (Mace 1994). In Chapter 3, the impact of

removing rare species is assessed relative to other multivariate methodological choices, such as

the choice of distance measure and multivariate method, which are all common to multivariate

bioassessment approaches.

Metapopulation Viability Models

Trajectories of populations through time are often needed to extrapolate population viability.

Approaches such age or stage based matrix population models require parameterization of

several species life history characteristics (e.g. fecundity, age/size of maturation), which may be

limited or unknown for species with conservation concern. As such, simplifications in how many

life history characteristics are needed to model population trajectories are needed.

Metapopulation viability analyses offer simplification in over traditional population viability

analyses as they only require parameterization of patch occupancy and estimates of rates of

colonization and extinction (Hanski 1999). One popular method of metapopulation viability

analysis is stochastic patch occupancy models (SPOM, Hanksi 1999; Moilanen 1999; 2004).

Stochastic patch occupancy models provide a spatially realistic model for patch dynamics as they

incorporate estimates of patch location and species dispersal into the modeling procedure. For

example, estimates of patch colonization (i.e. “reachability”) are quantified using the distance

between a starting patch and a target patch, and the ability of species to disperse (Hanksi 1994;

Hanski et al. 1996; Heniz et al. 2005). Most often, this measure is estimated by assuming that

colonization potential declines exponentially with distance (i.e. exponential decay; Hanski 1994;

7

Vos et al. 2001; Frank and Wissel 2002). However the ability of how well even simple formulae

are able to model patch dynamics remains an open question (Heinz et al. 2006; Marsh 2008).

Chapter 4 aims to better understand the impact of estimating population viability using direct

versus indirect parameterization of dispersal in metapopulation models.

II) Using Consensus Methods to Reduce Uncertainty from

Methodological Choices

Conservation-based models provide a useful tool for understanding consequences of species loss

(e.g. functional diversity), anthropogenic impacts (e.g. bioassessments) and the predictions for

population viability and habitat suitability. However, the difficulty with using the standard sets

of conservation-based models is that they are often inappropriate when analyzing data limited to

a few sites and/or species (Ellison and Agrawal 2005). This condition of sparse data, in turn,

contributes to many statistical limitations, including zero-inflated bias, increased collinearity

between variables and inflated coefficient of variation (Dixon et al. 2005; Edwards et al, 2006;

Dormann et al. 2008). However, uncertainties may arise during all stages of modeling such as

obtaining species level data obtaining accurate species counts, and linking species to

landscapes/habitats (Burgman et al. 2005; Cunningham and Lindenmayer 2005; Edwards et al.

2005; Elith et al. 2006). Therefore, developing methods that may reduce impacts from

methodological choices may enhance the utility of conservation-based models.

The use of consensus methods has been highlighted as one way to address problems related to

using conservation-based models (Araújo and New 2007; Marrimon et al. 2009). Laplace

(1820) demonstrated that the probability of error will decline as more models are included (i.e.

an ensemble or consensus approach). This old idea has only recently been applied to problems

of uncertainty with the influential work of Bates and Granger (1969) that in part, contributed to

the award of the Nobel Memorial Prize in Economics to Clive Granger in 2003. Recently,

ecologists have applied consensus models to issues with uncertainty in climate-based predictions

(Thuiller 2004; Araújo et al. 2005; Thuiller et al 2005; Buisson and Grenouillet 2009) and, to

some extent, to functional diversity (Mouchet et al. 2008). Yet, the application of consensus

methods seems due to other problems in ecology.

8

The application of consensus methods to resolve issues of methodological choices is in its

infancy. Issues such as how to build the best ensemble model have yet to be quantitatively

evaluated. The second section of the thesis applies consensus models to impacts from

methodological choices including whether consensus methods can improve prediction over

several singular methodological approaches (Chapter 5) and whether consensus methods can

identify instances where methodological choices may be an issue as in the measure of functional

diversity (Chapter 6). In Chapter 5, a quantitative evaluation of consensus models is applied to

the habitat modeling of the endangered fish, Redside Dace (Clinostomus elongatus). Chapter 5

aims to evaluate the ability of consensus models to improve model performance (e.g. model

specificity, sensitivity and overall classification). In Chapter 6, consensus methods are used to

evaluate whether or not methodological choices are an issue for measuring functional diversity.

Scope

The difficulty with developing any model for conservation-based purposes are the linkages

needed across many facets of ecological knowledge, from how species traits relate to the

environment, to the types of environments undergoing impact. These relationships require

knowledge not only of the species, but each system, the suite of species found there, the

relationships between those species and their related habitats, and the relationship between each

habitat and large scale (e.g. landscape) and small scale (e.g. in-stream habitat) ecological

processes (Smith and Powell 1971; Matthews 1998; Jackson and Harvey 1989; Jackson et al.

2001). Therefore, there is no easy starting point (e.g. scale, method or species) for determining

modeling approaches appropriate for conservation-based issues.

The scope for identifying the influence of methodological choices in conservation-based models

is broad by its very nature. As such, this thesis is not restricted to a given study organism, study

system, data series, or methodology. Instead, data is used that best addresses the specific

questions and objectives of the various chapters. For example, in Chapters 2, 3 and 6 data is

from published studies to demonstrate the impact of methodological issues. In Chapter 4,

empirical information was gathered over the course of a one-year field study. In Chapter 5, over

420,000 fish records were gathered from various not-for-profit and government agencies,

universities and other academic institutions and studies.

9

In addition, this thesis examines different spatial and temporal scales for identifying

methodological choices. In Chapters 4 and 5, data was collected on dispersal and habitat

suitability of Redside Dace (Clinostomus elongatus). In Chapter 4, the spatial scale is the

province of Ontario and temporal scale are data points within the last decade. In Chapter 5, the

spatial scale is one watershed and the temporal scale is one year. These spatial and temporal

scales were chosen to highlight the approaches being compared (e.g. predictive models in

Chapter 4, and metapopulation models in Chapter 5). The goal of this thesis is that through the

consistency of the results within this thesis, across study organisms, study systems, data series,

methodologies, and spatial and temporal scales, it will provide ecological insight into making

methodological decisions in ecology. Each chapter is linked based on the influence of

methodological choices and an overall interest on how methodological choices can alter

conclusions in ecological studies.

Statement of Contribution

Parts of this thesis would not have been possible without the contribution of many collaborators.

Chapters 2 and 6 were completed in collaboration with Steve Walker, who helped in the

derivation of Equation 1 (Appendix 2.1), the development of ideas, and in writing code for

assessing the robustness of the functional diversity measure. Chapter 4 was completed with the

help of staff from the Toronto Region Conservation Authority, which provided valuable field and

logistical assistance. Chapter 5 was completed with the help of various governmental, non-

governmental and not-for profit agencies who contributed data.

Publication of Thesis Material

Chapter 2 has been published and can be found as:

Poos, M.S., S.C. Walker and D.A Jackson. 2009. Functional diversity indices can be

driven by methodological choices and species richness. Ecology 90(2): 341-347. (doi:

10.1890/08-1638.1)

Chapters 3 and 5 are currently submitted and undergoing peer review. All published material is

provided under approved copyright from the related journal.

10

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18

Section I:

The importance of methodological choices in ecological models

19

Chapter 2: The importance of methodological choices in influencing the measure of functional diversity across ecological communities

Abstract Functional diversity is an important concept in community ecology because it captures

information on functional traits absent in measures of species diversity. One popular method of

measuring functional diversity is the dendrogram-based method, FD. To calculate FD, a variety

of methodological choices are required and it has been debated about whether biological

conclusions are sensitive to such choices. We studied the probability that conclusions regarding

FD were sensitive, and that patterns in sensitivity were related to alpha and beta components of

species richness. We developed a randomization procedure which iteratively calculated FD by

assigning species into two assemblages and calculating the probability that the community with

higher FD varied across methods. We found evidence of sensitivity in all five communities we

examined, ranging from a probability of sensitivity of 0 (no sensitivity) to 0.976 (almost

completely sensitive). Variations in these probabilities were driven by differences in alpha

diversity between assemblages and not by beta diversity. Importantly, FD was most sensitive

when it was most useful (i.e. when differences in alpha diversity were low). We demonstrate

that trends in functional diversity analyses can be largely driven by methodological choices or

species richness, rather than functional trait information alone.

Keywords: multivariate statistics, functional diversity, community ecology, species richness,

biodiversity, ecological organization, dendrogram.

20

Introduction

Functional diversity is the amount of inter-specific variation in functional traits in an ecological

community. The concept of functional diversity has received considerable attention recently,

largely because of the following intuitive argument. Species diversity indices treat all species

identically, whereas functional diversity indices do not. Therefore, it is reasonable to expect that

functional diversity is likely to be more ecologically relevant because species differ from one

another in functionally important ways (Petchey and Gaston 2002). For example, several studies

have concluded that measures of ecosystem function tend to correlate more strongly with

functional diversity indices than with species diversity indices (Loreau et al. 2001). These

studies have spurred continued interest in developing new and improved functional diversity

indices (Mouchet et al. 2008; Villeger et al. 2008).

Despite the conceptual simplicity of functional diversity, ecologists wishing to measure it must

choose from a number of approaches. Mason (2005) developed a typology of functional

diversity indices with three types: functional richness; functional evenness; and, functional

divergence. This typology is similar in spirit to the distinction between species richness and

evenness in species diversity studies. For example, functional richness indices measure the

amount of trait space filled by the species in a community, whereas functional evenness indices

measure the evenness in the distribution of abundance in trait space (Mason 2005; Villeger et al.

2008). Using rarefaction techniques, functional richness and evenness can also be thought of as

extremes along a gradient of functional diversity indices (Walker et al. 2008). Rarefaction also

makes clear the close relationship between species and functional richness. It is therefore

important to ensure that accepted indices of functional richness provide information beyond that

of species richness, as data on functional traits can be costly to obtain.

One approach to measuring functional richness, which has shown promise as a predictor of

ecosystem function, is the dendrogram-based approach known as FD (Petchey and Gaston 2002).

This approach consists of measuring functional richness as the total branch-length of a

dendrogram that clusters species based on the similarity of their functional-trait characteristics.

There are numerous methods for constructing a dendrogram; in particular, both a resemblance

measure, which measures the difference between species in their functional-trait characteristics,

21

and a clustering algorithm, which specifies the manner in which similar species are grouped

together, must be chosen. There is the possibility that ecological conclusions drawn from an

analysis of dendrogram-based functional diversity may be sensitive to the methodological

choices that are required for producing a dendrogram. This may be a serious issue given that

dendrogram topology may change considerably with changes in the methods used (e.g. Sneath

and Sokal 1973, Jackson et al. 1989).

There has been considerable recent debate about the importance of the method of dendrogram

construction for the measurement of dendrogram-based functional diversity (Petchey and Gaston

2006, 2007, Podani and Schmera 2006, 2007; Mouchet et al. 2008). To facilitate resolution, we

conducted a detailed analysis of the sensitivity of dendrogram-based functional diversity

measures to differences in species richness and methodological choices. For this resolution, we

need a quantitative definition of sensitivity. Given a pair of species assemblages and set of

dendrogram construction methods, we make the following definition: conclusions are insensitive

if all construction methods result in the same assemblage being identified as having higher

functional diversity. Conclusions are sensitive if at least one construction method identifies a

different assemblage as having higher functional diversity.

With this definition, we seek answers to the following questions. First, through a systematic

study of previously analyzed data from ecological communities (Petchey and Gaston 2002,

Podani and Schmera 2006, Petchey and Gaston 2007), what is the probability that conclusions

regarding FD are sensitive to methodological choices? Second, if sensitivity is found to be likely

in many communities analyzed, is the probability of sensitivity related to the difference in local

species richness (i.e. alpha diversity) between the two assemblages? We hypothesize that the

probability of sensitivity should be low when differences in alpha diversity are very high.

Intuitively, we expect species richness to drive functional richness patterns in these cases, no

matter how it is measured. This is a null hypothesis; it assumes that functional richness (as

measured by FD) does not provide information beyond that provided by species richness.

Failure to reject this hypothesis would suggest that FD and alpha diversity are largely redundant,

provided that the most species rich assemblage also tends to have the highest FD. Third, is the

probability of sensitivity related to the amount of species turnover (i.e. beta diversity) between

assemblages? As species turnover can be measured in numerous ways, hereafter we use the

term beta diversity to refer to Lande’s species turnover (Lande 1996). We hypothesize that the

22

probability of sensitivity should be high when beta diversity is high. High beta diversity will

tend to lead to lower redundancy across traits between assemblages, in comparison to low beta

diversity. Therefore, high beta diversity produces conditions under which we intuitively expect

small differences in functional diversity. Small differences will presumably be more sensitive to

methodological choices. Fourth, is the probability of sensitivity related to certain types of

methodological choices? We hypothesize that conclusions will be more sensitive to the choice

of distance measure than to the choice of clustering algorithm, because the distance measure can

completely change the order of functional similarity amongst the species whereas the clustering

algorithm is more limited in that it can only alter how groups of species relate to one another in

multivariate space. We note that there are reasons to believe that FD will also be quite sensitive

to the choice of a clustering algorithm. Indeed, different clustering algorithms can generate quite

different tree topologies, which may translate into FD sensitivity. We address these questions by

assessing the probability of sensitivity of pairs of randomly drawn sub-assemblages from five

ecological communities.

Methods All of these analyses were based on data from ecological communities obtained from the

literature. We used the same five datasets used in previous studies of FD (Petchey and Gaston

2002, 2007, Podani and Schmera 2006). These datasets represent variation in the number (from

13 to 37) and type of species, and the number and type of functional traits (from 6 to 27). For

example, the three vertebrate datasets use characteristics ranging from foraging behavior to the

consumption of prey species as their functional traits (Holmes et al. 1979, Jaksic and Medel

1990, Munoz and Ojeda 1997), whereas the remaining two datasets rely on vegetative

characteristics, such as rooting depth and herbivore palatability, of the plants being studied

(Golluscio and Sala 1993, Chapin et al. 1996).

The general approach to assessing the sensitivity of FD to methodological choices was as follows

(see Fig. 2.1 for an example). For each community (i.e. dataset), we organized all of the species,

γ, into two groups, hereafter referred to as assemblages. Let the average species richness over

the two assemblages be α . Each species in the community was included in either one of the

assemblages or in both. For a given level of beta diversity,β = γ −α , and difference in alpha

diversity between the assemblages, Δα, the total number of unique pairs of assemblages is

23

. (1)

The numerator is the total number of ways that one can order γ species. The three factorials in

the denominator are, respectively, the total number of ways that (i) the number of shared species

can be ordered, (ii) the species that are unique to assemblage 1 can be ordered, (iii) the species

that are unique to assemblage 2 can be ordered (see Appendix 2.1). In Fig. 2.1, we give two

examples of such orderings when γ = 11, Δα = 1 and β = 4.5. Note however, that assemblage

pairs for which (2β – Δα) is an odd number are not possible given the inter-dependencies of these

parameters. For each possible combination of β and Δα, we randomly selected 1000 pairs of

assemblages using code programmed in MATLAB version 7.1 (see Appendices 2.1 – 2.4). For

each of these randomly selected assemblages, we calculated FD based on several different

dendrogram construction methods. FD was considered insensitive to methodological choices for

a particular pair of assemblages if the assemblage with the higher FD was the same for all

construction methods; FD was otherwise considered sensitive. We then calculated the

proportion of the 1000 random iterations that were sensitive. We refer to this proportion as the

probability of sensitivity. When the probability of sensitivity is high for a particular combination

of Δα and β, it is very likely that the conclusions drawn from an FD analysis in this context will

be dependent on methodological choices, rather than on the data alone.

In order to calculate FD, two methodological choices must be made. First, a distance (or

resemblance) measure must be chosen. Distance measures quantify the difference between two

entities based on their characteristics (e.g. species based on their functional traits). There are a

large number of resemblance measures from which to choose (Jackson et al. 1989; Legendre and

Legendre 1998). We used three distance measures: Euclidean distance as suggested by Holmes

et al. (1979); cosine distance; and, Gower’s distance as it allows mixed and missing data types

24

Figure 2.1 –Measuring the sensitivity of FD in a hypothetical eleven species community. The procedure consists of randomly dividing the community into two assemblages, noting how FD orders the two assemblages and assessing how sensitivity effects this order (i.e. did assemblage 1 always have higher FD given differences in methodology?). Each species is represented as a letter and the assemblages are represented as overlapping rectangles that contain the letters associated with their component species. The first set of rectangles represents one random division of the community into two assemblages. A new random division can be obtained by leaving the overlapping rectangles fixed and simply permuting the order of the species. The second set of overlapping rectangles gives an example of such a permutation.

(Gower 1971; Podani 1999; Podani and Schmera 2006, 2007). For Euclidean distance, we

standardized all trait matrices so that all traits have a mean = 0 and variance =1 (i.e. z-scores;

Holmes et al. 1979, Gaston and Petchey 2002). We used cosine distance because it more

accurately reflects proportional changes in traits whereas the Euclidean distance emphasizes

absolute differences. For the Patagonian forb and Arctic vegetation datasets we used only

Gower’s distance because these datasets contained missing values and mixed data types; the

Euclidean and cosine distances were not appropriate for such datasets (e.g. Podani and Schmera

2006). Second, a clustering algorithm must be chosen. We used three clustering algorithms in

this analysis: 1) unweighted pair group method with arithmetic mean (UPGMA); 2) single

linkage (i.e. nearest neighbor); and, 3) complete linkage (i.e. maximum or farthest neighbor).

These algorithms represent natural endpoints across a methodological continuum of dendrogram

construction methods, where single linkage lies on one end, complete linkage on the other and

UPGMA lies somewhere in the middle (Podani and Schmera 2006).

We considered several different collections of construction methods because the sensitivity of

FD is defined in terms of a particular set of construction methods. For cases where multiple

comparisons could be made (e.g. several distance measures), we calculated four separate

25

probabilities of sensitivity: 1) sensitivity with respect to all nine construction methods; 2)

sensitivity with respect to the three distance measures with UPGMA clustering (i.e. clustering

algorithm is held constant); and, 3) sensitivity with respect to the three clustering algorithms with

Gower’s distance measure (i.e. distance measure is held constant). In cases where data were

deficient and only Gower’s distance could be used, only overall probabilities of sensitivity were

calculated. The sensitivity when the clustering algorithm is held constant could be calculated,

but these results would be identical to the overall values. Finally, we also calculated

probabilities of sensitivity holding other distance measures and clustering algorithms constant

and consider pairs of assemblages that do not contain all of the species in the complete datasets.

However, we do not present these additional results because they do not alter any of the

conclusions.

There exists an ongoing debate regarding a standard procedure for calculating FD. Petchey and

Gaston (2002) based their measure of FD on a dendrogram derived from a dataset that included

all species that were of interest (i.e. the entire community). For an assemblage that does not

contain all of the species in the entire community, FD is measured as the total branch length of

the dendrogram minus the branch lengths of the species that are not included in the assemblage

(see Petchey and Gaston 2002, 2007 for more details). We refer to this approach as the Petchey-

Gaston (PG) method. Alternatively, Podani and Schmera (2006) suggested that FD should be

calculated as the total branch length of a dendrogram that is unique to each assemblage, i.e.

recalculated from the reduced dataset. We refer to this measure as the PS (Podani-Schmera)

method. As this debate remains unresolved, we tested whether FD was sensitive using both

methods. To calculate FD using the PG method, we calculated a species-by-branch matrix and a

vector of branch lengths for the complete community using the code of Petchey and Gaston

(2002) for the R programming language. We then used this code to calculate FD using the PG

method for each assemblage (see Petchey and Gaston 2002 for more details). We repeated this

approach for each of the nine construction methods (i.e. three distance measures for each of the

three clustering algorthms). To calculate dendrograms using the PS method, we calculated

unique dendrograms for all assemblages and construction methods. We used MatLAB (v.7.1) to

calculate the sum of dendrogram lengths for each assemblage and construction method.

To display all of these results, we constructed image plots with the R programming language.

Image plots can be used to show how a variable changes over a two-dimensional grid. The

26

shading of each square on the grid represents the value of the variable at that grid location. In

this case, the variable of interest is the probability of sensitivity and the grid is defined by beta

diversity, β, and the difference in alpha diversity, Δα, between the two assemblages. However,

only certain combinations of β and Δα are possible. For example, for an eleven species

community, it is not possible to create two assemblages such that β = 6, Δα = 3 and all of the

species are in at least one of the two assemblages. Therefore, for identification purposes, these

impossible grid locations are plotted in white whereas all other levels of sensitivity are some

shade of grey. Higher levels of sensitivity are represented by darker shades of grey. This results

in a checkerboard pattern. However, it is important to keep in mind that the checkerboard pattern

is solely an artifact of the impossibility of certain combinations of β and Δα.

Results

We identified numerous cases for which FD had a high probability of sensitivity across all

communities; that is, it is easy to find cases for which conclusions derived from FD analyses will

be driven primarily by methodological choices. In the worst-case scenario, FD sensitivity

reached probabilities of 0.976 using the PS method and 0.594 using the PG method (Table 1).

Variation in the probabilities of sensitivity was largely driven by variation in alpha diversity,

with the highest probabilities of sensitivity found when assemblages were similar in alpha

diversity (Figures 2.2 & 2.3). In every case where the probability of sensitivity was zero, FD

was larger for the assemblage with more species; this result indicates that FD and alpha diversity

lead to identical conclusions about the diversity of assemblages in these cases. Therefore, the

hypothesis presented here concerning the relationship between alpha diversity and probability of

sensitivity is consistent with these results. Contrary to hypotheses presented here, there were no

consistent patterns in the relationship between beta diversity and probability of sensitivity

(Figures 2.2 & 2.3).

Decisions about distance measures were more important than decisions about clustering

algorithms. For example, when UPGMA clustering was kept constant and only distance

measures were compared, FD was more sensitive than when Gower’s distance was held constant

and clustering methods were compared (Figures 2.2 & 2.3). These results were not altered by

the distance measure held constant (e.g. Euclidean, cosine or Gower’s) or by the clustering

27

algorithm held constant (e.g. UPGMA, single linkage and complete linkage), and so we only

present the results for holding constant Gower’s distance and UPGMA respectively (Figure 2.2).

D

iffer

ence

in a

lpha

div

ersi

ty

0.0

0.2

0.4

0.6

0.8

1.0

Beta diversity

Overall PS Overall PG Gower PS UPGMA PS

Probability of Sensitivity

A)

B)

C)

15

5

0

10

8

4

0

6

6

2

0

4

2

118.563.50.5118.563.50.5118.563.50.5118.563.50.5

6.54.52.50.56.54.52.50.56.54.52.50.56.54.52.50.5

5.53.51.55.53.51.5 5.53.51.55.53.51.5

Figure 2.2 - The effect of alpha and beta diversity on the probability of FD sensitivity for three communities crossed with four sets of construction methods. Darker shading represents a higher probability of sensitivity. The communities are A) Insectivorous birds (Holmes et al. 1979), B) Intertidal fish (Munoz and Ojeda 1997) and C) Predatory vertebrates (Jaksic and Medel 1990). Each column is for a different set of construction methods. For the first and second columns, overall PS and PG, all nine methods of dendrogram construction (three distance measures times three clustering algorithms) were used with the PS and PG methods respectively. For the third column, all three clustering algorithms were used with Gower’s distance and the PS method. For the fourth column, all three distance measures were used with UPGMA and the PS method.

28

Diff

eren

ce in

alp

ha d

iver

sity

0.0

0.2

0.4

0.6

0.8

1.0

Beta diversity

Probability of Sensitivity

D)

E)

20

0

10

30

20

016

Overall PS Overall PG

10

2 7 122 7 12

1161161161

Figure 2.3 –The effect of alpha and beta diversity on the probability of FD sensitivity for two communities crossed with two methods of FD calculation. Darker shading represents a higher probability of sensitivity. The communities are D) Patagonian forbs (Golluscio and Sala 1993) and E) Artic vegetation (Chapin et al. 1996). For these communities, only Gower’s distance could be calculated and so only three construction methods could be compared, corresponding to the three clustering algorithms. Each column is for a different method of FD calculation. The first is for the PS method and the second is for the PG method.

