Ecological Models and Data in R

211 Pages·2017·1.37 MB·English

book August 29, 2007
Ecological Models and Data in R
book August 29, 2007
book August 29, 2007
Ecological Models and Data in R
Ben Bolker
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
book August 29, 2007
iv
ACKNOWLEDGMENTS
Lots of people have helped me start and finish this book. I would like to
thank:
The R community, for building such a useful tool.
Various research institutions that have provided me room to work
over the years: the Department of Zoology at the University of Florida, the
Mathematical Biosciences Institute, and the NERC Centre for Population
Biology at Silwood Park.
All the students and colleagues who have brought me such interesting
and challenging problems over the years, especially those who took the time
to find errors or make suggestions: Jorge Ahumada, Chad Brassil, David
Buck, Lin Cassidy, Lew Coggins, Rick Condit, Mick Crawley, Ian Dworkin,
Ian Fiske, Gregor Gorjanc, Tom Hobbs, Rico Holdo, Holly Kindsvater,
Aaron MacNeil, Julien Martin, Jeremy Mendoza, Mike McCoy, Jess Met-
calf, Toshinori Okuyama, Stuart Sandin, Nat Seavy, Darren Shaw, Adrian
Stier, Don Strong, Maria Uriarte, Will Wilson, and several anonymous re-
viewers. I apologize if I forgot your name, or if your suggestions slipped
through the cracks or were just too hard to implement.
Researchers who generously contributed their original data sets as
examples: James D. Thomson, Russ Schmitt and Sally Holbrook, Mike
Dodd and Jonathan Silvertown, James Vonesh, and Jackie Wilson.
MyPh.D.andpostdoctoraladvisors,BryanGrenfellandStevePacala,
for introducing me to mathematical and statistical ecology.
Myparents,JoanandEthanBolker,forthoroughandthoughtfulcom-
ments on the manuscript and for establishing the expectation that writing
books is just what people do.
And last but not least Aidan and especially Tara for their love, pa-
tience and support.
Essentially, all models are wrong
but some are useful.
George Box
book August 29, 2007
Contents
Chapter 1. Introduction 1
1.1 Introduction 2
1.2 What this book is not about 4
1.3 Frameworks for modeling 6
1.4 Frameworks for statistical inference 12
1.5 Frameworks for computing 23
1.6 Outline of the modeling process 26
1.7 R supplement 31
Chapter 2. Exploratory data analysis and graphics 41
2.1 Summary 42
2.2 Introduction 42
2.3 Getting data into R 43
2.4 Data types 48
2.5 Exploratory data analysis and graphics 56
2.6 Conclusion 82
2.7 R supplement 84
Chapter 3. Deterministic functions for ecological modeling 99
3.1 Summary 100
3.2 Introduction 100
3.3 Finding out about functions numerically 102
3.4 Finding out about functions analytically 106
3.5 Bestiary of functions 119
3.6 Conclusion 134
3.7 R supplement 135
Chapter 4. Probability and stochastic distributions for ecological
modeling 139
4.1 Introduction: why does variability matter? 140
4.2 Basic probability theory 141
4.3 Bayes’ Rule 145
4.4 Analyzing probability distributions 155
book August 29, 2007
vi CONTENTS
4.5 Bestiary of distributions 160
4.6 Extendingsimpledistributions; compoundingandgeneralizing 182
4.7 R supplement 188
Chapter 5. Stochastic simulation and power analysis 197
5.1 Introduction 198
5.2 Stochastic simulation 199
5.3 Power analysis 209
Chapter 6. Likelihood and all that 227
6.1 Introduction 228
6.2 Parameter estimation: single distributions 228
6.3 Estimation for more complex functions 243
6.4 Likelihood surfaces, profiles, and confidence intervals 249
6.5 Confidenceintervalsforcomplexmodels: quadraticapprox-
imation 260
6.6 Comparing models 267
Chapter 7. Optimization and all that 293
7.1 Introduction 294
7.2 Fitting methods 294
7.3 Markov chain Monte Carlo 310
7.4 Fitting challenges 319
7.5 Estimating confidence limits of functions of parameters 330
Chapter 8. Likelihood examples 347
8.1 Tadpole predation experiments 348
8.2 Goby survival analysis 363
8.3 Seed removal 374
Chapter 9. Standard statistics revisited 393
9.1 Introduction 394
9.2 General linear models 397
9.3 Nonlinearity: nonlinear least squares 404
9.4 Non-normal errors: generalized linear models 406
Chapter 10. Modeling variance 417
10.1 Introduction 418
10.2 Variable variance 420
10.3 Correlations: time-series and spatial data 423
10.4 Multi-level models: special cases 428
10.5 General multi-level models 431
10.6 Challenges 438
10.7 Conclusion 440
book August 29, 2007
CONTENTS vii
Chapter 11. Dynamic models 445
11.1 Introduction 446
11.2 Simulating dynamic models 447
11.3 Observation and process error 452
11.4 Process and observation error 455
11.5 SIMEX 458
11.6 State space models 459
11.7 Conclusions 470
Chapter 12. Afterword 477
Appendix A. Algebra and calculus basics 479
A.1 Exponentials and logarithms 480
A.2 Differential calculus 481
A.3 Partial differentiation 482
A.4 Integral calculus 482
A.5 Factorials and the gamma function 482
A.6 Probability 483
A.7 The delta method: formula and derivation 483
A.8 Linear algebra basics 484
book August 29, 2007
book August 29, 2007
Chapter One
Introduction and background
book August 29, 2007
2 CHAPTER1
SUMMARY
This chapter gives a broad overview of the philosophy and techniques of
ecological modeling. A small data set on seed removal illustrates the three
most common frameworks for statistical modeling in ecology: frequentist,
likelihood-based, and Bayesian. The chapter also reviews what you should
know to get the most out of the book, discusses the R language, and spells
out a step-by-step process for building models of ecological systems.
If you’re impatient with philosophical discussion, you can read Sec-
tion 1.4 and the R supplement at the end of the chapter and move on to
Chapter 2.
1.1 INTRODUCTION
This book is about combining models with data to answer ecological ques-
tions. Pursuing this worthwhile goal will lead to topics ranging from basic
statistics, to the cutting edge of modern statistics, to the nuts and bolts of
computer programming, to the philosophy of science. Remember as we go
along not to miss the ecological forest for the statistical trees; all of these
complexities are in the service of answering ecological questions, and the
most important thing is to keep your common sense about you and your
focus on the biological questions you set out to answer. “Does this make
sense?” and“What does this answer really mean?” are the two questions
you should ask constantly. If you cannot answer them, back up to the last
point you understood.
If you want to combine models with data, you need to use statisti-
cal tools. Ecological statistics has gotten much more complicated in the
last few decades. Research papers in ecology now routinely refer to likeli-
hood, Markov chain Monte Carlo, and other arcana. This new complexity
arises from the explosion of cheap computing power, which allows us to
run complicated tests quickly and easily — or at least more easily than
before. But there is still a lot to know about how these tests work, which
is what this book is about. The good news is that we can now develop
statistical methods that directly answer our ecological questions, adapting
statistics to the data rather than vice versa. Instead of asking“what is the
probability of observing at least this much variability among the arcsine-
square-root-transformed counts of seeds in different treatments?”, we can
ask“is the number of seeds removed consistent with standard foraging the-
ory, and what are the attack rates and handling times of predators? Do the
attack rates or handling times increase with mean seed size? With the time
that the seeds have been available? Is there evidence for variability among
seeds?”. By customizing statistical tests we can squeeze more information,