Bayesian Data Analysis in Ecology Using Linear Models with R, BUGS and Stan

329 Pages·2015·4.04 MB·English

by  Franzi Korner-Nievergelt, Tobias Roth, Stefanie von Felten, Jérôme Guélat, Bettina Almasi, Pius Korner-Nievergelt

Bayesian Data Analysis in Ecology Using
Linear Models with R, BUGS, and Stan
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Bayesian Data Analysis
in Ecology Using Linear
Models with R, BUGS,
and Stan
Fra¨nzi Korner-Nievergelt
Tobias Roth
Stefanie von Felten
Je´roˆme Gue´lat
Bettina Almasi
Pius Korner-Nievergelt
AMSTERDAM lBOSTON lHEIDELBERGl LONDON
NEWYORK lOXFORDlPARIS lSANDIEGO
SANFRANCISCOl SINGAPORElSYDNEYlTOKYO
AcademicPressisanimprintofElsevier
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Contents
DigitalAssets x
Acknowledgments xi
1. Why do we Need Statistical Models and What
is this Book About? 1
1.1 WhyWeNeedStatisticalModels 1
1.2 WhatThisBook isAbout 2
FurtherReading 4
2. Prerequisites and Vocabulary 5
2.1 Software 5
2.1.1 WhatIsR? 5
2.1.2 WorkingwithR 6
2.2 ImportantStatisticalTermsandHow
toHandleThem inR 7
2.2.1 DataSets,Variables,andObservations 7
2.2.2 DistributionsandSummaryStatistics 12
2.2.3 MoreonRObjects 15
2.2.4 RFunctionsforGraphics 16
2.2.5 WritingOurOwnRFunctions 17
FurtherReading 18
3. The Bayesian and the Frequentist Ways of Analyzing
Data 19
3.1 ShortHistoricalOverview 19
3.2 TheBayesianWay 20
3.2.1 EstimatingtheMeanofaNormalDistribution
withaKnownVariance 21
3.2.2 EstimatingMeanandVarianceofaNormal
DistributionUsingSimulation 22
3.3 TheFrequentistWay 27
3.4 ComparisonoftheBayesianandtheFrequentist Ways 30
FurtherReading 31
v
vi Contents
4. Normal Linear Models 33
4.1 LinearRegression 33
4.1.1 Background 33
4.1.2 FittingaLinearRegressioninR 36
4.1.3 DrawingConclusions 36
4.1.4 FrequentistResults 41
4.2 RegressionVariants:ANOVA,ANCOVA,andMultiple
Regression 42
4.2.1 One-WayANOVA 42
4.2.2 FrequentistResultsfromaOne-WayANOVA 47
4.2.3 Two-WayANOVA 49
4.2.4 FrequentistResultsfromaTwo-WayANOVA 53
4.2.5 MultipleComparisonsandPostHocTests 55
4.2.6 AnalysisofCovariance 56
4.2.7 MultipleRegressionandCollinearity 60
4.2.8 OrderedFactorsandContrasts 64
4.2.9 QuadraticandHigherPolynomialTerms 66
FurtherReading 67
5. Likelihood 69
5.1 Theory 69
5.2 TheMaximumLikelihoodMethod 72
5.3 TheLogPointwisePredictiveDensity 73
FurtherReading 74
6. Assessing Model Assumptions 75
6.1 ModelAssumptions 75
6.2 Independent andIdenticallyDistributed 76
6.3 TheQQPlot 79
6.4 TemporalAutocorrelation 80
6.5 SpatialAutocorrelation 85
6.6 Heteroscedasticity 91
FurtherReading 94
7. Linear Mixed Effects Models 95
7.1 Background 95
7.1.1 WhyMixedEffectsModels? 95
7.1.2 RandomFactorsandPartialPooling 96
7.2 Fitting aLinearMixedModelinR 98
7.3 RestrictedMaximumLikelihoodEstimation 101
7.4 AssessingModelAssumptions 102
7.5 DrawingConclusions 103
7.6 FrequentistResults 105
7.7 RandomIntercept andRandomSlope 106
Contents vii
7.8 NestedandCrossed RandomEffects 111
7.9 ModelSelectioninMixedModels 114
FurtherReading 114
8. Generalized Linear Models 115
8.1 Background 115
8.2 BinomialModel 117
8.2.1 Background 117