29

There are some additional trends worth mentioning. The PG method of FD calculation led to

lower probabilities of sensitivity than the PS method in all cases (Figures 2.2 & 2.3, Table 2.1).

Also, where greater numbers of dendrogram construction methods are compared, the

probabilities of sensitivity increase. For example, compare the overall probabilities (nine

construction methods) with the probabilities obtained by holding the clustering method at

UPGMA (three construction methods) (Figure 2.2). This difference makes intuitive sense

because as one considers more construction methods, it becomes more likely to find a method

that leads to different conclusions regarding the ranking of the assemblages in terms of FD. Table 2.1 – The maximum probability of FD sensitivity for five communities previously used to examine FD (Petchey and Gaston 2007; Podani and Schmera 2006). The number of species and the number of assemblage pairs tested are also shown. Data sources are as follows: A) Holmes et al. (1979), B) Munoz and Ojeda (1997), C) Jaksic and Medel (1990), D) Golluscio and Sala (1993), and E) Chapin et al. (1996).

Community No.

species

Maximum

Probability of

Sensitivity: PS

Maximum

Probability of

Sensitivity: PG

No. of

Assemblage

Pairs Tested

A) Insectivorous birds 22 0.818 0.497 134

B) Intertidal fish 13 0.976 0.366 46

C) Predatory vertebrates 11 0.610 0.594 32

D) Patagonian forbs 24 0.364 0.196 159

E) Arctic vegetation 37 0.244 0.142 370

Discussion These results demonstrate that FD is sensitive to choices of distance measure and clustering

algorithm in many cases. The major factor contributing to a high probability of sensitivity is low

variation in alpha diversity between the assemblages being compared. By contrast, beta diversity

between assemblages was a very poor predictor of sensitivity. This did not support the initial

hypothesis that lower beta diversity (i.e higher redundancy between traits across assemblages)

would lead to a higher probability of sensitivity. The consistency and severity of the results

30

suggest that this sensitivity is not likely to be unique to the examples we present. Indeed, we did

not actively search for atypical data to support this position; we merely used the same data that

have been used consistently by investigators when evaluating FD (Petchey and Gaston 2002,

2007, Podani and Schmera 2006).

If our results are so clear, why did others (e.g. Petchey and Gaston 2007) conclude that decisions

regarding methodological choices have only a minor affect on FD, especially given that they

used the same data that we use here? There are two possible reasons for this discrepancy. First,

to evaluate sensitivity, previous studies have shown that FD calculated using Gower’s distance

was strongly collinear with FD calculated using the Euclidean distance across many functional

trait matrices (Petchey and Gaston 2007). However, these trait matrices differed widely in

number of species. In this analysis, we demonstrate that FD becomes more sensitive as variation

in alpha diversity becomes small. Therefore, in the light of this new work, it is not surprising

that others have found low sensitivity to methodological choices; in their case, the results

strongly suggest that variation in FD was being driven largely by differences in alpha diversity,

no matter what methodological choices were made. Second, we compared more distance

measures than previously investigated (Podani and Schmera 2007; Petchey and Gaston 2007).

We feel this is a more appropriate comparison as there are a large number of distance measures

in the multivariate literature deemed to be appropriate. Additionally, when we restricted the

analysis to comparing only Gower’s distance and Euclidean distance (with PG dendrogram

construction and UPGMA held constant), we found that rates of sensitivity remained high when

differences in alpha diversity were low (maximum probability of sensitivity: 0.260 for the bird

data, 0.162 for the fish data, and 0.319 for the mammal data). Thus, FD did not provide much

additional information in this case, beyond that provided by alpha diversity.

The preceding discussion leads to the following important conclusion regarding FD. FD is most

sensitive to methodological choices when it genuinely provides new information beyond that

provided by alpha diversity. This is because conditions under which FD is sensitive coincide

with relatively little variation in alpha diversity between assemblages. Thus in these cases, FD

could potentially provide useful information about the differences between the assemblages and

ecosystem function. Unfortunately it is precisely in these cases, where FD would genuinely be

useful, that it is expected to be highly sensitive to the choice of a distance measure or clustering

algorithm. On the other hand, FD is not sensitive to methodological choices, in those cases when

31

it provides very little information beyond that already provided by species richness (alpha

diversity). This is because, when FD is insensitive, the results show that alpha diversity is

largely redundant with FD no matter what methodological choices are made. Newer approaches

to measuring functional richness (e.g. convex hull volume or consensus dendrograms) have been

proposed that may reduce the subjectivity of multivariate decisions (Cornwell et al. 2006;

Mouchet et al. 2008; Villeger et al. 2008); however, decisions are still required that may alter

results (e.g. trait scaling and transformations or what to include in the consensus). Further

research into understanding these methodological choices will likely enhance the ability to

measure functional richness. Here we wish to raise awareness about the importance of species

richness and methodological choices for calculating functional richness and identify cases for

which sensitivity is likely to be an issue.

What can be done to minimize the impact of sensitivity? One simple approach could be to

analyze data from ecological communities using several different construction methods to ensure

that sensitivity is not an issue. However, if sensitivity is an issue, a decision must be made. The

results suggest potential approaches for reducing the probability of sensitivity. First, we found

that probabilities of sensitivity were systematically lower for the PG method of FD calculation

than for the PS method. Therefore, one might be tempted to recommend the PG method for

general use. There is an important issue with this recommendation however. The PG method

assumes that the entire community is known whereas the PS method does not. In a recent paper

(Walker et al. 2008), they emphasized the importance of assuming that there may be species in

the community that are undiscovered or undetected in the study area when estimating FD from

field data. In some cases, this might not be a problem. For example, Barnett et al. (2007) have

recently published a list of species to be used in studies of FD in zooplankton communities.

However, in the vast majority of cases, there will typically be a high degree of uncertainty about

the composition of the entire community. The PG method does not provide the same estimate as

the PS method for a subset of the community. Given that the PS method provides the correct

dendrogram length for that particular subset, as it is based on a distance matrix constructed from

this subset, such differences between the methods remain a concern. Therefore, even though the

PS method is more sensitive than the PG method, the PS method is recommended for general use

and the PG method when the species list for the entire community is known. Second, we found

that FD is much more sensitive to the choice of a distance measure than to the choice of a

32

clustering algorithm. Therefore, one might be tempted to simply adopt a particular distance

measure as a standard. However, FD is not completely sensitive to the choice of clustering

algorithm (e.g. range in maximum probability of sensitivity across communities: 0.137 to 0.260

for PG method and 0.248 to 0.364 for PS Method). Furthermore, the choice of a distance

measure must be made very carefully. It is unlikely that a single distance measure can be found

that is justifiable in all situations; indeed, the history of multivariate statistics teaches us that

there is no distance measure that can be uniformly recommended in all cases (Sneath and Sokal

1973; Legendre and Legendre 1998).

To calculate functional richness, a method for quantifying inter-specific differences in functional

traits is required. However, the flexibility to use more than one trait is often required to

understand even simple natural systems (Villeger et al. 2008). Unfortunately in these

multivariate situations, complications arise as researchers have to make several key decisions

during data analysis (e.g. choice of a distance measure, clustering algorithm, data

transformations, scaling). Ideally, these decisions should have minimal impact on scientific

conclusions. Here we demonstrate that in the case of the popular index of functional richness,

FD, decisions inherent in multivariate analyses can drastically alter conclusions of functional

diversity and that sensitivity in FD is highest when alpha diversity is low. These results suggest

that in cases where information captured by dendrogram-based functional diversity would be

most useful, it is redundant with alpha diversity.

Acknowledgements

Funding was provided by NSERC Canada and OGS Scholarships to M.S.P and S.C.W., an

NSERC Discovery Grant to D.A.J., and the University of Toronto. We thank D.A.R. Drake, J.

Podani, D. Schmera, O. Petchey, and anonymous reviewers for comments on early drafts of this

paper.

33

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36

Appendices Appendix 2.1 – Derivation of Equation 1.

Definitions

Let γ be the total number of species in both assemblages.

Let α1 be the number of species in assemblage one.

Let α2 be the number of species in assemblage two.

Let Δα = α1 – α2.

Let <α> be the average of α1 and α2.

Let β = γ – <α>.

Let ν be the number of shared species (i.e., the number of species in both assemblages).

Assumption

All of the γ species are in assemblage 1, assemblage 2, or both.

Main result (Eq. 1)

For a given γ, Δα and β, the total number of unique assemblage pairs is [γ!] / [(γ – 2β)!(0.5(2β +

Δα))!(0.5(2β – Δα))!].

Deriving this result is much easier once there are three simpler results.

Simpler result 1

X! is the total number of ways to order X objects.

Simpler result 2

(γ – 2β) is the number of shared species, ν.

37

Derivation of simpler result 2

Given that all species are in at least one of the two assemblages (see assumption above), it

follows from the inclusion-exclusion principle that γ = α1 + α2 – ν. It must also be that β = γ –

0.5(α1 + α2) from the definitions of β and <α>. Eliminating the alphas and solving for ν leads to

simpler result 2.

Corollary of simpler result 2

2β = α1 + α2 – 2ν is the total number of unshared species (i.e., species that are only in either

assemblage 1 or assemblage 2 but not in both).

Simpler result 3

0.5(2β + Δα) and 0.5(2β – Δα) are the numbers of unshared species in assemblages 1 and 2,

respectively.

Derivation of simpler result 3

It is only necessary to derive the first claim; the second follows immediately after the first. Note

that one can write Δα = (α1 – ν) – (α2 – ν), from the definition of Δα, and 2β = (α1 – ν) + (α2 – ν),

from the corollary of simpler result 2. Eliminating α2 – ν and solving for α1 – ν one can obtain α1

– ν = 0.5 (Δα + 2β). Simpler result 3 follows once it is recognized that α1 – ν is the number of

unshared species in assemblage 1.

Derivation of main result

It follows from simpler result 1 that the numerator in Eq. 1 is the number of ways that γ species

can be ordered. From Fig. 1, one can see that the numerator gives the total number of ways that

iterations from the simulations could occur. However, many of these ways result in identical

assemblage pairs. This is because the order that species are listed in Fig. 1 determines a

particular assemblage pair, only insofar as it determines whether each species is in assemblage 1

only, assemblage 2 only or in both assemblages 1 and 2. Therefore one must divide by the total

number of ways to order the species within each of these three groups. It follows from simpler

results 1 through 3 that the denominator in Eq. 1 gives this number.

38

Appendix 2.2 – MatLAB Code for testing sensitivity of FD % X is the trait matrix (S by T) % S is species richness % T is # of traits % robustness measures % 1. overall % 2. euc % 3. cos % 4. gower % 5. UPGMA % 6. single % 7. complete S = length(X(:,1)); gamma = 10; iters = 10; n_measures = 9; % number of different ways to calculate FD G = zeros(iters,n_measures); output = -999.*ones((gamma+1),(gamma+1),iters,n_measures); % sign matrix nonrobust_probs = -999.*ones((gamma+1),(gamma+1),7);

% use when all dist measures %nonrobust_probs = -999.*ones((gamma+1),(gamma+1),1);

% use when only gowers (remove % and put one on the above) for B = 4:gamma for A = max(B,(gamma-B)):gamma delta = A - B overlap = A + B - gamma unsharedA = A - overlap unsharedB = B - overlap startA = 1; endA = unsharedA; startB = unsharedA + 1; endB = unsharedA + unsharedB; startShare = endB + 1; endShare = startShare + overlap - 1; Alist = zeros(iters,A); Blist = zeros(iters,B); Slist = zeros(iters,S);

39

FDA = zeros(iters,n_measures); FDB = zeros(iters,n_measures); if overlap == 0 for i = 1:iters currlist = randperm(S); Slist(i,:) = currlist; Alist(i,:) = [currlist(startA:endA)]; Blist(i,:) = [currlist(startB:endB)]; FDA(i,1) = FD(X(Alist(i,:),:),'euclidean','average'); FDB(i,1) = FD(X(Blist(i,:),:),'euclidean','average'); FDA(i,2) = FD(X(Alist(i,:),:),'euclidean','single'); FDB(i,2) = FD(X(Blist(i,:),:),'euclidean','single'); FDA(i,3) = FD(X(Alist(i,:),:),'euclidean','complete'); FDB(i,3) = FD(X(Blist(i,:),:),'euclidean','complete'); FDA(i,4) = FD(X(Alist(i,:),:),'cosine','average'); FDB(i,4) = FD(X(Blist(i,:),:),'cosine','average'); FDA(i,5) = FD(X(Alist(i,:),:),'cosine','single'); FDB(i,5) = FD(X(Blist(i,:),:),'cosine','single'); FDA(i,6) = FD(X(Alist(i,:),:),'cosine','complete'); FDB(i,6) = FD(X(Blist(i,:),:),'cosine','complete'); FDA(i,7) = FD(X(Alist(i,:),:),'gowers','average'); % change needed FDB(i,7) = FD(X(Blist(i,:),:),'gowers','average'); % change needed FDA(i,8) = FD(X(Alist(i,:),:),'gowers','single'); % change needed FDB(i,8) = FD(X(Blist(i,:),:),'gowers','single'); % change needed FDA(i,9) = FD(X(Alist(i,:),:),'gowers','complete'); % change needed FDB(i,9) = FD(X(Blist(i,:),:),'gowers','complete'); % change needed end else for i = 1:iters currlist = randperm(S); Slist(i,:) = currlist; Alist(i,:) = [currlist(startA:endA),currlist(startShare:endShare)]; Blist(i,:) = [currlist(startB:endB),currlist(startShare:endShare)];

40

FDA(i,1) = FD(X(Alist(i,:),:),'euclidean','average'); FDB(i,1) = FD(X(Blist(i,:),:),'euclidean','average'); FDA(i,2) = FD(X(Alist(i,:),:),'euclidean','single'); FDB(i,2) = FD(X(Blist(i,:),:),'euclidean','single'); FDA(i,3) = FD(X(Alist(i,:),:),'euclidean','complete'); FDB(i,3) = FD(X(Blist(i,:),:),'euclidean','complete'); FDA(i,4) = FD(X(Alist(i,:),:),'cosine','average'); FDB(i,4) = FD(X(Blist(i,:),:),'cosine','average'); FDA(i,5) = FD(X(Alist(i,:),:),'cosine','single'); FDB(i,5) = FD(X(Blist(i,:),:),'cosine','single'); FDA(i,6) = FD(X(Alist(i,:),:),'cosine','complete'); % change needed FDB(i,6) = FD(X(Blist(i,:),:),'cosine','complete'); % change needed FDA(i,7) = FD(X(Alist(i,:),:),'gowers','average'); % change needed FDB(i,7) = FD(X(Blist(i,:),:),'gowers','average'); % change needed FDA(i,8) = FD(X(Alist(i,:),:),'gowers','single'); % change needed FDB(i,8) = FD(X(Blist(i,:),:),'gowers','single'); % change needed FDA(i,9) = FD(X(Alist(i,:),:),'gowers','complete'); % change needed FDB(i,9) = FD(X(Blist(i,:),:),'gowers','complete'); % change needed end end output(overlap+1,delta+1,:,:) = sign(FDA-FDB); G(:,:) = output(overlap+1,delta+1,:,:); nonrobust_probs(overlap+1,delta+1,1) = sum(abs(sum((G(1:iters,:))'))<n_measures)/iters; nonrobust_probs(overlap+1,delta+1,2) = sum(abs(sum((G(1:iters,1:3))'))<3)/iters; nonrobust_probs(overlap+1,delta+1,3) = sum(abs(sum((G(1:iters,4:6))'))<3)/iters; nonrobust_probs(overlap+1,delta+1,4) = sum(abs(sum((G(1:iters,7:9))'))<3)/iters; nonrobust_probs(overlap+1,delta+1,5) = sum(abs(sum((G(1:iters,[1,4,7]))'))<3)/iters; nonrobust_probs(overlap+1,delta+1,6) = sum(abs(sum((G(1:iters,[2,5,8]))'))<3)/iters; nonrobust_probs(overlap+1,delta+1,7) = sum(abs(sum((G(1:iters,[3,6,9]))'))<3)/iters; end end

41

Appendix 2.3 – MatLAB Code for calculating FD via Podani and Schmera

function output = FD(X,distance,cluster) if strcmp(distance,'gowers') Y = gowers(X); else Y = pdist(X,distance); end Z = linkage(Y,cluster); output = sum(sum(branch_lengths(Z)));

42

Appendix 2.4 – MatLAB Code for calculating probabilities of sensitivity S = length(X(:,1)); iters = 1000; n_measures = 9; % number of different ways to calculate FD Amax = (ceil(S/2)-3); Bmax = (floor(S/2)-3); Omax = (S-8); dist.probs = zeros(Bmax,Amax,Omax,7); G = zeros(iters,n_measures); for BBB = 1:Bmax for AAA = BBB:Amax for OOO = 1:(S-5-AAA-BBB) G(:,:) = output(BBB,AAA,OOO,:,:); dist.probs(BBB,AAA,OOO,1) = sum(abs(sum((G(1:iters,:))'))<n_measures)/iters; dist.probs(BBB,AAA,OOO,2) = sum(abs(sum((G(1:iters,1:3))'))<3)/iters; dist.probs(BBB,AAA,OOO,3) = sum(abs(sum((G(1:iters,4:6))'))<3)/iters; dist.probs(BBB,AAA,OOO,1) = sum(abs(sum((G(1:iters,7:9))'))<3)/iters; dist.probs(BBB,AAA,OOO,1) = sum(abs(sum((G(1:iters,[1,4,7]))'))<3)/iters; dist.probs(BBB,AAA,OOO,1) = sum(abs(sum((G(1:iters,[2,5,8]))'))<3)/iters; dist.probs(BBB,AAA,OOO,1) = sum(abs(sum((G(1:iters,[3,6,9]))'))<3)/iters; end end end

43

Chapter 3: Addressing the removal of rare species in bioassessments with other choices in multivariate analyses

Abstract The inclusion or exclusion of rare species in the bioassessment of aquatic communities has been

greatly debated. Researchers may include rare species in bioassessments as they are likely better

indicators of ecosystem stress than more common species (i.e. the biological argument).

Alternatively, researchers may exclude rare species due to the potential influence on statistical

analyses (i.e. the statistical argument). As this debate remains unresolved, the objective of this

study was to determine the impacts of removing rare species in multivariate bioassessments.

These approaches were tested independently using multivariate comparisons of fishes from a

thoroughly sampled system. The biological argument was assessed using sites-level vector

residuals across treatments of species removal and demonstrated that the removal of rare species

had important site-level implications relative to full dataset, including up to a nine-fold change in

multivariate vector residuals at sites where single species were removed. The statistical

argument was assessed using variation partitioning of multivariate decisions such as ordination

method, distance measure and removal of rare species, and found that the removal of rare species

demonstrated similar levels of multivariate variation (e.g. 24.8% variation) as other choices

inherent in multivariate bioassessments, such as the choice of ordination technique (26%

variation) and similarity measure (11%). This study demonstrates that contrary to the common

held practice of removing rare species in multivariate bioassessments, that the removal of rare

species may be less important than previously thought, whereas other multivariate decisions may

be at least as equally important. Better justification for the removal of rare species, along with all

decisions in multivariate analyses, is needed to ensure bioassessments are developed in a

rigorous manner.

Keywords: Multivariate analysis; ordination; similarity measures; rare species; community

ecology; bioassessment; procrustes analysis.

44

Introduction The use of multivariate analyses has become an important tool in the biological assessment of

aquatic communities (Norris 1995; Wright et al. 2000). Several national bioassessment

programs are based on multivariate measures, notably those in the UK (e.g. RIPVACS; Wright et

al. 2000) and Australia (AUSRIVAS; Simpson and Norris 2000; Metzeling et al. 2006); and the

use multivariate analyses has become widespread elsewhere (Reynoldson et al. 2001; Joy and

Death 2002; Bailey et al. 2004). In multivariate analyses, researchers order entities in the data

(e.g. species or observations) on the basis of the similarity of their characteristics (e.g.

observations or species) (Wartenberg et al. 1987). The goal of such analyses is to determine the

basis for the order of entities; for example, differences in species abundance or occurrence

between impacted sites versus sites with little or no impact (Barbour et al. 1999; Wright et al.

2000). From these orderings, one may be able to infer causative relationships between species

and their environment so that site-level impacts can be identified and mitigated (Hawkins et al.

2000).

The application of multivariate analyses to bioassessments of aquatic communities has been

riddled with controversy. Debates in bioassessment literature include the use of multi-metric

versus multivariate approaches (Hannaford and Resh 1995; Wallace et al. 1996; Bowman and

Somers 2006); the amount of taxonomic resolution needed to determine site level impacts

(Somers et al. 1998; Hewlett 2000); the use of rapid assessment methods (Hannaford and Resh

1995; Somers et al. 1998); and issues with using reference conditions (Norris and Thoms 1999;

Bowman and Somers 2005; Bailey et al. 2008). In particular, the debate regarding the treatment

of rare species has received much attention (e.g. Faith and Norris 1989; Norris 1995; Cao et al.

1998; Cao and Williams 1999; Marchant 1999; 2002). On one hand, researchers remove rare

species with the perceived notion that they may add noise to multivariate analyses and provide

little additional information beyond more common species (Gauch 1982; Clarke and Green 1988;

Marchant 1990; 2006; McCune and Grace 2002; Paukert and Wittig 2002). On the other hand,

researchers retain rare species in multivariate analyses because they may be better indicators of

ecosystem stress than common species (Faith and Norris 1988; Cao et al. 1998; 2001), as they

may be more sensitive to the stressor(s). In either case, the debate regarding the treatment of rare

45

species has remained unresolved and researchers need to be aware of the impact of their decision

of how to treat rare species (among others).

There are many difficulties in attempting to resolve the debate regarding the treatment of rare

species in bioassessments. For example, most multivariate approaches require several decisions

beyond the removal of rare species, and these decisions may reduce insight into the effect of rare

species on resultant analyses. Researchers using multivariate methods typically must choose a

type of similarity coefficient and ordination technique, where such choices have been shown to

significantly alter results (Podani 2000; Podani and Schmera 2006; Hirst and Jackson 2007; Poos

et al. 2009). As such, the resolution to the debate regarding the impact of rare species cannot

proceed until the effect of removing rare species is placed into a context comparable to other

decisions inherent in multivariate bioassessments. Unfortunately, relatively little effort has gone

into comparing these methods of bioassessment (Norris 1995; Marchant et al. 2006), and few

studies have viewed decisions in analyses in a holistic manner (e.g. how do all of the decisions

inherent in multivariate bioassessments affect results?).

There are two general arguments for the inclusion or exclusion of rare species in multivariate

bioassessments. The first argument for removing rare species from bioassessments is that rare

species provide limited interpretative value (Marchant 1999). Proponents of this argument

suggest that rare species may simply reflect stochastic sampling effects and, as such, add noise to

the statistical solution (Gauch 1982; Clarke and Green 1988; Bailey et al. 2004). This argument

is referred to as the “statistical argument” for excluding rare species. Support for this argument

has come from results from multivariate methods could be driven by the inclusion of rare species

alone (Webb et al. 1967, Austin and Greig-Smith 1968, Day et al. 1971, Orloci and Mukkattu

1973). To some degree this argument has been examined in the literature with analyses of data

standardizations (Jackson 1993; Cao et. al 1999), similarity coefficients (Jackson et al. 1989),

ordination method (Marchant 1990), or their combinations (e.g. data standardization and

similarity coefficients; Jackson 1993; Hirst and Jackson 2007; taxonomic resolution and rarity,

Arscott et al. 2006). Unfortunately, a quantitative evaluation of the role of rare species in

community assessments is largely absent, including a more holistic evaluation that answers the

practical question of how important rare species are relative to other decisions in multivariate

analyses (but see Faith and Norris 1988). In this context, the statistical argument can be tested as

a hypothesis, with the prediction that multivariate analyses should show greater variation among

46

one another where rare species are removed than multivariate analyses where other decisions

have been altered (e.g. similarity measure, ordination method) – specifically, does the inclusion

or exclusion of rare species lead to greater changes in multivariate analyses than those arising

due to other decisions? If not, then the inclusion of rare species may be warranted as this

decision contributes similar or lower amounts of variation versus other decisions typical in a

bioassessment.

