Bayesian Methods for Ecology

✍ Scribed by Michael A. McCarthy

• Publisher: Cambridge University Press
• Year: 2007
• Tongue: English
• Leaves: 312
• Edition: 1
• Category: Library

✦ Synopsis

The interest in using Bayesian methods in ecology is increasing, however many ecologists have difficulty with conducting the required analyses. McCarthy bridges that gap, using a clear and accessible style. The text also incorporates case studies to demonstrate mark-recapture analysis, development of population models and the use of subjective judgement. The advantages of Bayesian methods, are also described here, for example, the incorporation of any relevant prior information and the ability to assess the evidence in favour of competing hypotheses. Free software is available as well as an accompanying web-site containing the data files and WinBUGS codes. Bayesian Methods for Ecology will appeal to academic researchers, upper undergraduate and graduate students of Ecology.

✦ Table of Contents

Cover......Page 1
front-matter......Page 2
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
1 Introduction......Page 17
Box 1.1 The Reverend Thomas Bayes, FRS......Page 19
Example 1: Logic in determining the presence or absence of a species......Page 20
Frequentist approaches......Page 21
Null hypothesis significance testing......Page 22
Information theoretic methods......Page 23
Bayesian methods......Page 24
Box 1.2 Conditional probability......Page 26
Box 1.3 Baye's rule for a finite number of hypotheses......Page 27
Random sampling from the posterior distribution using WinBUGS......Page 30
Box 1.4 The burn-in when sampling from Markov chains......Page 31
The frog surveying problem in WinBUGS......Page 32
Box 1.5 WinBUGS code for determining the presence of a species......Page 33
Box 1.6 Monte Carlo methods......Page 35
Example 2: Estimation of a mean......Page 36
Box 1.7 Baye's rule for continuous hypotheses......Page 37
The Bayesian solution for the normal mean......Page 38
Box 1.8 Estimating a mean for a normal model using WinBUGS......Page 39
Box 1.9 Estimating a mean for a normal model analytically......Page 41
Box 1.10 Estimating the mean of a normal model with an uninformative prior......Page 43
Concluding remarks......Page 45
Introduction......Page 46
Null hypothesis significance testing......Page 47
Box 2.1 Null hypothesis tests for a proportion......Page 48
Information-theoretic methods......Page 50
Null hypothesis significance testing......Page 51
Box 2.2 Bayesian analysis of a proportion......Page 52
Define the null hypothesis and its alternative......Page 53
Calculate the p-value......Page 55
Reject the null hypothesis if the p-value is small......Page 56
Box 2.3 Null hypotheses in the courts......Page 58
Summary of null hypothesis testing......Page 59
Select a set of candidate models......Page 61
Collect data......Page 62
Calculate the relative amount of information lost by each model......Page 63
Box 2.4 Information theoretic methods in the courts......Page 65
Box 2.5 Bayesian methods in the courts......Page 66
Summary of information-theoretic methods......Page 67
Bayesian methods......Page 68
Assign prior probabilities......Page 69
Use Bayes' rule to combine the prior and the data......Page 72
Estimating effect sizes......Page 74
Concluding remarks......Page 77
The average......Page 79
Box 3.1 A frequentist approach to estimating an average......Page 80
Box 3.2 A Bayesian approach to estimating an average......Page 82
Box 3.3 The Poisson distribution......Page 85
Box 3.4 Analysing the mean of the Poisson distribution......Page 86
Estimating differences......Page 87
Box 3.5 The Poisson model with extra variation......Page 88
Required sample sizes when estimating means......Page 89
Box 3.6 Hierarchical models......Page 90
Box 3.7 Estimating the difference between paired observations......Page 92
Propagating uncertainty in the required sample size......Page 93
Box 3.8 More on sample sizes......Page 94
Box 3.9 An example of calculating required sample sizes......Page 95
Box 3.10 Uncertainty in the required sample size......Page 96
Estimating proportions......Page 97
Box 3.11 Estimating a proportion......Page 98
Annual mortality of powerful owls......Page 99
Box 3.12 Estimating a proportion with a subjective uniform prior......Page 101
Box 3.13 Estimating a proportion with a beta prior......Page 103
Multinomial models......Page 104
Using proportions......Page 105
Box 3.14 Analysing multinomial proportions......Page 106
Box 3.15 Uncertainty in complex functions-diversity of a pond community......Page 107
Concluding remarks......Page 108
4 How good are the models?......Page 110