8.2.2 FittingaBinomialModelinR 118
8.2.3 AssessingModelAssumptions:Overdispersion
andZero-Inflation 121
8.2.4 DrawingConclusions 126
8.2.5 FrequentistResults 127
8.3 Fitting aBinary Logistic RegressioninR 128
8.3.1 SomeFinalRemarks 131
8.4 PoissonModel 132
8.4.1 Background 132
8.4.2 FittingaPoisson-ModelinR 134
8.4.3 AssessingModelAssumptions 135
8.4.4 DrawingConclusions 136
8.4.5 ModelingRatesandDensities:PoissonModel
withanOffset 137
8.4.6 FrequentistResults 138
FurtherReading 139
9. Generalized Linear Mixed Models 141
9.1 BinomialMixedModel 141
9.1.1 Background 141
9.1.2 FittingaBinomialMixedModelinR 142
9.1.3 AssessingModelAssumptions 143
9.1.4 DrawingConclusions 145
9.2 PoissonMixedModel 146
9.2.1 Background 146
9.2.2 FittingaPoissonMixedModelinR 146
9.2.3 AssessingModelAssumptions 148
9.2.4 DrawingConclusions 149
9.2.5 ModelingBirdDensitiesbyaPoissonMixed
ModelIncludinganOffset 151
FurtherReading 159
10. Posterior Predictive Model Checking and Proportion
of Explained Variance 161
10.1 PosteriorPredictiveModelChecking 161
10.2 MeasuresofExplainedVariance 169
FurtherReading 174
viii Contents
11. Model Selection and Multimodel Inference 175
11.1 When andWhyWeSelectModelsandWhyThis
isDifficult 175
11.2 Methods forModelSelectionandModelComparisons 179
11.2.1 Cross-Validation 179
11.2.2 InformationCriteria:AkaikeInformation
CriterionandWidelyApplicable
InformationCriterion 181
11.2.3 OtherInformationCriteria 183
11.2.4 BayesFactorsandPosterior Model
Probabilities 184
11.2.5 Model-BasedMethodstoObtainPosterior
ModelProbabilitiesandInclusion
Probabilities 184
11.2.6 “LeastAbsoluteShrinkageandSelection
Operator”(LASSO)andRidgeRegression 185
11.3 MultimodelInference 188
11.4 Which MethodtoChooseandWhichStrategytoFollow 193
FurtherReading 196
12. Markov Chain Monte Carlo Simulation 197
12.1 Background 197
12.2 MCMCUsingBUGS 201
12.2.1 UsingBUGSfromOpenBUGS 202
12.2.2 UsingBUGSfromR 204
12.3 MCMCUsingStan 206
12.4 Sim,BUGS,andStan 211
FurtherReading 212
13. Modeling Spatial Data Using GLMM 213
13.1 Background 213
13.2 Modeling Assumptions 214
13.3 Explicit ModelingofSpatialAutocorrelation 214
13.3.1 StartingtheModelFitting 214
13.3.2 VariogramModeling 216
13.3.3 BayesianModeling 216
13.3.4 OpenBUGSExample 222
FurtherReading 224
14. Advanced Ecological Models 225
14.1 HierarchicalMultinomialModeltoAnalyzeHabitat
SelectionUsingBUGS 225
14.2 Zero-InflatedPoisson MixedModelforAnalyzing
Breeding SuccessUsingStan 231
Contents ix
14.3 OccupancyModeltoMeasureSpeciesDistribution
UsingStan 240
14.4 TerritoryOccupancyModeltoEstimateSurvival
UsingBUGS 246
14.5 AnalyzingSurvivalBasedonMark-RecaptureData
UsingStan 252
FurtherReading 263
15. Prior Influence and Parameter Estimability 265
15.1 HowtoSpecifyPriorDistributions 265
15.2 PriorSensitivityAnalysis 269
15.3 ParameterEstimability 273
FurtherReading 278
16. Checklist 279
16.1 DataAnalysisStepbyStep 279
FurtherReading 287
17. What Should I Report in a Paper 289
17.1 HowtoPresent theResults 289
17.2 HowtoWriteUptheStatisticalMethods 293
FurtherReading 296
References 297
Index 309