The second argument for the inclusion or exclusion of rare species is that such species should be

included in multivariate analyses because they are better indicators of ecosystem stress than are

common species (Cao et al. 1999; 2001a), i.e. common species tend to have broad ranges of

tolerance to many conditions and, therefore, are not good indicators. This argument is referred to

as the “biological argument” for the inclusion of rare species. Support for this argument has

come from empirical studies that note the importance of including rare species for conservation

issues (Margules 1986; DeVelice et al. 1988; Norris and Hawkins 2000). Proponents of this

argument suggest that exclusion of rare species may lead to an underestimation of differences

between impacted and un-impacted sites (Cao et al. 1999; 2001a). As one of the main goals of a

bioassessment is to determine site-level impacts (Barbour et al. 1999; Wright et al. 2000), this

argument assumes that sites with rare species represent the strongest signals, such as decreases in

species diversity or changes in community composition (Cao et al. 1998). Therefore, the

biological argument can be tested also as a hypothesis, with the prediction that once rare species

are removed from multivariate analyses, sites which were chosen for the removal of rare species

should be more affected across multivariate analyses than sites that were not (i.e. when rare

species are removed from the analysis, the site-level signal will change in greater proportion at

sites with rare species than those sites without). If this result is not found, it would indicate that

the exclusion of rare species is warranted as they do not provide information beyond that

captured by more common species.

Given these hypotheses, the objective of this study is to determine the biological and statistical

effect of removing rare species relative to other methodological decisions inherent in

multivariate analyses (e.g. choice of ordination method and distance measure). For this

evaluation, the Sydenham River is used as a model system because it has the highest diversity of

aquatic fauna in Canada, as well as the highest number of species at risk in Canada (Staton et al.

2003). Also, the Sydenham River has undergone detailed sampling (Poos et al. 2007; 2008),

47

which provides a high-quality dataset. As there are many types of rarity (Raboniwitz et al. 1986;

Gaston 1994), for clarity rare species are defined as those which occur infrequently (i.e. at few

locations or low prevalence, e.g. <1%, 5% and 10% occurrence). As well, species that are rare

due to declines in population sizes or number of locations and have a conservation designation

(e.g. endangered, threatened, special concern) are referred to as species at risk.

Methods Fishes were collected from the Sydenham River using the Ontario Stream Assessment Protocol

(OMNR 2007). The Ontario Stream Assessment Protocol is a typical bioassessment protocol for

monitoring impacts to aquatic systems (e.g. Barbour et al. 1999; Wright et al. 2000) and has been

used to monitor the changes in riverine communities (Stanfield and Jones 1998; Poos et al.

2008). Fishes were used as model organisms rather than benthic macroinvertebrates as fish are

relatively easy to identify, and enumerate and have been used extensively in bioassessments (e.g.

Fausch et al. 1990; Joy and Death 2002; Boys and Thoms 2006; Kennard et al. 2006; Mugodo et

al. 2006); however, this approach is equally well suited to macroinvertebrates or any other group

of species. Further, most bioassessments using benthic macroinvertebrates (Marchant 1990;

Marchant et al. 1997; Cao et al. 1998) are sensitive to sampling method and taxonomic

resolution (Arscott et al. 2006; Nichols and Norris 2006).

Fishes were sampled using a variety of approaches (see Poos et al. 2007 for sampling protocol);

however, for this analysis only electrofishing data were used as it is the most commonly used

bio-monitoring protocol and regarded as the most effective gear type for sampling stream-fish

assemblages (Bohlin et al. 1989; Reynolds 1996). As sample representativeness may be an issue

(Cao et al. 2001a; 2002), electrofishing was assessed relative to other methods and determined to

be the most effective method for evaluating fish species at risk (Poos et al. 2007). Species were

collected at 50 sites in 2002 and 25 additional sites in 2003. Sampling sites were chosen at

random across the entire watershed, with the exception of non-wadeable sections of the river in

the lower portions of the watershed which were not sampled due to constraints with using

wadeable sampling gear.

48

Evaluating Decisions in Multivariate Bioassessments

Prior to analysis, four treatments of the removal of species were applied to the species matrix

(Table 3.1). Data transformation and standardization have been previously shown to influence

multivariate analyses (Austin and Greg-Smith 1968; Jackson et al. 1993; Cao et al. 1999) and,

not wanting to provide potential bias, data were reduced to presence/absence as it removes one

additional source of variation from the comparisons and provides focus on specific comparisons

of multivariate methods.

Traditionally, researchers decide what characteristics define a rare species within a sample (Faith

and Norris 1989; McCune and Grace 2002). Some researchers suggest eliminating species that

occur at single sites because of the inflated correlations created by attempting to relate

potentially random features at that site to its lone occurrence (Legendre and Legendre 1998).

Others suggest removing species that occur at less than five percent (McGardial et al. 2000) or

ten percent of sites (Marchant 1990; McCune and Grace 2002) or at even higher thresholds

(Boulton et al. 1992; Marchant et al. 1997). The following treatments of removing rare species

were used: all species included, single-occurrence species removed; species found at less than

five percent of sites removed; and, species found in less than 10% of sites removed. These

criteria removed 0, 2, 8, and 21 species respectively of the 67 species dataset.

49

Table 3.1 – Summary of ordination techniques, similarity coefficients and exclusion of rarely sampled species being compared. Abbreviations are indicated in parentheses and used in subsequent figures and tables. All four approaches described in the “Exclusion of Rarely Sampled Species” were used in each of the four “Similarity Coefficient” combinations with both PCoA and NMDS. As Correspondence Analysis has the implicit chi-squared distance measure, only the four approaches used in the “Exclusion of Rarely Sampled Species” were included in that set of analyses.

Ordination Technique Similarity Coefficient Exclusion of Rarely Sampled Species

Principal Co-ordinates Analysis (PCoA)

• Jaccard’s (J) • Phi (Φ) • Russell and Rao (RR) • Simple Matching (SM) *

• No species removed (All) • Single occurrences removed (M1) • < 5% occurrences removed (M5) • < 10% occurrences removed (M10)

Non-metric multidimensional scaling (NMDS)

• Jaccard’s (J) • Phi (Φ) • Russell and Rao (RR) • Simple Matching (SM)

• Same as above.

Correspondence analysis (CA)

• X2 distance (no other choice of similarity coefficient)

• Same as above.

Note: * Principal Co-ordinates Analysis (PCoA) using a simple matching similarity is identical to a Principle Components Analysis (PCA) using a correlation matrix (Gower 1966).

50

Similarity coefficients were calculated from each of the four matrices of rarely sampled species

(Table 3.1). Several dozen similarity coefficients have been developed for use with

presence/absence data (Gower 1966; Gower and Legendre 1986; Legendre and Legendre 1998;

Podani 2000). The choice of similarity coefficient has been largely subjective and is often

based on tradition or on a posteriori criteria without clear justification (Jackson et al. 1989;

Krebs 1998). As different similarity coefficients emphasize different aspects of the relation

between observations, the exclusion of rare species may alter species relationships and

subsequent results of analyses such as ordinations (McGargal et al. 2000). Treatments of

similarity coefficient included: Jaccard’s, phi (Φ), Russell and Rao, and simple matching

coefficients, and were chosen because they represent standard examples amongst the continuum

of similarity coefficients. Jaccard’s similarity does not consider joint absences; the phi

coefficient is the correlation coefficient for binary data; and, Russell and Rao and simple

matching are variations that consider joint absences (Jackson et al. 1989; Legendre and Legendre

1998; Podani 2000). All similarity coefficients were transformed into metric distances having

Euclidean properties for subsequent analysis (Jackson et al. 1989; Legendre and Legendre 1998).

Three types of ordination technique were compared for each combination of treatments

excluding rare species and using different similarity coefficients: principal co-ordinates analysis

(PCoA), non-metric multidimensional scaling (NMDS), and correspondence analysis (CA). As

PCoA measured from simple matching similarity is identical to PCA measured using a

correlation coefficient (Gower 1966), the results can be used to interpret both ordination types.

These ordination techniques were chosen because they represent typical multivariate methods

used by the majority of biologists (Legendre and Legendre 1998; Podani 2000; McCune and

Grace 2002). In addition, more current approaches (e.g. NMDS, CA) may provide robust

alternatives to previous methods (e.g. PCA) where non-linear relationships occur between

variables (Cao et al. 2001a). Whereas both NMDS and PCoA allow the user to choose a

similarity coefficient, CA do not provide the same option given its inherent resemblance measure

(chi-square distance). For NMDS, a random set of 20 starting configurations were used as input

configurations, and the solution having the lowest stress was retained. In NMDS, stress was

measured as an objective function of a regression analysis where the goodness of between the

fitted values and forecasted values was fit using a least square criterion (Legendre and Legendre

1998). A broken-stick model was used to compare the eigenvalues from PCoA and CA to those

51

expected from random relationships. This method has been shown useful in identifying non-

random patterns of association in multivariate analyses (Jackson 1993; King and Jackson 1999;

Peres-Neto et al. 2003). Axes not representing a meaningful contribution of the variation were

removed from resultant analyses. All analyses were completed using the R programming

language v2.70 plus statistics libraries simba (Jurasinski 2007), vegan (Oksanen et al. 2008) and

ecodist (Goslee and Urban 2007).

Assessing the Statistical Argument

The statistical argument in multivariate bioassessments was assessed in several ways. First, all

variants in ordination method, similarity coefficients and exclusion of rare species were

compared using Procrustes analysis (Gower 1971; Jackson 1995). Procrustes analysis is

appropriate for comparing separate ordination results because it is an orthogonal rotation that

best matches two or more ordinations (Olden et al. 2001; Peres-Neto and Jackson 2001; Paavola

et al. 2006). The first three dimensions from each ordination solution were retained for

comparisons as they represented the greatest portion of variance explained using a broken-stick

model (Legendre and Legendre 1998) and the majority of the ordination methods were best

represented by three-dimensional solutions, including the NMDS results. The sum-of-squared-

deviations (i.e. m2 statistic) was used as a metric of association, with lower sum-of-squared-

deviations representing greater similarity of multivariate configurations (Gower 1971; Jackson

1993; Peres-Neto and Jackson 2000) and was calculated between each pair of three-dimensional

ordination solutions to produce a matrix of m2 distances between all 36 exclusion-distance-

ordination combinations. The resultant 36-by-36 matrix of m2 distances was analyzed using a

PCoA to determine the relative effect of each methodological choice. This type of “ordination of

ordinations” (see Digby and Kempton 1987; Jackson 1993; Hirst and Jackson 2007) provides a

useful characterization of methodological decisions, where larger distances between objects

represent more dissimilar associations. A minimum spanning tree was calculated to determine

the most similar groups and super-imposed onto the first two axes of the ordination diagram.

Partitioning of variation of multivariate data can provide quantitative and objective

determination of the influence of methodological choices. For multivariate datasets, partitioning

of variation is often thought of in a spatial or temporal context, where the influence of variables

can be partitioned across various spatial or temporal scales (e.g. Borcard et al. 1992; Borcard et

52

al. 2004; Dray et al. 2006). Yet, partitioning of variation in multivariate data is also possible

through the analysis of residuals across various matrix comparisons (e.g. Rundle and Jackson

1996; Olden et al. 2001; Paavola et al. 2006). For example, the total of among-group variation of

removing rare species can be summarized relative to all treatments using the sum of squared

deviations from a Procruste’s analysis, and represents the amount of multivariate variation

explained. In this study, the variation of all treatments was quantified using a partitioning method

of multivariate matrices (see Rundle and Jackson 1996), which separated the sum-of-squared-

deviations for within- and among-treatment groups (e.g. removal of rare species, ordination

technique, and similarity measure).

Assessing the Biological Argument

To assess the biological argument of removing rare species from bioassessments, changes at sites

where rare species were removed were evaluated across the various multivariate analyses. Site-

level differences were calculated for each pair-wise Procrustes analysis using vector residuals

from PROtest (Jackson 1995). Vector residuals provide a means of investigating deviations in

position of individual samples between two superimposed ordinations (Olden et al. 2001;

Paavola et al. 2006), i.e. the degree to which any given observation changes from one ordination

to another ordination. The length of the vector residual represents a lack of fit of scores for an

individual sample between two ordinations, with low values indicating close agreement between

multivariate methods. Vector residuals were separated between sites where rare species were

removed with sites where rare species where no species were removed. For example in Figure

3.1, a typical example of site level vector residuals is shown across a comparison of multivariate

analyses (e.g. PCoA with Jaccard’s distance) between all data and with species occurring at 5%

of sites removed. From this comparison, the effect of the removal of rare species can be assessed

as the ratio of site-level vector residuals for sites where rare species were removed (i.e. grey bars;

Fig. 3.1) versus the sites where species were not removed (i.e. black bars; Fig. 3.1). Ratios over

1 indicate situations where site-specific differences are more related to the removal of rare

species than those that not (i.e. bioassessments may be affected by species removal as sites

where rare species occur, change in greater magnitude than sites where rare species do not occur

when compared to a full dataset).

53

Figure 3.1 – Example of rank-ordered, site-level vector residuals of Procrustean multivariate comparison. The length of a vector residual indicates an overall lack of fit for a site between two multivariate analyses. Shown is a comparison of full dataset of Principal Coordinates with Jacaard’s distance and the same dataset where species occurring at 5% of sites were removed. Vectors shown in grey indicate those sites where at least one species was removed, whereas vectors in black indicate sites where no species were removed. The ratio of mean vector residuals between sites where species were removed versus those sites that did not have species removed indicates the distribution of impacts of the removal of rare species across multivariate analyses. Where most vector residuals for sites having species removed are largest, they indicate that these observations (sites) have been changed the most in their position between two ordinations.

0

0.006

0.012

1 7

Vect

or R

esid

ual

Ordered Sites

5

54

Results The multivariate analyses used in this study provided generally good representation of the data.

Variance explained by the first three axes from all combinations of multivariate analyses ranged

from 24.6% (PCoA with Jaccard’s distance and all species included) to 38.4% (PCoA with

simple matching similarity and species occurring at less than 10% sites removed). In all cases,

the variance explained by each multivariate method increased with the exclusion of more

species.

For fish species in the Sydenham River, the removal of rare species had similar effects on

resultant analyses to those arising from other decisions in multivariate analyses, such as the

choice of ordination type or similarity coefficient. Recall that in considering the relative role of

the various decisions to be made, the partitioning of variation provides a measure summarizing

the relative importance. Variation across multivariate analyses was highest for ordination

method (26.15%) across all comparisons, followed by the removal of rare species (24.81%) and

similarity measure (10.99%; Table 3.2). These results were also evident by the well-defined

clustering of treatments of rarely sampled species (All, M1, M5) in close proximity to one

another relative to the clustering of the differences in ordination technique (PCoA, NMDS, CA,

PCA) or in similarity coefficients (J, Φ RR, SM) in the ordination of m2 distances, i.e. the

comparison of the various ordination results (Fig. 3.2). One clear exception to this result was the

removal of 10% of the least occurring species, which showed deviations from the general

multivariate groupings (Fig. 3.2a,b), and variation that exceeded most other choices (6.82%;

Table 3.2).

There was large variation between individual choices across multivariate methods. Whereas CA

and PCA demonstrated overall low amounts of variation among analyses (0.19%; 0.039%

respectively), there was large variation among analyses based on NMDS (21.60%; Table 3.2).

These differences may be influenced by the smaller number of comparisons for correspondence

analysis (as a choice of similarity measure is implicit and not selected); however, NMDS also

showed an almost seven-fold increase in variation over PCoA, which involved same number of

55

a) Axis 1 (47.3%)

-0.45 0 0.45

Axi

s 2 (2

0.7%

)

-0.3

0

0.3

M10

M10

All, M1, M5

M10

M10

NMDS-RR

All, M1, M5

NMDS-J

NMDS-Φ

All, M1, M5

NMDS-SM

CA

All, M1, M5

M10

PCoA-J

PCoA-Φ

M10M1 M5

All, M1, M5

PCoA-SM

PCoA-RR

All, M1, M5M10

All, M1, M5M10

M10All

AllM1M5

PCA

b) 

-0.3 0.30-0.3

0.15

M10

M5,M1,AllM10

M10

All,M1,M5

M10

CA

NMDS-Φ

NMDS-SMAllM1

M5

M5M1All NMDS-J

M10

NMDS-RRAllM1M5

PCoA-SM

PCoA-Φ

M10 M10

PCoA-J

PCAPCoA-RR

M10 M10

M1M5

All

M1M5

All

M5M1All

M5M1All

Axis 2 (20.7%)

Axi

s 3 (1

5.6%

)

0

0

Figure 3.2 –Principal Coordinates Analysis (PCoA) of the sum of squares deviations (m2 statistic) comparing the concordance between solutions based on different ordination techniques, similarity coefficients and treatments of excluding rarely sampled species. A minimum-spanning tree was overlaid on Axes 1 and 2 to highlight connections between groups of points. Dashed lines indicate deviations from group membership in cases where clear groupings do not exist (e.g. M10 for Axes 2 and 3). Short forms are continued from Table 3.2.

56

comparisons (Table 3. 2). Decisions, such as, which ordination method to choose, may be as

important (or more important) than other choices like the removal of rare species. In fact, the

inclusion or exclusion of rare species did not impact the resulting multivariate analyses any more

than the choice of ordination method (i.e. subtotals of 24.8% vs 26.2%, respectively) and likely

less given that four comparisons regarding species inclusion are used and only three comparisons

for ordination method. Levels of variation were similar between the removal of single

occurrence species (6.03%) and species occurring at 5% of the sites (5.93%) as they were for

using the entire dataset (6.03%; Table 3.2). Finally, the choice of similarity measure showed

lower levels of variation in general. Variation in X2 values was lowest (0.19%), followed by

simple matching (1.53%), Jaccard (2.03%) and the Phi (2.16%) coefficients (Table 3.2). Table 3.2 –Partitioning of variation in sum of squared deviations of Procrustes analyses (m2 statistic) across various choices in multivariate analyses, including: i) removal of rare species; ii) ordination technique; and, iii) choice of similarity measure. Abbreviations are those noted in Table 1.

Variation component %

I) Removal of Rare Species All 6.03 M1 6.03 M5 5.93 M10 6.82 Subtotal 24.81 II) Ordination Technique PCoA 4.37 NMDS 21.60 CA 0.19 Subtotal 26.15 III) Dissimilarity Measure J 2.03 Phi 2.16 RR 5.08 SM 1.53 X2 0.19 Subtotal 10.99

57

There was high site-level effect of removing rare species. In most cases, when rare species were

removed from the analysis, the effects were driven by differences in sites that contained species

that were removed. For example, the site-level residuals were much higher in sites impacted by

the removal of species than the full data (Figure 3.3; Appendix 3.1). Recall that site-level vector

residuals represent the degree to which any given observation changes from one ordination to

another (Olden et al. 2001; Paavola et al. 2006). Therefore, the ratio of site-level residuals

between sites impacted by species removal and those sites that did not have species removed

provides an indication as to magnitude that rare species may alter site-level assessments.

Overall, sites impacted by the removal of rare species changed in multivariate space over nine

fold when single-occurrence species were removed from the analysis, relative to the full data set,

and over two fold when species having prevalence less than 5% were removed from the analysis.

Interestingly, once species that occurred at less than 10% of sites were removed from the

analyses, there was virtually no difference between the two categories of sites (1.13 difference;

Figure 3.3), and in some cases represented less of an impact (e.g. NMDS-J, NMDS-RR;

Appendix 3.1).

0

3

6

9

12

15

M1 M5 M10

Proc

rust

e's v

ecto

r res

idua

l

Removal of Missing Species

Figure 3.3 – Site-level impact of the removal of rare species. Shown are box and whisker plots of the ratios of Procrustes vector residuals between sites for which rare species were removed and those sites that did not have any species removed. All comparisons were done by comparing site-level Procrustes vector residuals from the full datasets and with the removal of rare species across all similarity coefficients and ordination methods.

58

Discussion The treatment of rare species in multivariate bioassessments has been debated widely.

Advocates for the inclusion of rare species argue that rare species are useful indicators of

environmental stressors and their removal may result in the unnecessary loss of ecological

information (i.e. biological argument; Cao et al. 1999; 2001). On the other hand, advocates for

the exclusion of rare species argue that when rare species are removed from analysis, the

resultant analyses (and conclusions) are not altered (i.e. statistical argument; Gauch 1982;

Marchant 1999; 2002). To date, the debate surrounding the removal of rare species in

bioassessments has remained controversial and resolution is needed.

One of the difficulties with assessing the importance of removing rare species in bioassessments

is the lack of context from which to judge the consequences of the decision. For example, how

can one evaluate whether the inclusion of rare species provides redundant information with more

common species or provides undue influence (Gauch 1982; Marchant 1999; 2002; Bailey et al.

2004)? Alternatively, how can one determine whether rare species are more sensitive to

ecosystem stress than more common species (Faith and Norris 1989; Cao et al. 1999; 2001)?

Here, both the statistical and biological arguments were tested as separate hypotheses using data

collected from fish species in a well-sampled aquatic system. This study demonstrates contrary

to the previously held notions that rare species provide redundant information as compared to

more common species, or unduly influence multivariate analyses (Marchant 1999; Marchant et

al. 2006); neither was supported. In the case of fish species in the Sydenham River, the

hypothesis for the biological argument for the inclusion of rare species in bioassessments was

supported, while the hypothesis for the statistical argument was not.

The removal of rare species may have large biological consequences for bioassessments. First,

as rare species may not be as rare as perceived simply as a result of sampling bias (Preston 1948,

Resh et al. 2005; Arscott et al. 2006), the removal of rare species may limit the number of

species from which to assess the biological community. Second, and perhaps more importantly,

the removal of rare species may fundamentally change conclusions of multivariate

bioassessments. In this study, when rare species were removed from the analysis, sites impacted

by this removal shifted in multivariate space to a greater degree than those not directly changed

by this decision (e.g. nine fold change between the full data set and when single occurring

59

species were removed; Fig. 3.3). Interestingly, these results were reduced as more species were

removed (Fig. 3.3), demonstrating that as species are removed from the analyses, site-level

species assemblages become more homogenized and differences across multivariate analyses are

minimized. Therefore, the removal of rare species may also remove important indicators of

ecosystem stress. In the case of the Sydenham River, large-scale agricultural activity and

increases in turbidity has led to declines in several species (Staton et al. 2003; Poos et al. 2007),

with issues of turbidity shown to be more related to rare species (e.g. species at risk; Poos et al.

2008). Despite claims to the contrary (e.g. Marchant et al. 2006) this study demonstrates a

scenario where the choice of removing rare species can have limited statistical effect on the

overall multivariate analyses but, at the same time, have large biological effect on particular

observations (i.e. sampling sites). Therefore, the assumption that more common species can

sufficiently define impact or reflect the response of the whole community may not be justified

(Marchant 2002; Marchant et al. 2006).

The importance of choices in multivariate analyses need to be better justified for bioassessments.

For example, this study demonstrates that the removal of rare species had similar (and often less)

influence in multivariate analyses as other choices inherent in its calculation. Previous research

has noted that differences in similarity coefficients (Jackson et al. 1989; Jackson 1997; Cao et al.