How good is the fit?......Page 111
Box 4.1 Analysing the mean of a Poisson distribution, using a gamma prior......Page 112
Box 4.2 Credible intervals for proportions......Page 113
A measure of fit......Page 115
How complex is the model?......Page 117
Box 4.3 Calculating D as a measure of model fit......Page 118
Akaike's information criterion......Page 121
How different are the models?......Page 122
The Bayes factor and model probabilities......Page 124
Box 4.4 Comparing different models......Page 125
Box 4.5 Model probabilities and Bayes factors......Page 128
Evaluating the shape of distributions......Page 132
Concluding remarks......Page 134
Simple linear regression......Page 135
Box 5.1 Simple linear regression for coarse woody debris......Page 137
Box 5.2 Uncertainty of a prediction in frequentist analysis......Page 139
Multiple linear regression......Page 140
Box 5.3 Multiple linear regression......Page 141
Interaction terms......Page 143
Box 5.4 Linear regression with an interaction term......Page 144
Non-linear regression......Page 146
Box 5.5 Non-linear transformation of the dependent variable......Page 148
Logistic regression......Page 150
Box 5.6 Non-linear regression for coarse woody debris......Page 151
Box 5.7 Logistic regression: The occurrence of Leionema ralstonii in rock outcrops......Page 153
Box 5.8 Centring data for better sampling......Page 155
Imperfect detection......Page 157
Box 5.9 Accounting for imperfect detection......Page 158
Box 5.10 Poisson regression: ant species richness......Page 162
Correlation......Page 164
Box 5.11 Correlation analysis......Page 166
Model-based priors for correlations......Page 167
Box 5.12 Model-based priors for correlations......Page 168
Concluding remarks......Page 172
One-way ANOVA......Page 174
Coding of variables......Page 175
Box 6.1 A simple one-way ANOVA......Page 176
Box 6.2 Simple ANOVA using a reference class......Page 177
Fixed and random factors......Page 178
Box 6.3 One-way ANOVA with a random factor......Page 180
Two-way ANOVA......Page 181
Box 6.4 Two-way ANOVA......Page 182
Variance partitioning......Page 183
Box 6.5 Including an interaction term in ANOVA......Page 184
An example of ANOVA: effects of vegetation removal on a marsupial......Page 186
Box 6.6 Partitioning variation in the data......Page 187
Box 6.7 An example of repeated-measures ANOVA......Page 191
Box 6.8 An ANOVA to establish an informative prior......Page 193
Box 6.9 Back-transforming the predictions of a model......Page 195
Analysis of covariance......Page 196
Box 6.10 Analysis of covariance......Page 197
The first experiment......Page 198
Box 6.11 ANCOVA example for goby mortality......Page 199
The second experiment......Page 201
Box 6.12 ANCOVA using an informative prior......Page 202
Log-linear models for contingency tables......Page 206
Box 6.13 Analysis of contingency tables......Page 207
Concluding remarks......Page 209
Case studies......Page 211
Methods......Page 213
Box 7.1 Mark-recapture analysis......Page 215
Box 7.2 The 'ones trick' in WinBUGS......Page 217
Box 7.3 Explicit calculation of the likelihood for mark-recapture models......Page 219
Box 7.4 Mark-recapture analysis of female European dippers......Page 221
8 Effects of marking frogs......Page 223
Logistic regression......Page 225
Model A......Page 226
Models B and C......Page 227
Box 8.1 Analysing return rates with a non-linear model......Page 228
Mountain pygmy possums......Page 233
Box 9.1 Estimating parameters of a population model......Page 235
Box 9.2 Posterior distributions for predictions of a population model......Page 238
Eliciting probabilities......Page 241
Handling differences of opinion......Page 242
Using the consensus of experts......Page 243
Representing differences of opinion with subjective priors......Page 246
Using Bayesian networks to represent expert opinion......Page 252
Concluding remarks......Page 259
Prior information......Page 260
Bayesian methods make us think......Page 261
A Bayesian future for ecology......Page 262
Appendices......Page 263
The steps in more detail......Page 265
How to write WinBUGS code......Page 269
Discrete random variables......Page 271
Continuous random variables......Page 273
Binomial......Page 277
Poisson......Page 278
Negative binomial......Page 279
Categorical......Page 280
Beta......Page 282
Normal......Page 284
Lognormal......Page 285
Exponential......Page 286
Gamma......Page 287
Multinomial......Page 288
Multivariate normal......Page 289
Wishart......Page 290
Conjugacy......Page 291
C MCMC algorithms......Page 293
Why does it work?......Page 296
References......Page 298
Index......Page 309