1998; Legendre and Gallagher 2001; Podani 2005; Podani and Schmera 2007; Poos et al. 2009)

and ordination techniques (Jackson 1993; Podani 2000; Heino et al. 2003) can lead to divergent

results. Researchers often select methods based on past experience and assume that the resultant

summary adequately models the underlying data, or they choose solutions that are most

interpretable with regard to a priori hypotheses (Jackson et al. 1989; Jackson 1997; Podani

2000). This approach may have severe consequences for the ultimate goal of inferring

community responses for bioassessment. Here, comparisons of choices inherent in multivariate

analyses demonstrated that choices, such as ordination methods (e.g. NMDS), can provide

largely divergent results (Fig. 3.2; Table 3.2). As a result, the removal of rare species may be

less of a concern than previously noted (e.g. Marchant 1999; Marchant et al. 2006), whereas

other choices (e.g. type of ordination) may be more important. These results indicate that

researchers must be mindful of the statistical decisions they make including ordination

technique, similarity coefficient and the exclusion of rarely sampled species, as each choice may

have potential to influence community responses and meaningful conclusions. Other issues,

60

such as sample size (Cao et al. 2001b; 2002), seasonal effects (Furse et al. 1984), experience

(Metzeling et al. 2003), data standardization (Jackson 1993; Cao et al. 1999), taxonomic

resolution (Arscott et al. 2006) and data quality (Cao et al. 2003; Nichols and Norris 2006), also

require adequate justification.

There are issues with the inclusion of rare species in multivariate analyses that should be taken

into account for the bioassessment of aquatic communities that this study could not assess.

Species found infrequently, but with varying abundances have been shown strongly to influence

multivariate analyses (Legendre and Legendre 1998). Researchers who wish to minimize the

impacts of rare species can choose from a variety of data transformations that can downweight

the influence of rare species, although caution is warranted when using small sample sizes

(Jackson et al. 1989; Jackson 1993; Cao et al. 1999). For example, Legendre and Gallagher

(2001) have suggested the use of Helinger transformation for reducing the impacts of rare

species. In addition, authors can choose to reduce their data from abundance to presence-

absence (as done here), which represents the strongest form of standardization and is less likely

to influence analyses (Cao et al. 1999) than many other decisions. The decision to include or

remove rare species will be context dependant. This study demonstrates that, contrary to the

common practice of removing rare species in multivariate analysis, that the impact of leaving

rare species in may be minimal relative to other methodological choices while maintaining

important site-level information on species, including some with species at risk.

Ultimately, the decision to include or remove rare species should be justified by the goals of the

bioassessment. In cases, such as the Sydenham River, where rare species are used as targets for

evaluating ecosystem recovery, rare species should remain in the analyses as their inclusion did

not alter the results (e.g. levels of variation were similar among All, M1 and M5 treatment

groups; Table 2) and they also represent components of the community being assessed.

Naturally, researchers wish to limit their data to reflect the most appropriate number of species,

the most practical similarity coefficient and the most useful ordination technique; however, no

such set of criteria exists. One alternative is to use a consensus approach where several methods

(and choices within methods) are used and compared (Green 1979; Jackson et al. 1989; Jackson

1993). If the methods produce similar results then one can have greater confidence that the

results are more robust and representative rather than being dominated by the set of choices used

61

in the analysis (Jackson 1993). In summary, better justifications of all of decisions in analyses

are needed to ensure bioassessments are rigorous.

Acknowledgements

Funding was provided by NSERC and OGS Scholarships to M.S.P, an NSERC Discovery Grant

to D.A.J, and funding from the Ontario Ministry of Natural Resources and University of Toronto.

This manuscript was greatly enhanced by conversations with C. Harpur.

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Appendices Appendix 3.1 – Site-level effects of methodological choices in bioassessments. Shown are the ratios between mean site-level vector residuals from Procrustes analyses of sites having species removed and those sites having no species removed. Mean site-level vector residual values were separated by sites which had rare species removed (M1: n=2; M5: n=19; and M10: n=63); and compared with those sites that not.

M1 M5 M10 PCoA-J 13.83 2.58 1.10PCoA- Φ 2.40 1.45 1.03PCoA-RR 9.64 1.70 1.26PCoA-SM 12.61 3.47 1.11NMDS-J 8.66 2.41 0.93NMDS- Φ 9.25 2.00 1.03NMDS-RR 7.91 1.91 1.27NMDS-SM 9.58 1.21 0.79CA 13.97 2.46 1.61PCA* 12.61 3.47 1.11 Overall Average 9.76 2.13 1.13

Note: Abbreviations of various treatments are carried forward from Table 1. *PCA values are shown for comparison, but are not included in overall averages.

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Appendix 3.2 – Summary of three-dimensional ordination results. Shown are eigenvalues for Principal Coordinates Analyses (PCA) and Correspondence Analyses (CA), with percent variance explained shown in parentheses. Stress values are shown for Non-Metric Multidimensional Scaling (NMDS).

Ordination Technique Axis 1 Axis 2 Axis 3

A-PCoA-J 3.43 (10.67) 2.79 (19.37) 1.67 (24.58) 1-PCoA-J 3.43 (10.67) 2.83 (19.48) 1.68 (24.71) 5-PCoA-J 3.49 (10.99) 2.89 (20.05) 1.70 (25.40) 10-PCoA-J 3.69 (12.01) 3.11 (22.12) 1.70 (27.62) A-PCoA- Φ 4.87 (15.05) 3.97 (27.31) 2.14 (33.93) 1-PCoA- Φ 4.93 (15.09) 4.01 (27.37) 2.17 (34.00) 5-PCoA- Φ 5.23 (15.51) 4.21 (28.00) 2.30 (34.82) 10-PCoA- Φ 6.19 (17.25) 4.40 (29.51) 2.61 (36.79) A-PCoA-RR 2.21 (13.00) 2.06 (25.12) 1.62 (27.02) 1-PCoA-RR 2.23 (13.12) 2.07 (25.29) 1.63 (27.21) 5-PCoA-RR 2.32 (13.65) 2.15 (26.29) 1.65 (28.23) 10-PCoA-RR 2.58 (15.18) 2.30 (28.71) 1.68 (30.68) A-PCoA-SM 2.03 (15.63) 1.65 (28.36) 0.86 (34.95) 1-PCoA-SM 2.10 (15.67) 1.71 (28.41) 0.89 (35.02) 5-PCoA-SM 2.36 (16.04) 1.92 (29.07) 1.00 (35.85) 10-PCoA-SM 3.06 (17.56) 2.37 (31.16) 1.25 (38.36) A-NMDS-J 0.4158 --- --- 1-NMDS-J 0.4183 --- --- 5-NMDS-J 0.4179 --- --- 10-NMDS-J 0.4235 --- --- A-NMDS- Φ 0.4186 --- --- 1-NMDS- Φ 0.4219 --- --- 5-NMDS- Φ 0.4245 --- --- 10-NMDS- Φ 0.4492 --- --- A-NMDS-RR 0.3994 --- --- 1-NMDS-RR 0.3990 --- --- 5-NMDS-RR 0.4012 --- --- 10-NMDS-RR 0.3950 --- --- A-NMDS-SM 0.3830 --- --- 1-NMDS-SM 0.3870 --- --- 5-NMDS-SM 0.3882 --- --- 10-NMDS-SM 0.4120 --- --- ACA--- 0.32 (12.17) 0.22 (20.62) 0.15 (26.17 1CA--- 0.31 (12.24) 0.22 (21.10) 0.14 (26.83) 5-CA--- 0.30 (14.03) 0.22 (24.31) 0.14 (30.99) 10-CA--- 0.23 (14.88) 0.21 (28.98) 0.12 (36.62)

73

Chapter 4: Contrasting direct versus indirect dispersal in metapopulation viability analyses

Abstract Species dispersal is a central component of metapopulation models. Spatially realistic

metapopulation models, such as stochastic patch-occupancy models (SPOMs), quantify species

dispersal using indirect estimates of colonization potential based on inter-patch distance. In this

study, indirect parameterization of SPOMs was compared with dispersal and patch dynamics

quantified directly from empirical data. For this purpose two metapopulations of an endangered

minnow, redside dace (Clinostomus elongatus), were monitored using mark-recapture techniques

across 43 patches, re-sampled across a one year period. More than 2,000 fish were marked with

visible implant elastomer tags coded for patch location and dispersal and patch dynamics were

monitored. Direct and indirect parameterization of SPOMs provided qualitatively similar

rankings of viable patches; however, there were differences of several orders of magnitude in the

estimated intrinsic mean times to extinction, from 24 and 148 years to 362 and >100,000 years,

depending on the population. In several cases, patches that were in close proximity (high

colonization potential) that were not used by redside dace. This study demonstrates the

importance of incorporating species and patch-specific data directly into metapopulation models,

especially given heterogeneous landscapes.

Keywords: metapopulations, dispersal, population viability analysis, stochastic patch-occupancy

models, parameterization.

74

Introduction

Species dispersal is a central component in the study of spatially structured populations. At a

landscape scale, population viability strongly depends on individual dispersal allowing re-

colonisation of empty habitats or patches (Hanski 1999a). For this reason, species dispersal is

considered the ‘glue’ for maintaining local populations within a network of suitable habitats

(Hansson 1991; Vandewoestijne et al. 2004). The degree of dispersal has an impact on local

population dynamics, on gene flow, and on adaptation to local conditions. For example, low

dispersal can foster isolation and local adaptations (Thomas and Hanski 1997; Resetarits et al.

2005). Alternatively, high species dispersal can have a stabilizing effect on metapopulation

dynamics (Hanski 1999a).

Many species with spatially structured populations are in decline, and population viability

models provide a statistical evaluation of species viability to facilitate informed management

decisions (Ackakaya 2000; Nowicki et al. 2007). Metapopulation viability analyses provide a

spatially realistic evaluation of the local population structure (Hanski 2001; March 2008). A

metapopulation is defined as a system of local populations (patches) connected by dispersing

individuals (Hanski and Gilpin 1991). By quantifying patch dynamics, metapopulation viability

analyses can be used to understand better the importance of ecological processes such as species-

specific dispersal, patch quality and landscape influences (Moilanen and Hanski 1998), and to

enhance management through evaluation of minimum amount of habitat or population size

needed to maintain viability (Hanski 1999b; Robert 2009).

Understanding how the parameterization of metapopulation viability analyses may impact results

can inform managers as to the potential areas of concern when developing management

decisions. One popular type of metapopulation viability analysis is stochastic patch-occupancy

models (i.e. SPOMs), which have been used extensively to model the viability of spatially

structured populations (Hanksi 1999; Moilanen 1999). For example, SPOMs were used in

studies of species with conservation concern, such as capercaillie (Grimm and Storch 2000),

American pika (Moilanen et al. 1998) and Glanville fritillary, and silver spotted skipper

butterflies (Hanski et al. 1994). As SPOMs provide a simplification over traditional population-

viability analyses (Akayaka and Sjögren-Gulve 2000), they do not require demographic or stage

75

data, but only occupancy, colonization and extinction rates, which can be estimated readily from

empirical data (Hanski 1994,1999; Moilanen 1999, 2004; Grimm et al. 2004).

The influence of dispersal on metapopulation viability is often evaluated using some

approximation of colonization potential (Verbroom et al. 1993; Hanski and Gilpin 1997; Frank

and Wissel 2002; Heinz et al. 2005). The easiest approach to describe colonization potential (i.e.

patch accessibility) is as a function of distance between a starting patch to a target patch and the

ability of species to disperse (Hanksi 1994; Hanski et al. 1996; Heniz et al. 2005). This

relationship can be quantified in several ways; however, most often this estimation is done by

assuming that colonization potential declines exponentially with distance (i.e. exponential decay;

Hanski 1994; Vos et al. 2001; Frank and Wissel 2002). It is uncertain how well the assumption

of exponential decay can model species-specific dispersal (Hill et al. 1996; Baguette et al.2000;

Heinz et al. 2005). Whether simple formulae are adequate in describing species- and patch-

specific movement in metapopulation models remains an open question (Heinz et al. 2005;

Marsh 2008).

The overall aim of this study is to assess whether direct parameterization of species dispersal can

impact estimates of metapopulation viability. For this assessment, a detailed mark-recapture

survey of the endangered fish, redside dace (Clinostomus elongatus), was conducted in the

Greater Toronto Area, Ontario, Canada. The redside dace is a spatially structured, pool-dwelling

species that is undergoing declines in the majority of its range due to impacts from urbanization

(COSEWIC 2007; Poos and Jackson, submitted). Two locations on the Rouge River were

monitored, including one location on Leslie Tributary, and the other location on Berczy Creek

(Figure 4.1). These locations were shown previously to have among the highest abundances of

redside dace recently sampled across its entire Canadian range (Reid et al. 2008).

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A) Leslie Tributary

L6

L8L9

L10

L12 L11L13L17

L18L19

L20

L5

L3 L2

L4

L1

L14

L7

100m

N

N

L15

B1 N100m

B) Berczy Creek

B3

B4

B2B5B6

B7

B8

B9

B10B11

B12B13B14

B15B16B17

B18 B19

B20B21

B23B22

L16

0   2    4          8  

Figure 4.1 – Study sites on Rouge River, Ontario where redside dace (Clinostomus elongatus) dispersal and patch dynamics were monitored. Study locations: A) Leslie Tributary, and B) Berczy Creek, were sub-divided into extensive sites (black), where redside dace were tagged with a color-coded visual implant elastomer tag, and extended sites (grey), which were monitored for tag movement.

77

Methods The metapopulation dynamics of redside dace were monitored by enumerating the dispersal of

tagged individuals on monthly intervals during a one-year period. For this study, each location

was sub-divided into two areas: intensively monitored sites where individuals were tagged; and,

extended sites, that were beyond those areas where fish were not tagged but where tagged fish

could have potentially moved (Figure 4.1). As meta-populations can be defined in a number of

ways (Hanski 1999a), a metapopulation was defined as an assemblage of local populations

inhabiting spatially distinct habitat patches (Moilanen and Hanski 1998). Redside dace live

primarily in clear water with well-defined pools (COSEWIC 2007); therefore, each spatially

distinct pool segregated by a well-defined riffle (e.g. a passable, but natural migratory barrier)

was selected as a habitat patch. Leslie Tributary was sub-divided into 20, connected and distinct

patches, with 10 intensive sites and five extended sites on both upstream and downstream ends.

Similarly, Berczy Tributary was sub-divided into 13 intensive sites with five extended sites on

each of the upstream and downstream ends (Figure 4.1).

Sampling was conducted using multiple-pass depletion surveys at each pool. Using a twenty-

foot bag seine (1/4” mesh), each site was surveyed until depletion of redside dace, with a

minimum of three sampling events conducted at each site per time period. At each pool, redside

dace were implanted with visual implant elastomer (VIE) tags colour-coded for their location

(Plates 1 and 2). VIE tags were chosen because they had good tag retention and negligible

effects on survival, growth and behavior when used on other species (Dewey and Ziegler 1996;

Goldsmith et al. 2003; Walsh and Winkelman 2004). Tags were injected subcutaneously near

the anal fin on the ventral surface (See Plates 1 and 2). All redside dace were held in well-

oxygenated flow-through bins for 2-4 hours to monitor for potential physiological stress, and

then returned to the river at the site of capture. Both intensive and extended sites were re-

sampled for redside dace at monthly intervals, except under winter-ice conditions (November-

March) and when redside dace were spawning (June) to not disrupt this important life stage for

an endangered species. All redside dace sampled in a recapture event were examined for the

presence of a VIE tag. Redside dace dispersal and metapopulation dynamics were tracked and

mark-recapture data were recorded. If redside dace were re-captured at a new location, they

were subsequently tagged posterior to the existing tag, with a new colour code for the recapture

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location. The dispersal patterns, such as average distance dispersed, and proportion of stationary

tags, of the metapopulations were compared using non-parametric Mann-Whitney U-tests and

log-linear models (G-tests with Yates continuity correction; Zar 1999) respectively, in the R

language v2.80 (R Development Team 2008).

Determining Metapopulation Viability

The viability of the two stream metapopulations was quantified using direct and indirect

parameterization of stochastic patch-occupancy model (SPOM). SPOMs use a time-continuous

Markov-chain model (Hanski 1999; Grimm et al. 2004). Each patch (i) is assumed to be in one

of two states, vacant (xi =0) or occupied (xi=1). Changes in these states can occur from a patch

becoming vacant due to local extinction (xi: 1 0) or correlated extinction (i.e. regional

stochasticity) from another patch (xj, xi: 1 0). Alternatively, a vacant patch can become

occupied (xi: 0 1) via colonization from another patch (j). The state of the whole

metapopulation (xi, … xn) is given by a vector of states xi of these individual patches. The

metapopulation models were parameterized using a combination of indirect (i.e. dispersal ability

from empirical data and incidence functions; Hanski 1999) and direct parameterization of

SPOMs (using rates of actual patch colonization from empirical data alone). The models were

quantified as follows.

Colonization Rate:

Colonization between two patches i and j (bij) was defined using an incidence-function model

(Hanski 1994):   ·     · exp / I , where y is a parameter, and Mi is the number of

emigrants from pool i. The mean number of emigrants leaving a pool was estimated using data

from the tagging study. To account for the potential uncertainty with missing emigrants leaving a

patch, the probability of detection at each pool (PDi) was quantified using maximum likelihood

from the n-pass depletion surveys (Zippin 1956, 1958) with the Bayesian modification from

Carle and Strub (1978). These were coded in the R v2.80 (R Development Team 2008) using the

fisheries-assessment package FAS (Ogle 2009). The total number of emigrants leaving each

patch per year was measured as ∑ … 1 1 , where is the

uncorrected number of emigrants. Similar to most metapopulation models, a distance-based

dispersal kernel using a negative exponential decay was used, where exp  / , and dij is the

im

79

distance from patch i to patch j and d0 is the mean dispersal ability of redside dace. This type of

dispersal kernel has been used extensively in metapopulation models and assumes that patch

accessibility is dependent on distance (Hansson 1991; Hanski et al. 1996; Hokit et al. 1999;

Moilanen 2004). As dispersal data are quantified indirectly using patch area or depth, hereafter,

this approach is referred to as indirect parameterization of the metapopulation model. This type

of indirect parameterization allows researchers to extrapolate relationships in patch occupancy,

often by using species life-history characteristics, without the need of labor-intensive field

studies (Hanski, 1994; Moilanen 2004; Heinz et al. 2005).

A non-linear (polynomial) regression of mean distance of dispersal of recaptured fish through

time was used to identify potential dispersal across the patches. The fits of these non-linear

regressions were highly significant (Leslie Tributary; r2 = 0.92, p < 0.01, Berczy Creek; r2 =

0.88, p < 0.01) and the average distance dispersed (dI) for a one-year period (one time step in

final SPOM model) was 210m for redside dace in Berczy Creek and 150m for Leslie Tributary

(Table 1). As there were several consecutive surveys, it was possible to estimate y from the

number of transitions (i.e. an empty patch becoming occupied and vice versa; Hanski 1999a).

For this, a GLM procedure was used which considered multiple snapshots of the sampling events

using a binomial distribution and logistic-link function developed in the R programming

language v2.80 (R Development Team 2008) using the incidence function (see Oksanen 2004 for

details). The value of the y parameter for Leslie Tributary was 0.0816 and Berczy Creek was

0.0713. Finally, for dij, a distance matrix was measured using the river distance between patches

as calculated by a geographic information system.

Extinction Rate:

Extinction rates can be quantified in many ways (Hanksi 1999a). The simplest form of

determining extinction rate (Ei) is using the area of the patch (Ai), and given by   ,

where e defines the extinction probability of a patch of unit size, and x defines the scaling of the

extinction risk with patch area (Hanski 1998; Moilanen 2004). This model assumes that

probability of extinction generally depends on population size, which, in turn, is usually given by

a simple linear or power function of patch area. This relationship has been demonstrated on both

empirical and theoretical grounds (Lande 1993; Foley 1994; Hanski 1994; 1999a; Hanski et al.

1996). Here extinction rates were calculated using patch (i.e pool) depth (d). Patch depth was

80

calculated by taking the average of 60 equidistant point measurements as suggested by the

Ontario Stream Assessment Protocol (OMNR 2007). Patch depth was used because redside dace

are known to be pool-dwelling species (COSEWIC 2007) and, therefore, depth may be more

relevant to model patch dynamics. Indeed, redside dace abundances were more correlated with

patch depth (r=0.44, p=0.0018) than patch area (r = 0.39, p = 0.0048; Poos and Jackson,

unpublished data). The extinction rate was fitted using an incidence function relating species

presence in the patches over time and depth. For this modeling, a GLM procedure was used that

considered multiple snapshots of sampling events using a binomial distribution developed in the

R programming language v2.80 (R Development Team 2008) using the incidence function

(Oksanen 2004). The parameters of the incidence function for Leslie Tributary and Berczy

Creek were x = 0.4926, 0.5652, and e = 3.685, 4.187, respectively.

Incorporating Dispersal Directly into the Metapopulation Model:

Recent theoretical studies on the impact of species movement have found that it can alter

metapopulation viability (Heinz et al. 2005; 2006; Revilla and Wiegand 2008). Therefore, the

incidence function models were extended by incorporating species dispersal directly into the

metapopulation model using empirical data of patch-specific movement. As dispersal data was

directly used to quantify the metapopulation model, hereafter, this is referred to as direct

parameterization of the metapopulation model. For colonization rate, a model developed by

Frank and Wissel (1998; 2002) was used, which (in this case) is identical to the incidence

function model and allows the incorporation of patch dynamics (Grimm et al. 2004; Heinz et al.

2006). This model took into account three processes; emigration of individuals from occupied

patches; dispersal to a target patch; and, the establishment of a new subpopulation on the target

patch. The rate of colonization, bij, was defined as

· · .

0 where, Mi was the number of emigrants leaving the occupied patch i per year

(previously defined), ni was the number of connections from patch i to other patches, rij was the

probability of an individual started at patch i successfully dispersing to patch j, and Ij was the

number of immigrants needed to establish a new subpopulation (Frank and Wissel 1998; 2002).

For the probability of dispersal between patches (rij), a patch-colonization matrix was developed

using the empirical tagging results for each time period. As the tags were colour coded for patch

81

location (at each time period), rij was defined empirically as the ratio of fish that started at patch i

that dispersed to location j, across all recaptured fish. In this instance, tag loss was not accounted

for as no tag-related behavioral response or tagging related mortality was assumed (i.e. this ratio

was adequate given equal likelihood of mortality of a tagged fish versus non-tagged fish). Given

the monitoring data (>75 hours), only minor amounts of tag-related mortality occurred (<0.01%),

and all occurred on the first sampling day (likely due to experience bias). Finally, to quantify Ij,

an incidence-function model was developing using the probability a patch persistence (across all

time periods) given the starting population size (StPopn) of each patch at the start of the study

(t1). For this approach a GLM function was used with binomial distribution and logit link in the

R programming language v2.80 (R Development Team 2008; Appendix 4.1).

Regional Stochasticity

Regional stochasticity refers to the level of correlated extinctions caused by factors influencing a

shared geographic location, such as weather or disease (Lande et al. 1988; Lande 1993; Foley

1994). Regional stochasticity has the ability to impact metapopulation viability by incorporating

the influence of the fate of proximal patches. The influence of regional stochasticity was

quantified at three levels: 0 (no influence of regional stochasticity); 0.1 (a moderate level of

regional stochasticity); and, 0.2 (more severe regional stochasticity).

Comparing Viability of Metapopulations Using Direct versus Indirect Parameterization

The ultimate viability of patch (i) was defined using the intrinsic mean time to extinction Tm =

1/λ, determined using the reciprocal value of the overall extinction rate λ calculated using a plot

of −ln(1 − P0(t)), where P0 is the probability of extinction at a given time (t) (Verbroom et al.

1991; Grimm et al. 2004; Grimm and Wissel 2004). Intrinsic mean time to extinction has been

previously shown to be an adequate currency in assessing the viability of metapopulations and

can be easily extracted from simulation data (Frank and Wissel 1998; Grimm and Wissel 2004;

Heinz et al. 2006). Transitions in metapopulations were simulated 10,000 times using ‘stochastic

time steps’ (Frank et al. 2004; Grimm et al. 2004) of transition probabilities of extinction and

colonization rates. For this estimation a manually created sub-routines was created in the

software program Meta-X (Frank et al. 2003; Grimm et al. 2004), a metapopulation program

flexible for incorporating behavior into metapopulation-viability analysis (Heinz et al. 2006).

82

Results In total, 2,141 redside dace were tagged and monitored across 43 patches during in a one-year

period from 2007-2008. Due to logistical issues, the stream systems were not sampled during

winter-ice conditions (November-March) or when redside dace were spawning in June given the

potential to disrupt spawning activities. Recapture rates for redside dace - calculated as the

proportion of fish marked during the preceding marking period that were recaptured - were

generally high (>25%) during the initial four monitoring events, ending in October 2007. These

numbers were greatly reduced by the following spring, with recapture rates < 10% likely due to

high over-winter mortality or the re-distribution of tagged fishes to areas beyond the study

location (Table 4.1). In addition, the capture efficiency – as determined by probability of

detection using n-pass depletion surveys (Zippin et al. 1954; 1956; Carle and Strub 1978) – was

also very high for both study systems: Leslie Tributary (mean 71%); and, Berczy Creek (mean

65.6%, Appendix 4.1).

Table 4.1 – Summary of mark-recapture information of the endangered fish the reside dace (Clinostomus elongatus) used to directly parameterize stochastic patch occupancy metapopulation models. Shown are two locations in the Greater Toronto Area, Ontario, Canada: A) Leslie Tributary, and B) Berczy Creek. Note: items denoted with a single asterisk (*) represent a significant difference between populations (Mann-Whitney U for average dispersal, G-test for stationary tags, p<0.05).

A) Leslie Tributary B) Berczy Creek

Time Cum. Tags

Recap. (%)

Avg. (m)

Max. (m)

Stationary Tags (%)

Cum. Tags

Recap. (%)

Avg. (m)

Max. (m)

Stationary Tags (%)

T1: July 133 0.46 21* 175 0.67 342 0.61 9* 125 0.74 T2: Aug. 305 0.26 91 227 0.19 511 0.26 84 290 0.23 T3: Sept. 404 0.28 71 547 0.52 770 0.26 66 357 0.44 T4: Oct. 483 0.26 139 680 0.03* 1,045 0.25 77 315 0.41* T5: Apr. 503 0.04 196* 680 0.00 1,137 0.03 129* 411 0.00 T6: May 542 0.09 182 649 0.14 1,376 0.03 174 411 0.10 T7: July 662 0.10 180* 547 0.17 1,479 0.06 82* 275 0.29 Avg. 0.14 105 0.31 0.18 53 0.49

Legend: Cum. Tags refers to the cumulative number of tags released in the population; Recap (%) refers to the recapture rate for cumulative tags, Avg. (m) refers to the average distance the recaptured tags were captured at, Max. (m) refers to the maximum distance recaptured tags were captures at, and Stationary Tags (%) refers to the percentage of tags, at each time interval, that were recaptured in the same location as tagged.

Metapopulation Dynamics

In all cases, dispersal was higher in Leslie Tributary as compared with the Berczy Creek;

however, this difference was only statistically significant in time periods 1, 5, 7 (Mann-Whitney

U-test; p=0.015, 0.004, 0.033, respectively; Table 4.1). The difference in dispersal was not due

83

to more tags dispersing as there was no significant difference in the proportion of stationary tags,

except for time period 4 (G-test, p=0.0003; Table 4.1). More likely, the increase in dispersal was

due to larger average dispersal per individual, as indicated by mean and maximum dispersal

through time (Table 4.1).

Metapopulation and Patch Viability

Metapopulation viability, as indicated by both probability of extinction through time and the

intrinsic mean time to extinction, were orders of magnitude different depending on whether the

patch-occupancy model was parameterized indirectly using colonization potential or directly

using observed colonization (Figure 4.2; Table 4.2). For example, when the patch-occupancy

models were parameterized directly using observed colonization, the Leslie Tributary population

was inviable long-term and the Berczy Creek showed much longer viability. The intrinsic mean

time to extinction for Leslie Tributary was 24 years, and occurred in as little as 12 years

(regional stochasicity set at 0, 0.2, respectively). The probability of extinction was plotted over

time and showed that 95% of the simulations were extinct in less than 100 years. Similarly, the

intrinsic mean time to extinction for Berczy Creek was calculated as a maximum of 148 years

and occurred in as little as 32 years (regional stochasticity = 0, 0.2 respectively), with 95% of

simulations showing metapopulation extinction in under 1000 years (Figure 4.2; Table 4.2).

84

Table 4.2 – Intrinsic mean time (in years) to extinction (Grimm and Wissel 2004) of two metapopulations of the endangered fish the redside dace (Clinostomus elongatus) for different levels of regional stochasticity.

A) Leslie Tributary B) Berczy Creek

Regional Stochasicity 0 0.1 0.2 0 0.1 0.2

Intrinsic mean time to extinction (Indirect Parameterization)

362 95 48 109,594 3,417 764

Intrinsic mean time to extinction (Direct Parameterization)

24 17 12 148 54 32

When the patch-occupancy models were parameterized using indirect colonization, based on

estimated dispersal ability, one population (Berczy Creek) was deemed as viable and quasi-

stationary (regional stochasticity = 0; Figure 4.2; Table 4.2), with 95% of simulations showing

viability beyond 250,000 years (Figure 4.2). The remaining population estimates varied

considerably in their viability, ranging in intrinsic mean times to extinction from 48 to 348 years

in Leslie Tributary and from 764 to >109,000 years in Berczy Creek (regional stochasticity 0.2,

0, respectively).

85

I) Direct parameterization of dispersal II) Indirect parameterization of dispersal

A)

B)

YEARS

Figure 4.2 – Metapopulation viability of the endangered species the redside dace (Clinostomus elongatus) in two stream metapopulations: A) Leslie Tributary, and B) Berczy Creek. Shown are the probabilities of extinction (y-axis) in years (x-axis) of a stochastic patch-based metapopulation model. Models were parameterized using: I) indirect parameterization of colonization and dispersal via patch distance, and; II) direct parameterization of colonization and dispersal using empirical estimates from a mark-recapture study. Legend: Vertical hashes represent a time interval of 100 years, solid lines indicate population trajectories where regional stochasticity was set to 0, dashed lines set to 0.1 and dotted lines set at 0.2.

86

Specific-patch viability mirrored overall metapopulation viability, with all patches showing

reduced viability when parameterized directly from empirical data (Figure 4.3). In all cases

(except L12) patch viability was over-estimated with the indirect parameterization relative to

direct parameterization of the SPOM (Figure 4.3). Mean patch viability was significantly higher

with indirect parameterization of the SPOM for both Leslie Tributary (mean indirect patch

viability = 0.51 ± 0.10, mean direct patch viability = 0.35 ± 0.08; Welch’s t-test; t= 26.85, p-

value << 0.0001) and Berczy Creek (mean indirect patch viability = 0.69 ± 0.09, mean direct

patch viability = 0.39 ± 0.08; Welch’s t-test; t = 11.204, p-value << 0.0001).

0

0.5

1

0 0.5

B15L13

B14

L14

L12

L9

B16B12

B9

B10 B8L10

L6

B7B13

B6B18 B11

L11

L7

1

Patc

h V

iabi

lity

(Ind

irect

Par

amet

eriz

atio

n)

Patch Viability (Direct Parameterization)

L15B17

L8

Figure 4.3 – Differences in patch viability parameterized using indirect (y-axis) and direct (x-axis) patch dynamics of the endangered species the redside dace (Clinostomus elongatus) in two stream metapopulations: A) Leslie Tributary (L6-L15), and B) Berczy Creek (B6-B18). Shown are the mean probabilities of persistence of a given patch across 10,000 simulations. To demonstrate the variability in patch viability, 25% quantiles are overlaid as the negative of both the vertical and horizontal axes, while 75% quantiles are overlaid as the positive vertical and horizontal axes. The dashed line is a 1:1 line.

87

Interestingly, the rankings of patch viability did not markedly differ based on the

parameterization of the SPOM. For example, both indirect and direct parameterization of the

SPOM identified the same five most-viable patches per population (overall) as: L6, L8, L10,

L11, L9; and, B7, B13, B6, B11, B18. One notable difference in patch viability was that several

patches that were in close proximity to good-quality patches had significantly lower viability

when parameterized using direct parameterization (Figure 4.3). Patches such as L9, L11, B6 and

B9 had reduced viabilities when directly parameterized (mean indirect viability = 0.57, 0.84,

0.86, 0.90, respectively; mean direct viability = 0.28, 0.46, 0.63, 0.17; Figure 4.3).

Discussion Metapopulation-viability models have a long history of use (e.g. Levins 1969, 1970) and provide

advantages over traditional population-viability analyses (PVAs). One clear advantage of using

stochastic patch-occupancy metapopulation models (SPOMs) over traditional PVAs (e.g.

structured models, demographic models; (Akcakaya and Sjorgen-Gulve 2000; Morris and Doak

2002) is that they require the parameterization of fewer variables (Ovaskainen and Hanski 2004).

This reduction is especially advantageous for modeling endangered species, where enumeration

is complicated by rarity and where greater uncertainty exists. Simplification is often needed as

incorporating species-specific or demographic data into ecological studies can be difficult, time

consuming, or not economically possible. Stochastic patch-occupancy models allow for

simplification of metapopulations, as only patch occupancy, colonization and extinction rates are

needed, even within a single snapshot (Moilanen et al. 2004; Marsh 2008).

There is a tradeoff between simplification of PVAs by using less parameters and with the added

value and information that those parameters may have (Shreeve et al. 2004). This study

demonstrates that differences in species dispersal patterns, a key component of metapopulation

models such as SPOMs, have the ability to dramatically impact estimates of metapopulation

viability. Further, this study demonstrates that direct parameterization of species-specific

dispersal can reduce the overall estimates of viability of redside dace metapopulations by several

orders of magnitude over estimates using indirect estimates (e.g. exponential-decay kernels). In

several cases (e.g. L9, L11, B6 and B9), patches that were in close proximity to good-quality

patches (i.e. had high dispersal potential and high viability with indirect parameterization) but

had significantly lower viability when parameterized using direct parameterization (Figure 4.3).

88

From these results it may be inferred that the study patches are likely distributed across a

heterogeneous landscape where occupancy rates or habitats may not be equal, as assumed by

SPOMs. Landscape and patch heterogeneity has been shown to impact colonization potential,

which alters metapopulation dynamics (Gustafson and Gardner 1996; Heinz et al. 2005). Even if

habitats were equal, studies of the rosyside dace (Clinostomus funduloides), a sister species,

demonstrated that dispersal could not be predicted by suitable habitats (Freeman and Grossman

1993). These conclusions are in agreement with others which indicate that better integration of

species-specific behaviour is needed into the analyses of metapopulations (Tischendorf 2001;

Tischendorf and Fahrig 2001; Vos et al. 2001; Heinz et al. 2005, 2006; Baguette and vanDyck

2007; Marsh 2008).

There is an ongoing debate whether SPOMs are useful in cases where the assumptions of classic

(i.e. Levin’s type) metapopulations are not met (see Levins 1969; Harrison 1994; Baguette 2004;

Hanski 2004; Shreeve et al. 2004). For example, empirical studies have demonstrated large

temporal variation in patch dynamics (i.e. colonization and extinction rates) can lead to

sensitivity in SPOMs (Crone et al. 2001; Thomas et al. 2002). In addition, there are few

empirical examples of metapopulations that meet the assumption of a constant pulse of

extinction-colonization (e.g. pool frog, Sjogren-Gulve 1991; Glanville fritillary butterfly, Hanski

et al. 1994; but see Nowicki et al. 2007); whereas, the vast majority do not (Harrison 1994;

Baguette et al 2004). However, recently SPOMs have been shown to be appropriate for use over

a range of spatially structured populations from classic metapopulations to species found in

fragmented landscapes with patchy distributions (Ovaskainen and Hanski 2004). For example,

using a SPOM with individual-based background, Ovaskainen and Hanski (2004) demonstrated a

unifying framework for incorporating metapopulation dynamics into SPOMs. Their study

suggested that instead of attempting to identify metapopulation types, research should focus on

relevant processes, such as dispersal. By understanding processes behind the variability shown

between (and within) patch dynamics, the conservation and management of endangered species

can be improved (Revilla and Wiegand 2008).

Comparative simulations of how metapopulation analyses perform when altered are important

tools for ensuring appropriate management action. Interestingly, despite finding large

differences in the prediction of intrinsic mean time to extinction (Table 4.2), this study

demonstrates that regardless of how the SPOMs were parameterized (directly or indirectly), they

89

identified qualitatively similar patches as having the highest viability (Figure 4.2). Such results

are reassuring given that SPOMs rarely are used for exact quantitative analyses, but rather used

to compare among several scenarios to develop decision support (Lindenmayer and Possingham

1996; Heinz et al. 2006). These results suggest that qualitative comparisons of SPOMs may still

be a fruitful management option; however, care should be taken with estimates of probability of

extinction through time, which may be over-estimated (as shown here). Indeed, others have

shown SPOMs to be comparable to other spatially realistic models (Kindvall et al. 2000; Keeling

2002; Ovaskainen and Hanski 2004). Modifications to SPOMs, such as consideration of patch

quality (Hanski and Moilanen 1998; Ovaskainen and Hanski 2002), improved dispersal metrics

(Ovaskainen 2004; Heinz et al. 2005), incorporation of transition states (Thomas and Hanski

2004), rigorous parameter-estimation techniques (Moilanen 1999; Dreschler et al. 2003), and/or

direct parameterization of dispersal data, will only ensure better integration of biological data

into SPOMs.

Studying metapopulations in stream settings is challenging and has limitations which should be

noted. Fagan (2002) demonstrated how a dendritic network can provide additional isolation of

patches not encountered in terrestrial landscapes. Further, Gotelli and Taylor (1999)

demonstrated how stream fishes may not fit Levin’s type metapopulation models as migration

may be asynchronous in upstream versus downstream movement. Finally, mark-recapture

studies have shown that tagged individuals may travel outside recapture territory, thereby,

reducing estimates of overall colonization (Ovaskainen 2004). All the above examples may

explain the lower viability using direct parameterization of SPOMs in this study. In response, the

study design accounted for such potential shortcomings in several ways. First, the sampling was

done sequentially, across the entire stream network (using block nets during sampling to

eliminate movement induced by the sampling of each pool). Therefore, unlike most studies in

terrestrial systems, this study was able to monitor the stream metapopulations within defined

boundaries, sampling the entire patch, as well as neighboring patches and connections. Second,

this study re-sampled the patches at 7 time intervals to determine intra-and inter- annual

asynchrony in movement and rate of patch fidelity. In general asynchrony in movement was

identified at a single time period (T4; G-test, p<0.05); however, there was also variation in patch

fidelity (Table 4.1). This variation in patch fidelity indicates that redside dace may be more

consistent with the patchy population model (Harrison 1991) than the with a Levin’s type

90

metapopulation model (Levins 1969). Temporal variation in patch fidelity was also shown in

metpopulations of rosyside dace (Freeman and Grossman 1993). Third, in this study the

boundaries of the recapture locations were extended to include patches beyond the marking

locations (~ 350 meters on upstream and downstream ends per location; Figure 4.1). These

extended patches allowed both average dispersal of tagging individuals and the rates of dispersal

out of their marking locations to be determined. Regardless of time period, > 75% of tags were

recaptured within 350 m meters of their starting pool for Leslie Tributary and > 90% of tags in

Berczy Creek, suggesting that the monitoring was done at the appropriate scale. Finally, this

study accounted for the probability of missing tags due to sampling bias by correcting for

abundance counts using probability of detection. These approaches suggest that, although

asynchrony (and perhaps over-winter mortality) may be an issue, the study design was adequate

to monitor species and patch specific movement.

Understanding how species- and patch-specific qualities can alter SPOMs is an important area

for advancing metapopulation-viability analyses, which are a crucial tool for the management of

endangered species (Ackakaya 2000; Morris and Doak 2002; McCoy and Mushinsky 2007).

Species- and patch-specific processes can alter metapopulation dynamics in many ways

(Roitberg and Mangel 1997; Hanski and Moilanen 1998; Schtickzelle et al. 2006). Patch quality

can affect both the probabilities of colonization and extinction of an empty patch (Hanski and

Moilanen 1998; Thomas et al. 2002). Changes in landscape structure can alter migration

pathways between patches, which may impact dispersal (Roitberg and Mangel 1997;

Schtickzelle et al. 2006; North and Ovaskainen 2007). Finally, the configuration of patches may

play a role in population viability (Robert 2009). Maintenance not only of high-quality patches,

but their connectivity, should be an important aspect of endangered species management,

including redside dace.

Accounting for regional and stochastic processes are important considerations for the

management of endangered species. In this study, when the rate of regional stochasticity was

altered, the intrinisic mean time to extinction quickly decreased (Table 4.2). The impact of

regional stochasticity was small in cases where populations were already considered to be

inviable (e.g. Leslie Tributary with 0 stochasticity), but it had a large impact in cases where a

population was viable or quasi-stationary (e.g. Berczy Creek with 0 stochasticity; Figure 4.2).

For example, the Berczy Creek population was considered to be viable if regional stochasticity

91

was ignored. Altering rates of regional stochasticity from 0 to 0.1 and 0.2 caused the populations

to move towards extinction (intrinsic mean time to extinction from >109,000 to 54 and 32 years,

respectively). These results indicate that regional stochastic factors (e.g. weather, drought) that

may alter patch dynamics and have undue influence on metapopulation viability, as others also

have shown (Grossman et al. 1982; 1985; Lande 1993; Foley 1994; Robert 2009).

The conservation applications of species-specific dispersal may help inform management

decisions. By incorporating species- and patch-specific data directly into metapopulation

models, managers may be better apt at determining the relative importance of spatial and

temporal factors, such patch connectivity and seasonal impacts. In this study, empirical

estimation of patch viability was shown to be qualitatively similar when parameterizing SPOMs,

but that estimates of metapopulation viability were shown to be significantly higher when using

indirect parameterization of dispersal. These results indicate that care is needed in ensuring that

even simplified metapopulation models, such as SPOMs, are consistent with biological data.

Comparisons of how species- and patch-specific data directly impact metapopulation models, as

done here, may be one way to accomplish this cautionary aspect (Dreschler et al. 2003; Grimm et

al. 2004). Further study into the impact of species-specific behavior and patch dynamics on

metapopulation viability will provide additional insight for both management and conservation

issues.

92

Acknowledgements

Funding was provided by NSERC Canada and OGS Scholarships to M.S.P., an NSERC

Discovery Grant to D.A.J., Interdepartmental Recovery Fund #1410 provided by Fisheries and

Oceans (DFO), the Ontario Ministry of Natural Resources (OMNR), and the University of

Toronto. All field work was conducted under an approved Animal Care Protocol (# 20006805)

from the University of Toronto Animal Care Committee; and under the approval of the redside

dace recovery team (chair A. Dextrase) and with the guidance of the Ontario Ministry of Natural

Resources (J. Pisapio). The Toronto Region Conservation Authority (TRCA) Aquatics Group

(Christine Tu, David Lawrie, Tim Rance) provided logistical support and help during this

project. Field work was conducted by a dedicated group of volunteers from- the University of

Toronto – A. Drake, C. Harpur, B. Edwards, M. St. John, M. Neff, P. Venturelli, A. Manning, M.

Granados, J. Ruppert, S. Sharma, N. Puckett, C. Hart, C. Howard, M. Luksenberg, J. Brett, S.

Walker – and the Toronto Region Conservation Authority – D. Lawrie, C. Tu, T. Parker, T.

Rance, B. Paul, E. Elton, B. Stephens, C. Hart, L. DelGiudice, B. Moyle, and M. Parish – and P.

Ng, M. Ken, C. Zehr, B. Edwards, Y. Nozoe, D. Trim, K. Lee, D. Morodvanschi and D. Forder

(Ontario Streams). This work was improved by discussion with Marie-Josee Fortin and review of

earlier drafts of this manuscript. Finally reviewers were helpful for in providing valuable

suggestions.

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Appendices

Appendix 4.1 –The endangered redside dace (Clinostomus elongatus). Photo credit: Mark Poos.

Appendix 4.2 –Visual implant elastomer (VIE) tag inserted subcutaneously on the ventral surface of the endangered redside dace (Clinostomus elongatus). Photo credit: Mark Poos.

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Appendix 4.3 – Model parameters used in the stochastic patch-occupancy models. Shown are the number of emigrants (mi), the mean probability of detection (PD), the number of emigrants adjusted for potentially missed tags (Mi), the number of immigrants needed to start a new sub-population, and the rate of extinction (Ei). Other parameters include: the incidence function for Leslie Tributary (dI = 210, x = 0.4926, e = 3.685, y = 6.12), and Berczy Creek (dI = 150, x = 0.5652, e = 4.187, y = 7.01). Not shown: dij given it is a pairwise estimate rather than unique for each pool.

Patch mi PD Mi Ij, 0.5 Ei

Leslie Tributary L6 35 0.710 60 4.89 0.2593 L7 7 0.619 11 2.48 0.2901 L8 38 0.767 67 5.48 0.1956 L9 10 0.740 17 8.21 0.2724 L10 37 0.647 61 5.10 0.2583 L11 38 0.780 68 5.74 0.1870 L12 14 0.727 24 8.03 0.3167 L13 8 0.792 14 2.88 0.2912 L14 0 0.000 0 7.69 0.3383 L15 3 0.800 3 7.69 0.3304 Overall 190 0.718 326

Berczy Creek B6 48 0.676 80 1.91 0.2645 B7 116 0.564 181 7.43 0.1918 B8 23 0.582 36 2.78 0.2231 B9 8 0.667 13 9.20 0.1394 B10 17 0.836 31 2.85 0.1639 B11 34 0.741 59 4.91 0.1683 B12 9 0.800 16 7.43 0.2026 B13 88 0.715 151 5.91 0.2025 B14 2 0.882 4 10.94 0.2110 B15 5 0.800 9 10.94 0.2726 B16 8 0.744 14 1.91 0.1569 B17 0 0.667 0 7.43 0.2388 B18 49 0.661 81 2.78 0.1417 Overall 407 0.656 670

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Section II:

Reducing uncertainty from methodological choices using consensus methods

103

Chapter 5: Reducing uncertainties in modeling the distribution of endangered species using habitat-based ensemble models

Abstract Modeling the distribution of endangered species is often problematic due to their rarity and the

consequential statistical issues. Recent studies have demonstrated the utility of using ensemble

models to reduce the uncertainty of singular predictions when modeling species distributions.

This study provides a quantitative evaluation of the efficacy of ensemble models for improving

the prediction of endangered species and their habitats using the endangered fish, the redside

dace (Clinostomus elongatus), as a model organism. Specifically this study asks: 1) how well do

ensemble models improve modeling metrics (e.g. specificity, sensitivity and overall

classification); 2) how many singular methods are needed to build the optimal ensemble; and, 3)

what scale(s) and type(s) of habitat are most important for modeling this species. For this

evaluation, five ensemble models were compared based on seven singular approaches. Habitat

variables were derived from 200 sites measured from 1997-2007 and divided hierarchically into

fine-, intermediate-, and broad-scale habitat. In all cases, the ensemble models were equal to or

better than any singular method across all modeling metrics, although there was large variation

with certain combinations of initial models. This study demonstrates how comparative analysis

of modeling types, ensemble approaches and scales can be useful for reducing uncertainty in the

modeling of endangered species and their habitats.

Keywords: ensemble model, consensus model, conservation, biodiversity, biogeography, species

distributions.

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Introduction Predictive models of species distributions have become important tools in modeling changes in

biogeography and biodiversity (Guisan & Zimmermann, 2000; Guisan & Thuiller, 2005; Sharma

and Jackson 2008). Studies have used predictive models in a wide range of applications

including to evaluate species distributions in relation to climate change (Thuiller 2004; Araújo et

al. 2005), evaluate the establishment and spread of expanding invasive species (Hartley et al.

2006), and model habitat suitability of endangered species (Rodriguez et al. 2007; Marmion et al.

2009; Franklin et al. 2009). With the increasing availability of remote sensing data and advances

of geographic information systems, researchers often are no longer data limited, and are

expanding the use of predictive models to include more applications (Guisan and Zimmerman

2000; Marmion et al. 2009).

Modeling existing and future habitats of endangered species has become a popular application of

predictive models. One reason for this popularity is that faced with limited data on the

distribution, abundance and dynamics of endangered species (Mace et al., 2005; Rodríguez,

2007), predictive models allow for the extrapolation of relatively few field samples to the entire

potential range of a species. However, the application of predictive models to endangered

species remains controversial (Loiselle 2003; Wilson et al. 2005; Thompson et al. 2007). One

difficulty with the standard sets of predictive models used by the majority of ecologists, is that

they are often inappropriate when analyzing data limited to a few sites and scales (Ellison and

Agrawal 2005; Araújo & Guisán, 2006). This, in turn, produces data sets that have many

statistical limitations, including zero-inflated bias, increased collinearity between variables, and

inflated coefficient of variation (Graham 2003; Dixon et al. 2005; Edwards et al. 2006; Guisan et

al. 2006; Dormann et al. 2008). However, uncertainties in predictive models of endangered

species may arise during all stages of modeling including obtaining species level data, obtaining

accurate species counts, and, linking species to landscapes/habitats (Elith et al. 2002; Loiselle

2003; Heikkinen et al. 2006). Therefore, approaches that reduce uncertainty in predictive models

are continually being sought, as are improvements to current approaches (Elith et al. 2006;

Austin 2007).

One way to overcome the difficulties of using singular predictive models has been the use of

ensemble-based approaches, which combine several predictive models (Aruajo and New 2007;

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Marimon et al. 2008). The goal of ensemble models is to reduce the uncertainty in any singular

approach by combining their predictions. The idea of ensemble approaches dates back to

Laplace (1820) who demonstrated that the probability of error will rapidly decline with the

inclusion of additional predictors. However, in comparison to other disciplines, the application

of ensemble models to ecological issues remains in its infancy (Bates and Granger 1969; Arujo

et al. 2005b). In particular, the use of ensemble models have been restricted to determining

distributions of species under climate-change scenarios (Thuiller 2004; Araujo et al. 2005b;

Thuiller et al 2005; Buisson and Grenouillet 2008), with less emphasis on establishing habitat

relationships. However, as the first step of building an ensemble model is developing singular

predictive models of initial conditions as filters, studies regarding uncertainties with the existing

singular approaches from which the ensembles are based remain relatively understudied.

The use of ensemble models for improving model predictions of species distributions and their

habitats has received little attention. Comparative studies of efficiencies of ensemble models are

scarce (Thuiller 2004; Marimon et al. 2008), and few studies provide guidance on how to build

the most suitable ensemble (e.g. how do models improve prediction, how many initial filters are

needed?). Given the paucity of both ensemble models and data with most endangered species

(Mace et al. 2005; Rodriguez et al. 2007), an evaluation is needed of the application of ensemble

models for identifying habitat characteristics of endangered species. The objective of this paper

is to provide a quantitative comparative analysis of both singular predictive models and

ensemble approaches for identifying existing habitat for the endangered species, the redside dace

(Clinostomus elongatus). Specifically, this study seeks to determine: 1) whether habitat-based

ensemble models improve modeling metrics (e.g. sensitivity, specificity, and overall

classification); 2) to what degree ensemble models actually improve predictive success; and, 3)

how many models should be used to build an optimal ensemble?

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Methods

Study Area and Species

The redside dace is a pool-dwelling minnow found only in Ontario within its Canadian range

(Figure 5.1). Redside dace is in decline due to changes in land-use through urbanization,

especially in the Greater Toronto Area, which includes 80% of its Canadian range, (COSEWIC

2007). To evaluate the efficacy of predictive models for correctly classifying redside dace

locations and their habitats, a dataset of 100 known redside dace locations sampled from 1997-

2007 was compiled from historical records from various government agencies, universities and

conservation authorities. Absence data were obtained from a larger dataset of 560 locations

obtained from the same agencies and randomly reduced to 100 locations as a means of

developing a balanced approach for assessing modeling metrics (Olden et al. 2004).

A hierarchical approach was used to predict the presence and absence of redside dace. In total

28 variables were used across multiple scales; fine-scale (i.e. site level habitat), intermediate-

scale (i.e. landscape level) and broad-scale (i.e. geologic and geomorphic variables) to help

quantify the importance of redside dace habitat (Table 5.1). These variables included factors

thought to influence redside dace, such as urban land cover (Scott and Crossman 1973; McKee

and Parker 1982; Daniels and Wisniewski 1994; Novinger and Coon 2000; COSEWIC 2007),

ranging to factors thought to influence stream fish in general (Grossman and Freeman 1987).

Fine-scale habitat features were derived using the Ontario Stream Assessment Protocol (OMNR

2007) and included: habitat type (e.g. percent pool, riffle, run); substrate type (e.g. percent

cobble, gravel or fines); stream depth (m); type of in-stream cover (e.g. percent flat rock, round

rock, wood, bank, or macrophytes); characteristics of the stream bank (e.g. percent eroding,

vulnerable, protected or depositional); overland temperature (degrees Celsius), and amount of

adjacent riparian cover (Table 5.1). Intermediate-scale habitat variables included percent

coverage by urban, forest, cropland, pasture and wetland landcover and were obtained from most

recent satellite imagery (2001) that was converted in shapefiles using a geographic information

system. Various spatial buffer sizes were used to characterize adjacent land-use; however a one

kilometer buffer was deemed most appropriate given a sensitivity analysis using various buffer

sizes. Broad-scale habitat variables were assessed using various buffer sizes in a similar fashion

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to landscape variables. The broad-scale variables were derived from The Canadian System of

Soil Classification and included type of soil geology in the vicinity (Newmarket Till, Elma Till,

Halton Till), as well as slope (Soil Landscapes of Canada Working Group 2007). Soil categories

that occurred at < 5% of sites were removed in order to not over-parameterize the models.

New York,

. U.S.A78°20’5

42°41’2

Ontario, CANADA

Lake Ontario

Greater Toronto Area

Lake Erie

Lake Huron

Figure 5.1 – Distribution of sampling locations between 1997-2007. Closed circles indicate redside dace occurrences, whereas, open circles indicate where redside dace were absent.

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Table 5.1 – Summary of the hierarchical habitat-based model used for predicting the presence of endangered minnow, redside dace (Clinostomus elongatus). Seven predictive models were used including: LR (logistic regression), CT (classification trees), MARS (multivariate adaptative regression splines), ANN (artificial neural networks), DA (discriminant analysis), RF (random forest), and BR (boosted regression trees). Variables were derived using forward-selection procedures on five independent datasets and are shown as a percentage of datasets where each variable was selected (parentheses indicate negative associations). Variables selected from only one dataset were omitted. A prioi predictions (ap) based on habitat predictors thought to influence the decline of the species are shown for reference, where + indicates a positive correlation, - negative correlation, 0 none.

Scale Variable ap LR CT MARS ANN DA RF BR Fine (Site)

Width - [100] [80] [100] [80] Pool Glides Fast Riffles Slow Riffles

+ 0 0 0

100

60

100

80

[80] [80]

100

[60]

100

[80]

100

[40]

Fine substrate Gravel substrate Cobble substrate

0 0 0

40

Shallow depth Intermediate depth Deep

0 0 0

80

80 Flat rock cover

Round rock cover Wood cover Bank cover Macrophyte cover

0 0 0 + 0

Eroding banks Vulnerable banks Protected banks Depositional banks

+ 0 0 0

[80]

40

40

Intermediate Urban - [100] [60] [100] [100] [100] [100] [80] (Landscape)

Cropland 0 80 Pasture 0 40

Forest 0 80 Wetland 0 Broad Newmarket Till 0 (Geologic) Halton Till 0 Elma Till 0 40 80 60 Slope 0 Temperature 0

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Building Individual Models

Previous comparisons of individual models have demonstrated that model comparisons

are needed for understanding the relative strengths and weakness of each modeling approach,

and this is especially true for modeling endangered species where greater uncertainty exists

(Olden and Jackson 2001; 2002a). In this stud, five ensemble methods were compared for

improving the predictive success of the redside dace and its habitat. For this assessment, seven

modeling approaches were used as initial filters to identify environmental variables linked with

the occurrence of redside dace (Figure 5.2). These modeling approaches were: logistic

regression (LR); classification trees (CT); multivariate adaptive regression splines (MARS);

artificial neural networks (ANN); discriminant analysis (DA); random forest (RF); and, boosted

regression trees (BR). Each of these modeling approaches has been shown to be useful for

describing species occurrences (Thuiller 2003; Elith et al. 2006; Thuiller et al. 2006; Marimon et

al. 2008) and were used to develop the output models (Figure 5.2). The modeling methods used

represent a continuum of use from logistic regression, the most prevalent and widespread

statistical method for modeling binary data (Hosmer and Lemeshow 1989; Pampel 2000), to

newer methodologies that may perform better, such as artificial neural networks, boosted

regression trees and random forest (Olden and Jackson 2001; Olden et al. 2004; Thuiller et al.

2004). In general, the modeling methods can be broadly broken down into three groups: two

regression-based methods (LR, MARS); three machine-learning methods (BR, RF, ANN); and,

two classification methods (CT and DA; Elith et al. 2006; Marrimon et al. 2009).

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Step 2: Analyze Training Data

Step 4: Build Ensemble Forecast (e.g. Mean, PCA, WA)

Multivariate Adaptive

Regression Splines

(MARS)

Boosted Regression

(BR)

Discriminant

Analysis (DA)

Random Forest (RF)

Artificial Neural

Networks (ANN)

A 1 2 3 B 4 5 6

Step 1: Partition Data C 7 8 9

Step 3: Produce Output Models Using Predictions of Independent Test Data

A = 1 B = 0 C = 1

A = 1 B = 1 C = 0

A = 1 B = 0 C = 0

A = 0 B = 0 C = 1

A = 1 B = 1 C = 0

A = 0 B = 0 C = 0

A = 0 B = 1 C = 1

Logistic Regression

(LG)

Classification Trees (CT)

A = 1 B = 0 C = 0

Figure 5.2 – Summary of ensemble forecasting approach. Five-fold cross validation was used by partitioning data into five individual test and validation datasets. Using each training dataset, data was analyzed individually across seven initial approaches: logistic regression (LR), classification trees (CT), multivariate adaptive regression splines (MARS), artificial neural networks (ANN), discriminant analysis (DA), random forest (RF) and boosted regression (BR). Five ensemble models were built from initial seven predictions, including: a consensus model (CM), principal components analysis (PCA), weighted average using overall classification (WA), mean (Mn) and median (Md). Modeling metrics (specificity, sensitivity and overall classification) were obtained by comparing the individual or ensemble predictions on the independent test dataset.

111

Models were configured in the following ways. LRs were run using a logit function using

maximum likelihood (Allison 1999; Olden and Jackson 2002a). CTs used a chi-square distance

of 0.05 to determine significant group cut-offs. The number of terminal nodes in the leaf

structure were optimized by iteratively running all terminal nodes and choosing the leaf structure

with the smallest error (in this case a terminal structure of 5 was determined to have the lowest

error). This step was important as too many terminal nodes may artificially inflate correct

classification, whereas too few terminal nodes may not provide meaningful species groupings

(Vayssiéres et al. 2000). These models were developed in the SAS programming language v.9.1.

MARS provides an alternative regression based method using piecewise linear fits rather than a

smoothing parameter (e.g. as in general additive models; Friedman 1991; Elith et al. 2006).

ANNs used a one-hidden layer feed-forward network trained by the back-propagation algorithm

(Bishop 1995). This type of network is considered a universal approximator of any continuous

function, has low associated rates of error and is used most often in ecological studies

(Rumelhart et al. 1986; Hornik and White 1989; Olden and Jackson 2002b, Olden et al. 2004).

ANNs were optimized (optimal referring to minimizing the trade-off between network bias and

variance) for the number of hidden neurons in the neural network by determining empirically the

number of hidden neurons that produces the lowest misclassification rate (Bishop 1995); which

in this case was a layer that contained seven nodes. Random forests were run for each node of

the tree, randomly using m variables on which to base the decision at that node. The best split

was calculated based on these m variables in the training set. Two group (presence/absence)

discriminant function analysis was developed using a linear model. Models 3-6 were built using

the statistical program Statistica v.7. BR prediction was based on an accuracy-weighted vote

across estimated classifiers (Ridgeway 1999) and run using code provided by Elith et al. (2008)

in the R programming language v.2.8.0 (R Development Core Team, 2008)

Habitat variables were chosen using forward step-wise selection methods; except for the

machine-learning methods (ANN, RF, BR), which iteratively fit their own responses to the

habitat data (Olden and Jackson 2002b; Dormann et al. 2008). This approach was necessary to

compare the ability of individual models for selecting habitat variables with high explanatory

power. Although bias can be introduced by variable selection methods, recent studies have

demonstrated that selection procedures may actually have very small impact on resultant models

(Maggini et al. 2006; Meynard and Quinn 2007; Dormann et al. 2008). A forward-selection

112

procedure (p < 0.05) was used to determine variables with high explanatory power. In the cases

of machine-learning methods, the single ANN model with the highest overall classification (e.g.

highest area under the curve) was retained. With the RF and BR approaches the importance of

variables was determined by estimating the relative influence of each variable reducing the loss

function, based on a square error algorithm (Thuiller, 2003).

Model thresholds were calculated using receiver operator characteristic curves (ROC) with

thresholds balanced between misclassification of species presence and species absence (Olden

and Jackson 2001). It should be noted that the choice of model thresholds can impact the results

(Jiménez-Valverde & Lobo 2007; Lobo et al. 2008). For example, researchers must weigh the

need for correct classification of either presence or absence at the expense of inflated error (i.e.

misclassification). One could choose model-selection thresholds that prioritize correct

classification, at the expense of increased misclassification of species absences (or vice versa).

Such a decision may be well suited for conservation applications, where one would wish to

emphasize correct predictions of the true occurrence of an endangered species, but were willing

to accept higher commission rates (Loiselle et al. 2003). For the purpose of this study, a

balanced approach was used, where models were select that had equal likelihood of

misclassification of species presence and absence. This approach was well suited for this study

as model were evaluated based on their specificity, sensitivity and overall classification, which

require equal consideration (Hartley et al. 2006). In addition, previous studies (e.g. Marrimon et

al. 2008), which have used area-under-the-curve operations to evaluate predictive performance

of ensemble models may inappropriately produce models that have high overall classification

due solely to model specificity, which would not otherwise be known (Loiselle et al 2003;

Jiménez-Valverde. & Lobo 2007; Lobo et al. 2008).

For all of the model comparisons, data were split into two types: 80% of the data were used to

train each model, and 20% of the data were used to validate and test the predictive capability of

each model (Figure 5.2). A 5-fold cross validation procedure was used where five separate

models were calculated for each statistical method to build a complete validation data set (Figure

5.2). The validation data is an important component of model evaluation, as it provides an

independent unbiased evaluation of the predictive capability of each trained model (Olden et al.

2002a).

113

Evaluating Individual Models

Predictive models were compared in a number of ways. First, model outputs were compared to a

set of a prioi predictions based on habitat predictors thought to influence the decline of the

species (Scott and Crossman 1973; McKee and Parker 1982; Daniels and Wisniewski 1994;

Novinger and Coon 2000). Second, predictive models were evaluated by three modeling

metrics: model sensitivity (the ability of each model to correctly predict species presence); model

specificity (the ability of each model to correctly predict species absence); and, overall

classification (the ability of each model to correctly classify both species presence and absence).

The use of these metrics provides an alternative means to evaluate each model, their comparative

successes, their associated errors, and their relationship to ecological relevant variables, for both

species presence and absence (Olden and Jackson 2002a). The distinction between modeling

metrics is an important one as models for predicting imperiled species are often difficult to

assess because they tend to artificially inflate correct classification of species absence (model

specificity), where the majority of sites are associated and the habitats do not reflect the ecology

of the model species, whereas, model sensitivity measures the ability to classify species

presence, which may be more relevant for conservation applications. Finally, predictive outputs

were compared to the observed data using unweighted pair-group method (UPGMA) cluster

analysis using phi similarity. Cluster analysis has been used widely in ecological literature

(Legendre and Legendre 1997) as the resultant output produces a dendrogram which connects

closely matching neighbors. For example, statistical methods connected by branches proximal to

one another match closer than methods connected by branches more distal (Podani 2000). The

phi coefficient is the binary correlation coefficient and is not biased by frequency of occurrence

as has been shown for other coefficients (Jackson et al 1989).

Building Ensemble Models

Ensemble models allow for a decrease in predictive uncertainty of singular models, by using a

combination of their predictions (Figure 5.2). Various ensemble models have been proposed,

including those that use selective algorithms or basic mathematical functions such as the mean

and median predictions (Araújo et al. 2005; Marrimon et al. 2008). Here, five ensemble

approaches currently in favor were compared: a consensus model (CM), Principal Components

Analysis (PCA), weighted average using overall classification (WA), Mean (Mn) and Median

(Md) ensemble approaches. Each of these approaches have been used extensively in building

114

ensemble models (e.g. Gregory et al., 2001; Johnson and Omland 2004; Thuiller, 2004; Araújo et

al 2005a; 2005b, 2006; Thuiller et al 2005; Araújo and New 2007; Goswami and O’Connor,

2007; Marimon et al. 2008)

All ensemble models were built from reducing a matrix of model predictions (j) by sites (i) to a

vector of ensemble predictions. For example, the consensus ensemble model used a majority

rules criterion, where all model outputs were compared and the majority output (presence or

absence) was retained. For the mean ensemble model, all continuous model outputs were

averaged and the final output was rounded to either presence or absence. Similarly, the median

ensemble model took the median value of all seven individual outputs (either presence or

absence). The principal components analysis used the dominant axis of an eigenvalue

decomposition of a covariance matrix programmed in the R v.2.8.0 (R Development Core Team,

2008). The resultant eigenvalue was scaled to presence absence using the ROC approach

highlighted earlier. Finally, the weighted average ensemble model used the overall classification

(across all predictions) and multiplied it to each prediction to produce a singular ensemble

prediction.

Evaluating Ensemble Models

Ensemble models were evaluated using identical metrics as the individual models: specificity,

sensitivity and overall classification, and cluster analysis. The number of initial models needed

to build the most appropriate ensemble model was also evaluated. For this, a re-sampling

approach was developed using a sample-based rarefaction routine (Colwell et al. 2004), which

was coded using R v.2.8.0 (R Development Core Team, 2008). This re-sampling approach

randomly selected predictions at a given site from a matrix of model predictions by site. Each

prediction was matched to the observed data and model specificity, sensitivity and overall

classification was calculated across study sites. The set of predictions was then randomly

permuted and a new prediction was added from a competing model and evaluated as to whether

this additional prediction increased overall classification (Figure 5.2). This randomization

procedure was repeated 1000 times to determine 95% confidence intervals for all combinations

of ensemble models.

115

Results

I) Individual Models

The majority of the initial predictive models identified similar fine-scale habitat features,

including: positive associations with deep, pool habitats with gravel substrate, and negative

associations with stream width and fast riffles (Table 5.1). Only a negative association with

urban land cover and a positive association with Elma till, came out as a strong indicator at a

broader scale. These findings are in agreement with previous studies of current knowledge of

this species and its habitat (Scott and Crossman 1973; McKee and Parker 1982; Daniels and

Wisniewski 1994; Novinger and Coon 2000; COSEWIC 2007; Table 5.1).

All individual models generally performed well (> 80% overall classification). Performance was

generally better for model specificity (range 84-90%; Table 5.2) than for model sensitivity (75-

87%). When individual models were compared, logistic regression was most closely related to

the observed data, followed by the learning-based approaches, such as boosted regression,

random forest, and artificial neural networks (Fig. 5.3A). These methods also had the highest

rates of model specificity and generally the highest rates of model sensitivity, although artificial

neural networks dropped off in this area (Table 5.2). Alternatively, classification-based methods,

such as classification trees and discriminant analysis were least associated with the observed

data, followed by multivariate adaptive regression splines.

116

Table 5.2 – A comparison of model sensitivity (correct classification of species presence), specificity (correct classification of species absence), and overall classification (correct classification of both species presence and absence) for redside dace (Clinostomus elongatus). Single models are: LR (logistic regression); CT (classification trees); BR (boosted regression trees); MARS (multivariate adaptive regression splines); ANN (artificial neural networks); DA (discriminant analysis); and, RF (random forest). Ensemble forecasts are: consensus model (CM); principal component analysis (PCA); weighted average using overall classification (WA); mean (Mn); and, median (Md).

Singular Models Ensemble Models

LG CT BR MARS ANN DA RF CM PCA WA Mn Md

Sensitivity (%) 87 75 83 81 75 84 83 87 88 90 88 87

Specificity (%) 90 86 90 88 89 84 89 94 91 90 89 92

Overall Classification (%) 88 81 87 85 82 84 86 91 90 90 89 90

117

II) Ensemble Models

Ensemble models provided as good as or improved model performance over singular

methods. Overall classification increased from 1-10% with ensemble models (Table 5.2). On

average, ensemble models improved model sensitivity over model specificity. For example,

whereas, only logistic regression had model sensitivity above 85%, all ensemble models had

rates of model sensitivity at 87% or above and had model sensitivity equal or above to all the

individual models. With the exception of the mean ensemble result, all ensemble models

provided equal or superior specificity of 89% rather than the 90% achieved by the best individual

model. Overall, the various ensemble methods produced similar levels of model sensitivity,

model specificity and overall classification; however, the consensus ensemble model provided

the highest rate of classification, due to having both higher values of model sensitivity and

specificity. The various ensemble models show close association (i.e. phi correlation measure) to

one another (Fig. 5.3) and to the observed data, which were similar to the association between

the observed data and the best individual model (i.e. logistic regression).

Figure 5.3- Cluster analysis showing the relationship with the observed distribution (Obs.) of redside dace (Clinostomus elongatus) and: A) the seven individual modeling approaches alone, and B) with ensemble forecasts included. Model short forms are carried over from Table 5.2.

118

There was large variation when comparing all possible combinations of consensus ensemble

models. Whereas, model sensitivity and overall classification increased (on average) with the

addition of more initial models, the average model specificity decreased for combinations of

three and five initial models, before improving at the final seven-input ensemble model (Fig.

5.4). In addition, the variation (e.g. 25th and 75th percentiles, as well as minima and maxima) in

model sensitivity, specificity and overall classification increased when larger combinations of

ensembles were considered (e.g. there were 35 combinations of ensembles for three-input

models, and 21 for five-ensemble models). For the three-model consensus, the combination of

LG, MARS, RF had the highest model specificity, CT, ANN, BR had the highest model

sensitivity and LG, RF, BR had the highest overall classification. Alternatively, the combination

of learning-based methods ANN, RF, BR had the lowest model specificity and overall

classification, whereas, the combination of LG, CT, RF showed the lowest model sensitivity

(Appendix 5.1). Similarly, for the five-model consensus models, combinations with all three

learning-based methods performed the worst, where the combination of MARS, ANN, RF, BR,

DA had the lowest sensitivity and overall classification and LG, MARS, ANN, RF, BR had the

lowest specificity (Appendix 5.1).

119

A)

B)

C) Figure 5.4 – Box and whisker plots showing the variability in consensus ensemble forecasts for predicting the presence of an endangered redside dace. Consensus ensemble models were compared across all combinations of one (n =7), three (n =35), five (n =21) and seven (n=1) input models (x-axis). Boxes are 25th and 75th percentiles, horizontal lines indicate the median, vertical lines indicate the upper and lower values, diamonds indicate the mean and are connected by dashed lines. Modeling metrics were: A) model sensitivity (i.e. correct classification of species presence); B) model specificity (i.e. correct classification of species absence); and, C) overall classification. Dashed lines indicate 95% confidence intervals.

120

Discussion Ensemble models provide a useful method for reducing uncertainty when modeling distributions

of endangered species. Endangered species are not only rare, but they are also often difficult to

capture and enumerate, thereby, complicating the evaluation of their habitat. Thus methods

which can be used to reduce the uncertainty of model type should allow for improvement in the

management of endangered species. In this study, ensemble models improved model specificity

for the endangered redside dace (Clinostomus elongatus) by 6.9%, model sensitivity by 3.2% and

overall classification by 5.3% (on average); over individual models. Previous studies have

demonstrated that ensemble approaches can reduce uncertainty and improve model fit for

predicting future distributions (Thuiller, 2004; Thuiller et al., 2005; Araujo and New 2007;

Marrimon et al. 2008). Here, this study provides the first evidence that ensemble models can be

equally applied to modeling current distribution and provides increased model performance over

singular approaches.

Ensemble models may be especially useful for identifying the potential importance of habitat for

endangered species, and at the scale in which they function. Modeling the distributions of

endangered species and their habitats is an activity filled with uncertainty. Using the endangered

redside dace as an example, this study demonstrates that although predictive models varied in

their ability to identify correctly existing habitat, in all cases an ensemble approach improved

model prediction. In addition, whereas previous studies have demonstrated the utility of

ensemble models for identifying habitats at broader scales (e.g. landscape), this study

demonstrates the utility in developing a multi-scaled approach. With the increase in availability

of remote-sensing data and advances in geographic information systems, researchers are left with

a plethora of data from which to model distributions (Guisan and Zimmerman 2000). Often

climatic, topographic and landuse data are available readily (Guralnick et al. 2007), whereas

information on fine-scale variables is more scarce (Austin 2007; Dorman et al. 2008). This study

demonstrates a situation where fine-scale habitat models had a greater ability to describe factors

relevant to the redside dace than broader scale factors (Table 5.1).

The need for comparative approaches using many statistical methods has also been highlighted

as another way of reducing uncertainty with modeling endangered species (Guisan and

Zimmerman 2000; Olden and Jackson 2002a). The choice of modeling approach has been

121

shown to have severe consequences for the application to conservation decisions (Loiselle 2003;

Wilson et al. 2005; Rodriguez et al. 2007; Marrimon et al. 2008). For example, Pearson et al.

(2006) showed that distribution changes of South African plant species varied from 92% loss to a

322% gain depending on the model they used. In addition, Dorman et al (2008) demonstrated

that of several uncertainties in modeling species distributions, including variable selection and

collinearity between variables, the choice of model type had the largest impact. In this study,

there was variation in how each model type fitted predictions, with logistic regression most

closely resembled the observed distribution, followed closely by the three machine learning

methods: boosted regression, random forest and artificial neural networks. These results are in

general agreement with other studies showing the benefits of machine-learning methods such as

random forest, and general boosted regression (Cutler et al., 2007). One reason for the superior

performance of logistic regression is that more complex models (which iteratively fit a solution)

can be prone to overfitting (Olden and Jackson 2002a). As previous quantitative comparative

analyses have demonstrated, the success of predictive modeling approaches can be largely data

dependent and there is no clear indication of the preeminence of any singular approach (Olden

and Jackson 2002a; Araujo and New 2007). Comparative analyses can be used to identify

problems related to modeling approaches given the available data and to determine situations

where comparative analysis may work better (Guisan and Zimmerman 2000; Olden and Jackson

2002a).

Comparative approaches are needed not just to ensure that appropriate models are being

developed, but to ensure the best ensemble models are being built. Comparisons of ensemble

models are rare (e.g. Araujo et al. 2005; Marrimon et al. 2008) and only one (Marimon et al.

2008) has evaluated the relationship between ensemble models and predictive performance. In

this study, there was variation in the ability of ensemble models to improve prediction over

singular approaches. The consensus ensemble model (i.e. vote counting) provided the highest

model specificity and the best estimated overall classification, whereas the weighted average

ensemble model provided the highest model sensitivity (Table 5.2). The remaining ensemble

models performed similarly, including those based on principal components analysis, which have

been used most frequently in previous studies (Thuiller 2004; Araujo et al. 2005b). The reason

for this similarity is likely due to the lack of independence in the information provided by the

initial models. Future research is needed into understanding the study settings in which different

ensemble methods are likely to perform best (Marrimon et al. 2007).

122

The evaluation of the efficacy of combining initial models into ensemble models has received

little attention and remains an important knowledge gap (Loiselle et al. 2003; Wilson et al. 2005;

Marrimon et al. 2008). Decisions such as how many initial models are needed to build the most

useful ensemble model have not been addressed. Ensemble models work under the assumption

that the initial models provide some independent information (Araujo and New 2007): at some

stage, adding more initial models beyond those that provide independent information may

actually decrease the utility of the ensemble approach. Here a simple sub-sampling procedure

(Colwell et al. 2004) was used to demonstrate that an optimal ensemble model could be

developed using as few as three initial models (Figure 5.3). As understanding the contributions

of initial models to ensemble forecasts remains a limitation to the ensemble approach (Araujo

and New 2007), the use of sample based rarefaction techniques may allow substantial insight

into ensuring that appropriate ensemble models are being produced.

The application of ensemble models to ecological issues remains in its infancy (Araujo et al.

2005b). Previous studies on ensemble models have demonstrated their utility in reducing

uncertainty with singular methods (Thuiller et al. 2004; Araujo et al. 2005b; Marrimon et al.

2007), however several challenges remain to ensure ensemble models are applied appropriately,

especially for modeling distributions of endangered species. First, researchers should have

thorough understanding of predictive modeling, their uncertainties and conditions under which

they should be applied (Elith et al. 2002; Loiselle et al. 2003). Second, researchers should be

mindful of the limitations of working with endangered species, which provide additional

statistical problems (Thompson 2003; Ellison and Agrawal 2005). Third, comparative analyses

of not only singular methods, but ensemble methods are needed to ensure that the most

appropriate models are being constructed. The development of appropriate validation data

(Araujo and New 2007), decision of modeling metrics (e.g. AUC, sensitivity, specificity), and

consideration of how many initial models are needed to build the ensemble, are all needed to

ensure that ensemble models are actually improvements to singular methods. Fourth, the goals

of the ensemble model need to be explicitly testable. Such testing should include the use of

models to corroborate hypothesis with either a priori prediction or existing knowledge and

ensuring that habitat is modeled using appropriate scale/s.

123

Conclusion

This study demonstrates that ensemble models can provide marked improvements to singular

approaches in situations where there is uncertainty exists, such as modeling suitable habitat of

endangered species. Using the endangered redside dace as a model organism, this study shows

that an ensemble approach can improve model sensitivity, specificity and overall classification.

These findings have important consequences for improving species distribution models of

endangered species. In addition, this study demonstrates the utility of sub-sampling procedures

for determining how many initial models are needed to build an optimal ensemble. As ensemble

models are dependent on their initial inputs, improvements to future ensemble models can be

expected with greater consideration of how initial models perform best. This study provides an

example of how comparative analyses across many scales, initial modeling types and ensemble

approaches can be used to improve the prediction of endangered species and their habitats.

Acknowledgements

Funding was provided by NSERC Canada and OGS Scholarships to M.S.P., an NSERC

Discovery Grant to D.A.J., Interdepartmental Recovery Fund #1410 provided by Fisheries and

Oceans (DFO), the Ontario Ministry of Natural Resources (OMNR), and the University of

Toronto. The following agencies and individuals were helpful for provided data for this research:

Royal Ontario Museum, Fisheries and Oceans, Ontario Ministry of Natural Resources, Toronto

Region Conservation Authority (TRCA), Conservation Halton (CH), Lower Lake Simcoe

Conservation Authority (LLSCA), Credit Valley Conservation Authority, D Forder (Ontario

Streams), J Anderson (LLSCA), L Stanfield (OMNR), E Holm (ROM), D Lawrie (TRCA), S

Jarvie (TRCA), S Watson-Leung (CH), and S Reid (DFO). In addition, M Neff and G Rawnsley

provided field assistance. This manuscript benefited from discussions with C Harpur and A

Drake. Finally anonymous reviewers were helpful for providing comments on early drafts of

this paper.

124

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Appendices Appendix 5.1 – Model metrics for all combinations of consensus ensemble models. Models are: LR (logistic regression), CT (classification trees), MARS (multivariate adaptive regression splines), RF (random forest), ANN (artificial neural networks), BR (boosted regression trees, and DA (discriminant analysis).

Model/s Included Sensitivity Specificity Overall Classification

LG 0.8700 0.9000 0.8800 CT 0.7500 0.8600 0.8100 MARS 0.8100 0.8800 0.8500 RF 0.8300 0.8900 0.8600 ANN 0.7500 0.8900 0.8200 BR 0.8300 0.9000 0.8700 DA 0.8400 0.8400 0.8400 LG,CT,MARS 0.8822 0.8516 0.8669 LG,CT,ANN 0.8428 0.8650 0.8539 LG,CT,RF 0.8878 0.7922 0.8400 LG,CT,BR 0.8784 0.8254 0.8519 LG,CT,DA 0.9350 0.8198 0.8774 LG,MARS,ANN 0.8640 0.8796 0.8718 LG,MARS,RF 0.9318 0.8188 0.8749 LG,MARS,BR 0.9222 0.8250 0.8736 LG,MARS,DA 0.9088 0.8228 0.8658 LG,ANN,RF 0.8222 0.8646 0.8434 LG,ANN,BR 0.7900 0.8730 0.8315 LG,ANN,DA 0.8594 0.8396 0.8495 LG,RF,BR 0.9164 0.8656 0.8909 LG,RF,DA 0.9286 0.8102 0.8694 LG,BR,DA 0.9118 0.8610 0.8864 CT,MARS,ANN 0.9058 0.8532 0.8795 CT,MARS,RF 0.9150 0.8292 0.8721 CT,MARS,BR 0.8836 0.8408 0.8622 CT,MARS,DA 0.9296 0.8400 0.8848 CT,ANN,RF 0.9056 0.8626 0.8841 CT,ANN,BR 0.8688 0.8894 0.8791 CT,ANN,DA 0.8980 0.8804 0.8892 CT,RF,BR 0.8670 0.8280 0.8475 CT,RF,DA 0.9174 0.8106 0.8640 CT,BR,DA 0.8970 0.8372 0.8671 MARS,ANN,RF 0.7368 0.8718 0.8043

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MARS,ANN,BR 0.7182 0.8446 0.7814 MARS,ANN,DA 0.7430 0.8284 0.7857 MARS,RF,BR 0.7808 0.8068 0.7938 MARS,RF,DA 0.7982 0.8018 0.8000 MARS,BR,DA 0.8052 0.8274 0.8163 ANN,RF,BR 0.6836 0.8766 0.7801 ANN,RF,DA 0.7234 0.8670 0.7952 ANN,BR,DA 0.7018 0.8750 0.7884 RF,BR,DA 0.7818 0.8424 0.8121 LG,CT,MARS,ANN,RF 0.8744 0.8826 0.8785 LG,CT,MARS,ANN,BR 0.8654 0.8892 0.8773 LG,CT,MARS,ANN,DA 0.8624 0.8558 0.8591 LG,CT,MARS,RF,BR 0.9064 0.8502 0.8783 LG,CT,MARS,RF,DA 0.9194 0.8344 0.8769 LG,CT,MARS,BR,DA 0.8908 0.8730 0.8819 LG,CT,ANN,RF,BR 0.8574 0.9074 0.8824 LG,CT,ANN,RF,DA 0.8656 0.8660 0.8658 LG,CT,ANN,BR,DA 0.8432 0.9006 0.8719 LG,CT,RF,BR,DA 0.8888 0.8256 0.8572 LG,MARS,ANN,RF,BR 0.8232 0.8244 0.8238 LG,MARS,ANN,RF,DA 0.8792 0.8366 0.8529 LG,MARS,ANN,BR,DA 0.8238 0.8686 0.8462 LG,MARS,RF,BR,DA 0.9072 0.8344 0.8708 LG,ANN,RF,BR,DA 0.8094 0.8720 0.8407 CT,MARS,ANN,RF,BR 0.8762 0.8834 0.8798 CT,MARS,ANN,RF,DA 0.9140 0.8562 0.8851 CT,MARS,ANN,BR,DA 0.8804 0.8944 0.8874 CT,MARS,RF,BR,DA 0.9078 0.8462 0.8770 CT,ANN,RF,BR,DA 0.8504 0.8802 0.8653 MARS,ANN,RF,BR,DA 0.7368 0.8424 0.7896

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Appendix 5.2 – R Code for testing configurations of 1,3, 5 and 7 prediction consensus models data<-read.table ("F:\\R\\Predict.txt",h=T) # load prediction matrix iters<-10000 # define number of iterations model<-matrix(nrow=iters,ncol=8) for (i in 1:iters){ # start loop (data[1:200,1:8][sample(200,1,replace=TRUE),])->temp # from sample of 200, randomly pick 1 row (with replacement) as.matrix(temp)->temp2 # make temporary matrix temp2->model[i,] # fill this matrix with model outputs from n number of random sites } # end loop model[1:iters,1]->OBS # define where obs is (e.g. col 2) model[1:iters,2]->LG # name model in column 3 model[1:iters,3]->CT # name nmodel in column 4 model[1:iters,4]->MARS # name model in column 5 model[1:iters,5]->RF # name model in column 6 model[1:iters,6]->ANN # name model in column 7 model[1:iters,7]->BR # name model in column 8 model[1:iters,8]->DA # name model in column 9 # define combination # THREE MODEL COMBINATIONS combo<-cbind(LG,CT,GLM) #combo<-cbind(LG,CT,ANN) #combo<-cbind(LG,CT,RF) #combo<-cbind(LG,CT,BR) #combo<-cbind(LG,CT,DA) #combo<-cbind(LG,MARS,ANN) #combo<-cbind(LG,MARS,RF) #combo<-cbind(LG,MARS,BR) #combo<-cbind(LG,MARS,DA) #combo<-cbind(LG,ANN,RF) #combo<-cbind(LG,ANN,BR) #combo<-cbind(LG,ANN,DA) #combo<-cbind(LG,RF,BR) #combo<-cbind(LG,RF,DA) #combo<-cbind(LG,BR,DA)

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#combo<-cbind(CT,MARS,ANN) #combo<-cbind(CT,MARS,RF) #combo<-cbind(CT,MARS,BR) #combo<-cbind(CT,MARS,DA) #combo<-cbind(CT,ANN,RF) #combo<-cbind(CT,ANN,BR) #combo<-cbind(CT,ANN,DA) #combo<-cbind(CT,RF,BR) #combo<-cbind(CT,RF,DA) #combo<-cbind(CT,BR,DA) #combo<-cbind(MARS,ANN,RF) #combo<-cbind(MARS,ANN,BR) #combo<-cbind(MARS,ANN,DA) #combo<-cbind(MARS,RF,BR) #combo<-cbind(MARS,RF,DA) #combo<-cbind(MARS,BR,DA) #combo<-cbind(ANN,RF,BR) #combo<-cbind(ANN,RF,DA) #combo<-cbind(ANN,BR,DA) #combo<-cbind(RF,BR,DA) # FIVE MODEL COMBINATIONS #combo<-cbind(LG,CT,MARS,ANN,RF) #combo<-cbind(LG,CT,MARS,ANN,BR) #combo<-cbind(LG,CT,MARS,ANN,DA) #combo<-cbind(LG,CT,MARS,RF,BR) #combo<-cbind(LG,CT,MARS,RF,DA) #combo<-cbind(LG,CT,MARS,BR,DA) #combo<-cbind(LG,CT,ANN,RF,BR) #combo<-cbind(LG,CT,ANN,RF,DA) #combo<-cbind(LG,CT,ANN,BR,DA) #combo<-cbind(LG,CT,RF,BR,DA) #combo<-cbind(LG,MARS,ANN,RF,BR) #combo<-cbind(LG,MARS,ANN,RF,DA) #combo<-cbind(LG,MARS,ANN,BR,DA) #combo<-cbind(LG,MARS,RF,BR,DA) #combo<-cbind(LG,ANN,RF,BR,DA) #combo<-cbind(CT,MARS,ANN,RF,BR) #combo<-cbind(CT,MARS,ANN,RF,DA) #combo<-cbind(CT,MARS,ANN,BR,DA) #combo<-cbind(CT,MARS,RF,BR,DA) #combo<-cbind(CT,ANN,RF,BR,DA) #combo<-cbind(MARS,ANN,RF,BR,DA) # SEVEN MODEL COMBINATION

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#combo<-cbind(LG,CT,MARS,RF,ANN,BR,DA) combo1<-apply(combo,1,mean) # take mean of models being combined (e.g. combo#) as.matrix(combo1)->t # load mean predictions from combos as matrix round(t)->RoundAll # round predictions (consensus) matrix(nrow=iters,ncol=1)->OC # make matrix for overall classificaiton (OC) matrix(nrow=iters,ncol=1)->presence # make matrix for overall presence matrix(nrow=iters,ncol=1)->absence # make matrix for overall absence cbind(model[,1],RoundAll)->temp3 # combine obs with ensemble rowSums(temp3)->temp4 # sum rows (obs + ensemble), if = 2 then true presence, if = 1 then disagree, if = 0 then true absence as.matrix(temp4)->tempsum # place into matrix called tempsum for (k in 1:iters){ # start loop OC[k,]<-if (RoundAll[k,]==model[k,1]) 1 else 0 # compare combo (ie. ensemble) to obs for each observation in matrix (iters by predictions), if they agree then 1, otherwise 0 presence[k,]<- if (tempsum[k,]==2) 1 else 0 # make vector of true presences absence[k,]<-if (tempsum[k,]==0) 1 else 0 # make vector of true absences } (sum(OC/iters))->TrueOC # rate of overall classification sd(combo1)->SDModels # SD of ensemble #error <- qt(0.95,df=length(combo1$vals)-1)*sd(combo1$vals)/sqrt(length(combo1$vals)) (sum(presence/iters*2))->Truepr # sensitivity rate (sum(absence/iters*2))->Trueab # specificity rate Truepr Trueab TrueOC

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Chapter 6: Using consensus methods to identify (and reduce) sensitivity from methodological choices when measuring functional diversity

Abstract Functional diversity indices have become important tools for measuring variation in species

characteristics that are relevant for ecosystem services. Recently, a popular method for

measuring functional diversity, FD, was shown to be sensitive to methodological choices in its

calculation. The objective of this study was to determine whether consensus methods can be used

to identify situations where methodological choices may be an issue when measuring

dendrogram-based functional diversity, FD. To calculate FD, a distance measure and a

clustering method must be chosen. Using data from natural communities, this study

demonstrates that consensus methods were able to determine instances where the choice of

distance measure (Euclidean and cosine) and clustering method (UPGMA, complete and single

linkage) produced qualitatively different relationships across communities and markedly

different dendrogram topologies. In particular this study highlights how consensus methods

may aid in the choice of a particular index of functional diversity with the hope that such

discussions may improve biodiversity-ecosystem studies.

Keywords: functional diversity, clustering, distance measures, community composition,

biodiversity, ecosystem productivity, ecological organization; index.

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Introduction The association between ecosystem properties and levels of species diversity is well studied

(Tilman 1997, Symstad 1998, Loreau et al. 2001). This association was thought to be driven by

the tendency for species-rich communities to have wider variation in functional traits (Diaz and

Cabido 2001, Heemsbergen et al. 2004, Hooper et al. 2005). The importance of functional traits

has led to the development of indices of functional diversity aimed at quantifying functional trait

variation; the evaluation of functional diversity indices continues to be an area of active research

(Walker 1999, Symstad 2000, Petchey and Gaston 2002, Mason et al. 2003, 2005, Naeem and

Wright 2003, Botta-Dukat 2005, Mouillot et al. 2005). Several studies (Cardinale et al. 2000,

Petchey and Gaston 2002) have concluded that ecosystem function tend to correlate more

strongly with functional diversity indices than with species diversity indices. Results such as

these have spurred interest in developing new and improved functional diversity indices that

incorporate ecosystem functions (Wright et al. 2006).

To calculate most functional diversity indices, a method is required for quantifying interspecific

differences in functional traits. In cases where there is only one trait of interest, simple

approaches may be appropriate, such as the weighted-trait variation (FDVar; Mason et al. 2003)

or the functional evenness (known as functional regularity; Mouilet et al. 2005). However, the

flexibility to use more than one trait often is required to understand even simple natural systems

and in such cases, the inclusion of trait matrices, distance measures, and sometimes

dendrograms, is required. Unfortunately, the use of these multivariate statistical procedures

introduces complications that require researchers to make several key decisions for data analysis.

Ultimately, these decisions should have minimal effect on patterns of species characteristics as

they relate to ecosystem function. However, recent studies (e.g. Poos et al. 2009) have shown

that these methodological decisions may be more important than thought here to fore. Not

surprisingly there are countless ways to incorporate multivariate functional variation into

measures of diversity.

A popular index, known as FD (Petchey and Gaston 2002), measures functional diversity as the

total branch length of a dendrogram based on functional traits. To produce a dendrogram several

decisions need to be made. First, the number and type of traits important to ecosystem function

need to be identified. Second, a distance measure needs to be chosen that characterizes the

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relative differences among species based on their traits. Finally, a clustering algorithm is needed

to produce a dendrogram that hierarchically segregates species into functional groups based on

their relative distances (Petchey and Gaston 2002). This method has been criticized recently for

having the additional subjective step of clustering traits onto a dendrogram (Podani and Schmera

2006; Poos et al. 2009). Although standard methods may provide one way to reduce

subjectivity, it is unlikely that a single distance measure or clustering algorithm can be used in all

circumstances (Poos et al. 2009). Therefore, methods should be sought which can be used to

identify sensitivity.

Despite the common use of FD as a measure, claims related to whether FD can be used to derive

ecologically robust conclusions have only been quantitatively evaluated recently (Podani and

Schmera 2006; Poos et al. 2009). For example, Poos et al. (2009) showed that the probability of

two random assemblages showing contrasting levels of functional diversity ranged from 0 to as

high as 97.6%. Recently, consensus methods have been suggested as a means for providing a

standardized approach for dealing with variation in methodological issues (Mouchet et al. 2008;

Poos and Jackson, submitted). However, there are difficulties when employing consensus

methods to methodological choices in functional diversity, such as how many initial methods are

needed to build the optimal consensus (Poos et al. 2009), and to what degree adding poorly fitted

models may undermine a consensus approach (Poos and Jackson, submitted). Therefore the

objective of this study is to determine whether consensus methods can be used to understand

when sensitivity in measuring FD may be an issue. For this purpose, sensitivity is defined in two

ways. First, a sensitive index is defined as one where the same qualitative trends in functional

diversity across communities do not persist despite methodological choices (e.g. distance

measure and clustering algorithm). This definition is meant to compare the broad ecological

consequences of applying FD to ecological communities, with the implicit assumption that a

robust index should provide qualitatively repeatable trends. Second, a sensitive index is defined

as one where the identified dendrogram topologies persist. This definition requires that the

produced patterns of species groupings are maintained. If FD is not robust in this sense, it would

suggest that the implication of using FD may be unclear. Explicit recognition of the effects of

using a dendrogram, and the decisions needed to get there (e.g. choosing a distance measure and

clustering algorithm) need to be better understood, so appropriate guidelines for making these

decisions can be formulated.

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Methods

In this study, the same five data sets used in previous studies of FD (Petchey and Gaston 2002,

Podani and Schmera 2006; Poos et al. 2009). These datasets represent a variation in the number

and type of species (from 13 to 37), and the number and type of functional traits (from 6 to 27).

For example, the three vertebrate datasets use characteristics ranging from foraging behavior to

the consumption of prey species as their functional traits (Holmes et al. 1979, Jaksic and Medel

1990, Munoz and Ojeda 1997), whereas, the remaining two datasets rely on vegetative

characteristics, such as rooting depth and herbivore palatability, of the plants being studied

(Golluscio and Sala 1993, Chapin et al. 1996).

The measure functional diversity, FD, is based on the total branch length of a dendrogram of

functional traits. To obtain this dendrogram, species traits must be assigned a distance (or

resemblance) measure and clustering algorithm. Distance measures quantify the association

between two entities based on their characteristics (e.g. species based on their functional traits).

There are a large number of distance measures from which to choose depending on the data

(Jackson et al. 1989; Legendre and Legendre 1998; Podani 1999). Two distance measures were

used: Euclidean distance as suggested by Holmes et al. (1979), and cosine distance. Cosine

distance was used because it down-weights the potential over-fit created by covarying traits

(Legendre and Legendre 1998), a problem often encountered when analyzing functional traits of

species (Petchey and Gaston 2006), whereas, Euclidean distance emphasizes larger values, in

particular where positive covariance exists between traits. All trait matrices were standardized so

that all traits have a mean = 0 and variance =1 (i.e. z-scores; Holmes et al. 1979, Gaston and

Petchey 2002).

Variability in ecological data is often associated with just a few entities of which clustering into

key groupings can provide insight, such as the clustering of species based on functional traits

(Legendre and Legendre 1998; Podani 1999). Three clustering algorithms were used in this

analysis, unpaired pair group method with arithmetic mean (UPGMA), single linkage (i.e.

nearest neighbor) and complete linkage (i.e. maximum or farthest neighbor). These algorithms

represent natural endpoints across a methodological continuum of hierarchical clustering

algorithms, where single linkage lies on one end, complete linkage on the other, and UPGMA

lies somewhere in the middle (Gordon 1999, Podani and Schmera 2006; Poos et al. 2009).

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Using Consensus Methods to Identify Uncertainty when Measuring FD

To determine whether distance measure or clustering algorithm influenced dendrogram

topologies of FD, a routine was developed (in MatLAB version 7.1) to randomize the removal of

n species from the dataset and recalculate FD for each species combination, clustering algorithm

and distance measure. Each level of n species was replicated 1000 times. Functional diversity

(FD) was measured at each species richness interval as the total distance of branches in the

dendrogram. As FD measures the total branch lengths of a functional dendrogram, which relies

on clustering method and distance measure, all dendrograms were rescaled to value between 0-1

using the full species model. The range in FD at the full species level at each different clustering

method and distance measure was summarized.

The initial dendrogram was compared to each variation in distance measure using consensus

trees (Margus and McMorris 1981, Rohlf 1982) in NT-SYS (Rohlf 1997). Dendrograms were

compared using the consensus index CI(C) (Rohlf 1982, 1997). Unlike cophenetic correlation

(e.g. Mouchet et al. 2008), which compares a dendrogram to the un-modeled raw data, the

consensus index compares the similarity of dendrograms based on their cluster membership

(Sokal and Rohlf 1962, Shao and Soskal 1986, Legendre and Legendre 1998). The 50%

majority rules consensus index was used where a value of one indicates all subgroups share at

least 50% membership (i.e. the consensus tree is completely bifurcated indicating similar

topology between the original trees) and a value of zero indicates no subgroups are shared

(Jackson et al. 1989, Lapointe and Legendre 1990). Although a more strict measure of

consensus can be used, the use of a 50% majority rules leads to a more liberal assessment of the

similarity between trees than a strict measure would provide.

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Results

The Relationship between FD, Distance Measure & Clustering Algorithm

The relationship between FD and community type changed with distance measure and clustering

algorithm. In particular, a change in distance measure caused communities to be ranked

differently with respect to FD (Figure 6.1). Differences in levels of FD between communities

changed with either a change in distance measure or a change in clustering method. For

example, FD for predatory vertebrate communities was more similar to Patagonian forb

communities using UPGMA and Euclidean distance, but more similar to intertidal fish

communities when the clustering algorithm was changed to complete linkage (Figure 6.1).

Similarly, the Patagonian forb communities showed similar levels of FD with three different

communities, depending on the distance measure and clustering method (Figure 6.1). Rescaling

FD did not change these conclusions.

Altering the clustering algorithm had a greater impact on the measured variation of FD than

altering of distance measure. On average, the overall choice of distance measure and clustering

algorithm accounted for a range of 27.4% in the measured amount of functional diversity. At

maximum species richness measured functional diversity ranged between a minimum of 21%

and a maximum of 61% (mean 34.2%) across clustering methods. Similarly, at maximum

species richness the measured amount of functional diversity ranged between a minimum of 12%

and a maximum of 41% (mean 20.5%) based on distance measure (Figure 6.2). Qualitatively,

single linkage showed the smallest amount of functional diversity, whereas, complete linkage

showed the largest.

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Figure 6.1 - The relationship between distance measure (Euclidean or cosine) and clustering algorithm (SL = single linkage / nearest neighbor, UPGMA = unweighted pair group method with arithmetic mean, CL = complete linkage / farthest neighbor) with FD using five community data sets: A) Arctic vegetation (Chapin et al. 1996); B) Insectivorous birds (Holmes et al. 1979); C) Patagonian forbs (Golluscio and Sala 1993); D) Intertidal fish (Munoz and Ojeda 1997); and, E) Predatory vertebrates (Jaksic and Medel 1990). FD values were re-scaled relative to the Arctic vegetation data, which has the highest FD values. This standardization leads to the appearance of a constant outcome for the Arctic dataset, but this consistency is solely an artifact of using it as the reference point rather than the outcomes not differing depending on the resemblance measure or clustering algorithm.

There was large variation in the measured amount of functional diversity as it relates to the

removal of species (Figure 6.2). Not surprisingly, FD was related to species richness, with

communities with less species showing smaller ranges of FD. In general, there was a greater

similarity between the fitted functional curves for FD and clustering algorithms than the

functional curves for FD and similar distance measures. There were two exceptions to this

generality. First, the predatory vertebrate dataset showed functional relationships more closely

related to similar distance measures (Figure 6.2D), whereas, the curve fitted with UPGMA

clustering algorithm and Euclidean distance of the intertidal fishes dataset was more closely

related to the curve based on cosine distance and single linkage, than a similar clustering

algorithm or distance measure. Overall, conclusions regarding the qualitative relationship of

functional diversity among communities were not robust to methodological choices (i.e. FD was

not consistent across communities).

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A) B)

C) D)

E) Figure 6.2 - The relationship between distance measure (solid lines = Euclidean distance, dashed lines = cosine distance) and building a dendrogram using a clustering algorithm (1 = complete linkage / farthest neighbor, 2 = unweighted pair group method with arithmetic mean / UPGMA, 3 = single linkage / nearest neighbor) where species are individually removed when calculating FD. Five community data sets are shown: A) Arctic vegetation (Chapin et al. 1996), B) Insectivorous birds (Holmes et al. 1979), C) Patagonian forbs (Golluscio and Sala 1993), D) Intertidal fishes (Munoz and Ojeda 1997), and E) Predatory vertebrates (Jaksic and Medel 1990). Shown inset are 50% majority rule consensus trees demonstrating lack of between group fidelity of species where calculating functional diversity using different distance measures, but the same clustering approach.

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Identifying Sensitivity in FD Using Consensus Methods

The overall low value of the CI(C) index indicates that the decision of choosing a distance

measure influences the overall dendrogram to such an extent that there was little resemblance

between the dendrogram based on Euclidean distance and the dendrogram based on cosine

distance (Table 6.1; Figure 6.2 inset). A comparison of consensus values indicates that FD

identified different dendrogram groups (only 46-51% similarity) depending on the distance

measure or clustering algorithm (unpublished results) used (Table 6.1).

Table 6.1 – Consensus measures of dendrogram group fidelity across distance measures (Euclidean and cosine) for each clustering algorithm: single linkage, unweighted pair group method with arithmetic means (UPGMA), and complete linkage. Group fidelity was determined by majority rules consensus tress using CI(C) consensus index.

Data set No. species

Complete Linkage UPGMA Single

Linkage Insectivorous birds 22 0.55 0.35 0.35 Intertidal fish 13 0.55 0.45 0.55 Patagonian forbs 24 0.45 0.50 0.41 Predatory vertebrates 11 0.22 0.55 0.33 Arctic vegetation 37 0.69 0.71 0.66 Average --- 0.49 0.51 0.46

The clustering algorithm did not improve the similarity between functional topologies. For

example, single linkage, unweighted pair group methods using arithmetic mean, and complete

linkage all showed similar rates of consensus tree resemblance, regardless of the size of the tree

or the dataset used (Table 6.1). Overall, dendrogram topologies were not robust to the choice of

distance measure or clustering algorithm.

Discussion Functional diversity has become an important, but controversial focus of research at the

boundary between community and ecosystem ecology (Tilman 2000, Mason et al. 2003, Leps et

al. 2006; Poos et al. 2009). This study focused on a popular measure of functional diversity: the

total branch length of a functional dendrogram, known as FD (Petchey & Gaston 2002). The

results demonstrate that consensus methods were able to identify instances where FD was not

robust to the choice of distance measure or clustering algorithm. Specifically, consensus

methods indicated that both definitions of robustness were not supported. First, the qualitative

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relationships did not persist across communities (Figure 6.1). Second, the dendrogram

topologies of communities measured using FD differed with the decision of clustering method

and distance measure (Figure 6.2). These results were all in agreement with previous

quantitative analysis of FD (Poos et al. 2009).

The ability of consensus methods to determine qualitative differences of measuring functional

diversity across communities have ecological repercussions to many biodiversity-ecosystem

functioning studies. It is commonly assumed that diversity is a relative concept; that is, the

diversity of a community can only be judged vis-à-vis another community (Magurran 2004). In

this case, the conclusions reached by comparing communities based on functional diversity

would depend on the distance measure used to create the dendrogram and, therefore,

relationships among communities would be altered (Figure 6.1). For example, the FD for the

Patagonian forbs community was identical to predatory vertebrate community using single

linkage and Euclidean distance. However, if the distance measure was altered to cosine distance,

the Patagonian forbs community more closely resembled the insectivorous bird communities,

whereas, a change in clustering algorithm would show the Patagonian forbs community more

closely resembled the intertidal fish community. In this study, a difference in clustering

algorithm or distance measure altered community relationships altogether, and these

relationships persisted regardless of the scaling used. These differences have a high likelihood

of altering ecological interpretations of functional diversity across communities that, in turn, will

potentially confound ecosystem-based studies.

Different methods of calculating FD can lead to different dendrograms, and consequently

different measures of functional diversity. For example, data analyzed using different distance

measures and clustering algorithms varied by a range of 27% in the measured amount of

functional diversity at maximum species richness (Figure 6.2). Therefore, studies which use

dendrogram-based methods of measuring functional diversity may drastically under or over

estimate the amount of functional diversity, assuming some “true” value can be determined.

Therefore one may assume two communities are more similar using one method when they may

not be when based on another measure. Finally, decisions are required regarding how the data

are treated - for example, depending on how tied values in a similarity matrix are treated, a

number of different dendrograms can be produced (Jackson et al. 1989). As the clustering

technique will produce dendrograms (with the underlying goal of determining group structure)

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whether or not true groups exist (Jackson et al. 1989), the cumulative effect of these decisions to

the relevance of the identified groups may be unknowingly large. For example, this study

suggests that the decision of adding distance measure and clustering algorithm to produce a

dendrogram will alter the range in the measure of FD (e.g. 34.2% for clustering algorithm and

20.5% for distance measure) and the ecological conclusions reached (Fig. 6.1).

Identifying areas where sensitivity may be an issue can aid in developing more robust or

representative indices of functional diversity. Recently, studies have attempted to use consensus

methods, which take the average of several different methodological approaches, to reconcile

differences in FD from methodological issues (Mouchet et al. 2008). However, previous

applications of consensus methods may not be sufficient in determining where sensitivity may be

an issue. In this paper, only 46-51% of the topologies were in agreement (Table 6.1) and there

was large variation in the measurement of FD across methodological choices and species (Figure

6.2). Although consensus methods were useful in identifying issues of sensitivity, using

consensus methods as a standardized approach may not be appropriate because averaging three

poorly associated models together may actually produce a worse outcome (Poos and Jackson

submitted). In addition, previous applications of consensus methods provide several more

decisions – such as how many initial models are needed to build the optimal consensus, and

which consensus method is most appropriate (Poos et al. submitted) – which may further impact

FD (Poos et al. 2009). There is considerable debate regarding the most appropriate measure of

functional diversity and the qualities that metric should possess (Loreau et al. 2001, Mason et al.

2003, Ricotta 2005, Leps et al. 2006). Clearly, regardless of the index used, any index of

functional diversity must be robust (e.g. qualitatively similar across methods and dendrograms)

to decisions inherent in its calculation or one must decide upon a common statistical

methodology in order to permit comparisons amongst studies. Quantitative comparisons of how

functional diversity indices differ are rare (e.g. Petchey et al. 2004; Walker et al. 2008), and

evaluations of other functional diversity indices are needed. In calculating FD, the decisions

inherent in its calculation represent two additional difficulties aside from previous criticisms of

which species, what kind of diversity, and which ecosystem function (Bengtsson 1998, Symstad

et al. 1998, Cardinale 2000, Jax 2005) are to be included and therefore should also define the

choice of similarity measure and clustering algorithm. Further criticisms, such as how many

functional traits (Walker et al. 1999, Podani and Schmera 2006), what qualifies as a functional

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group (Petchey and Gaston 2006), and what type of consensus approach, also apply. Explicit

recognition and justification of each of these decisions is warranted for improving functional

diversity research.

Acknowledgements Funding NSERC & OGS Scholarships to M.S.P and S.C.W., NSERC Discovery Grant to D.A.J.,

Ontario Ministry of Natural Resources, and University of Toronto.

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152

Appendices

Appendix 6.1 – MatLAB Code for calculating total branch lengths of dendrograms from various species combinations

function branches = branch_lengths(Z) linkages = size(Z,1); species = linkages+1; branches = zeros(linkages, 2); for i = 1:linkages if (Z(i,1) < (1+species)) & (Z(i,2) < (1+species)) branches(i,1) = Z(i,3); branches(i,2) = Z(i,3); elseif (Z(i,1) > species) & (Z(i,2) < (1+species)) branches(i,1) = Z(i,3) - Z((Z(i,1)-species),3); branches(i,2) = Z(i,3); elseif (Z(i,1) < (1+species)) & (Z(i,2) > species) branches(i,1) = Z(i,3); branches(i,2) = Z(i,3) - Z((Z(i,2)-species),3); else branches(i,1) = Z(i,3) - Z((Z(i,1)-species),3); branches(i,2) = Z(i,3) - Z((Z(i,2)-species),3); end end

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Chapter 7: General Conclusions

This thesis provides novel advancement in determining the influence of methodological choices

in conservation-based models and on how consensus methods may reduce uncertainty in such

models. In particular, this thesis demonstrates that regardless of the scale (local as in Chapter 5

or regional in Chapter 6), of the species, or study system in question, methodological choices

have the ability to dramatically impact resultant analyses and ecological inquiry. For example, in

Chapter 2, methodological choices had the ability to provide levels of sensitivity over 97% in the

measure of functional diversity. Similarly, in Chapter 4, methodological choices changed

estimates of population viability by several orders of magnitude. In Chapter 3, the addition or

removal of rare species did not impact multivariate analyses as much as choice of distance

measure or multivariate method, which had the ability to drastically alter bioassessments. As

these findings are novel, this chapter will highlight some conclusions regarding the importance

of methodological choices and attempt to provide recommendations for minimizing

methodological impacts.

A) Conservation-based Models in General

The importance of understanding the impact of methodological choices is not only timely, but

essential. With complex statistical software readily available, ecologists now have several dozen

approaches to choose from when developing conservation-based models. Perhaps not

surprisingly, this thesis demonstrates, as have previous authors, the importance of model type in

impacting results (e.g. Jackson et al. 1989; Jackson 1993; Guissan and Zimmerman 2000;

Thuiller 2005; Elith et al. 2006; Dormann et al. 2008; Marrimon et al. 2009). For example, in

Chapter 3, multivariate technique impacted analyses more than choice in distance measure or the

removal of rare species. This is in agreement with others who have demonstrated that rare

species may provide useful information (e.g. Cao et al 1998; 2001) and that selection of

multivariate method may influence interpretation strongly (e.g. Jackson et al. 1989; Jackson

1993). However, this thesis also demonstrates that despite laborious efforts in developing

modeling comparisons (Olden and Jackson 2002; Thuiller et al. 2005; Elith et al. 2006; Sharma

and Jackson 2008; Marrimon et al. 2009; this thesis), there has been no evidence of the pre-

eminence of any singular methodological approach. Issues related to methodological impacts

154

will vary depending on the dataset and objectives in question, such as which distance measure,

clustering algorithm, ordination (or other). Therefore, methods that can be used to reduce the

influence of methodological choices, or highlight when methodological choices may be an issue

- such as the use of consensus models - will help recognize and reduce the impact of

methodological choices.

B) Functional Diversity

Incorporating functional traits can be another way of improving modeling approaches for species

with conservation concern. Species’ functional characteristics strongly influence ecosystem

properties (Loreau et al. 2001; Hooper et al. 2005), and the understanding of the relationships

between functional diversity and community structure has been important for identifying

mechanisms of biodiversity effects. To date, the focus on functional traits has been on dominant

species (Grime 1998); although rare (presumably including imperiled) species can have a large

influence on ecosystem processes (Power et al. 1996).

Methodological choices in the measure of functional diversity remain a controversial topic

(Mason et al. 2003, Ricotta 2005, Podani and Schmera 2006). For example, this thesis (Chapters

2 and 6) demonstrates that the measure of functional diversity is greatly complicated by

methodological choices. Therefore, discussion is warranted on the future use of FD as a metric

of functional diversity and some of the qualities this metric poses. Ultimately, the choice of what

to do regarding the impact of methodological choices on measuring functional diversity may rely

on the advantages and disadvantages of the features of each approach. Many of these features

have been discussed previously (Mason et al. 2003, Hooper et al. 2005, Mouillot et al. 2005,

Ricotta 2005, Leps et al. 2006, Petchey and Gaston 2006; Schmera et al. 2009a; 2009b) and here,

properties of metrics of functional diversity are discussed plus are new insights into areas worthy

of future research.

One issue with metrics of functional diversity is whether or not they increase monotonically with

species richness (known as ‘set monotonicity’; Ricotta 2005; Schmera et al. 2009a). For

example, unlike Rao’s quadratic entropy (Rao 1982, Botta-Dukat 2005), FD has the intuitive

property of a monotonic relationship with species richness (Petchey and Gaston 2006). FD

cannot decrease when a species is added to a community, and when a species is removed FD

155

cannot increase (Petchey and Gaston 2002). However, in some cases FD violates this feature

(Podani and Schmera 2006) and Walker et al. (1999, 2008) have noted that functional attribute

diversity (FAD) previously has been misidentified as lacking this feature. Recent advancements

in measuring FAD (Schmera et al. 2009a) have helped to clarify the issue of set monotonicity in

FAD; however, future research into whether set monotonicity is a requirement for robust

estimation of functional diversity remains to be determined. For example, Rao’s quadratic

entropy is gaining currency as a flexible index of functional diversity as it is an intuitive

extension of Simpson’s index of diversity (Simpson 1949, Botta-Dukat 2005, Leps et al. 2006),

yet does not have set monotonicity (Pavoine and Bonsall 2009).

Given that a change in distance measure caused a related change in the qualitative relationships

of functional diversity across communities (in Chapter 6), it bears asking to what degree a

distance measure is required. Although the choice of distance measure involves subjectivity that

may influence the analysis, the inclusion of a distance measure allows for a continuous

segregation of multiple species based on multiple functional traits (Legendre and Legendre

1998). Therefore, the question of whether or not to include a distance measure depends to what

extent ecological diversity is based on the trait dissimilarity among species in a community

(Tilman 1997, Petchey and Gaston 2002) and to what degree a distance measure can distinguish

those traits and/or species. Few studies relate trait dissimilarity among species in a community

(Petchey and Gaston 2002, Garnier et al. 2004, Heemsbergen et al. 2004, Leps et al. 2006), and

unfortunately, even fewer studies determine the degree from which distance measure can

distinguish those trends. One mechanism to lower the subjectivity of the calculation in the

choice of distance measure is to provide standards based on the purpose of the study. Where

traits have a mixture of data types the use of Gower similarity may more applicable (Podani and

Schmera 2006), although quantitative comparisons with other distance measures are still needed.

Furthermore, unlike Euclidean distance, Gower similarity can be used with missing values and

has the advantage of not being influenced by the unit of measure (Gower 1971, Legendre and

Legendre 1998, Podani 1999). Future research should focus on quantitative comparisons of pair-

wise distance measures and ensuring they retain strong linkages to raw data on species traits.

The treatment of functionally redundant species in functional diversity indices remains

controversial. Central to this debate is whether an index of functional diversity should change if

a species is added or lost that is identical to one already present; a feature known as functional

156

redundancy (Walker et al. 2008; Schmera et al. 2009b). Petchey and Gaston (2006) contend that

a good index of functional diversity should not increase if a redundant species is added.

However, this feature may be a product of the number of traits used to segregate species, and the

ecosystem function in question, and not necessarily because the species provide no additional

function (Rosenfield 2002). More recently, improvements to the calculation of functional

diversity indices have allowed for comparison of functional diversity across communities

without sampling bias (Walker et al. 2008). This feature may provide additional benefit to

functional diversity indices as it deemphasizes the current trend in identifying functionally

redundant species and emphasizes the need to compare functional traits across broader

assemblages (Rosenfeld 2002).

There is a growing consensus that functional diversity is likely to be the component of

biodiversity most relevant to ecosystem function (Wright et al 2006). Recently, biodiversity

theory and management perspectives have converged, where each has embraced the need for

incorporating species-specific biology into models (Srivastava and Velland 2005). For example,

large-bodied species that occupy high trophic positions in food webs and occur at low abundance

are thought to be particularly vulnerable to extinction (Lawton and May 1995). The application

of functional traits, such as these, to improve models is becoming more widespread (Cardillo and

Bromham 2001; Olden et al. 2006), and their ability to improve models for species with

conservation concern needs to e assessed. The consideration of both diversity and evenness of

species traits may be an important consideration as evenness measures not only the range of

functional variation, but how much of the functional variation is filled within that range (Mason

et al. 2003 Mouillot et al. 2005). Newer approaches that disassemble functional evenness from

functional richness (e.g. Walker et al. 2008) and allow the researcher to distinguish one from the

other, may allow for better application of functional diversity indices to research on species with

conservation concerns.

157

Recommendations

• Methodological choices should be closely scrutinized when modeling species with

conservation concern. As this thesis (Section I) demonstrates, regardless of scale, study

system or ecological question, methodological choices had the ability greatly to impact

results.

• The use of singular statistical methods should be avoided when dealing with species

where data are deficient (as often is the case of species with conservation concern). This

dependency was true regardless of the scale - community level as in Chapters 2, 3, and 6,

population level as in Chapter 5, or the metapopulation level in Chapter 4. The need for

comparative approaches using multiple statistical methods has been highlighted as one

means to reduce problems related to conservation-based models (Guisan and Zimmerman

2000; Olden and Jackson 2002).

• In the case of modeling species with conservation concerns, singular statistical methods

should not be interpreted in isolation. As Chapter 6 demonstrated, several methods

provided contrasting explanatory relationships. These relationships, if taken in isolation,

may bias future conservation efforts in areas where species are not showing strong habitat

relationships.

• This thesis demonstrates that consensus methods may provide reduced uncertainty in

modeling species with conservation concerns. The advantages of consensus modeling

over singular approaches are numerous :

o It is an intuitive extension of modeling using singular approaches;

o Advancements in computing power have accelerated the use of new statistical

models, which can be added to this approach (e.g. random forest, boosted

regression);

o It reduces biases based on choosing singular statistical approaches, and the

resultant limitations of fitting data to that approach (especially if assumptions are

not met);

158

o It produces a prioritized model output that can be used to address conservation

priorities, identify areas of high conservation value … etc., even if there are

differences across the models and data is noisy;

o Model thresholds can be adjusted readily to prioritize the correct classification of

species presence or absence (e.g. a balance approach was used here in); and,

o Misclassifications can be identified and plotted spatially to identify areas where

error rates were high and data can be collected to refine the models.

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