1Introduction1
1.1ObjectivesoftheBook ...................1
1.2BackgroundMaterial ....................5
1.2.1InferencefromData,GivenaModel ........5
1.2.2LikelihoodandLeastSquaresTheory .......6
1.2.3TheCriticalIssue:“WhatIstheBestModel
toUse?” .......................13
1.2.4ScienceInputs:FormulationoftheSetof
CandidateModels ..................15
1.2.5ModelsVersusFullReality .............20
1.2.6AnIdealApproximatingModel ...........22
1.3ModelFundamentalsandNotation .............23
1.3.1TruthorFullReality f ...............23
1.3.2ApproximatingModels gi (x|θ)...........23
1.3.3TheKullback–LeiblerBestModel gi (x|θ0).....25
1.3.4EstimatedModels gi (x| ˆ θ)..............25
1.3.5GeneratingModels .................26
1.3.6GlobalModel ....................26
1.3.7OverviewofStochasticModelsinthe
BiologicalSciences .................27
1.4InferenceandthePrincipleofParsimony ..........29
1.4.1AvoidOverfittingtoAchieveaGoodModelFit..29
1.4.2ThePrincipleofParsimony .............31
1.4.3ModelSelectionMethods ..............35
1.5DataDredging,OveranalysisofData,and
SpuriousEffects .......................37
1.5.1OveranalysisofData ................38
1.5.2SomeTrends ....................40
1.6ModelSelectionBias ....................43
1.7ModelSelectionUncertainty ................45
1.8Summary ...........................47
2InformationandLikelihoodTheory:ABasisforModel
SelectionandInference49
2.1Kullback–LeiblerInformationorDistanceBetween
TwoModels .........................50
2.1.1ExamplesofKullback–LeiblerDistance ......54
2.1.2Truth, f ,DropsOutasaConstant .........58
2.2Akaike’sInformationCriterion:1973 ............60
2.3Takeuchi’sInformationCriterion:1976 ...........65
2.4Second-OrderInformationCriterion:1978 .........66
2.5ModificationofInformationCriterionforOverdispersed
CountData ..........................67
2.6AICDifferences, i .....................70
2.7AUsefulAnalogy ......................72
2.8LikelihoodofaModel, L(gi |data).............74
2.9AkaikeWeights, wi .....................75
2.9.1BasicFormula ....................75
2.9.2AnExtension ....................76
2.10EvidenceRatios .......................77
2.11ImportantAnalysisDetails .................80
2.11.1AICCannotBeUsedtoCompareModelsof
DifferentDataSets .................80
2.11.2OrderNotImportantinComputingAICValues..81
2.11.3TransformationsoftheResponseVariable .....81
2.11.4RegressionModelswithDiffering
ErrorStructures ...................82
2.11.5DoNotMixNullHypothesisTestingwith
Information-TheoreticCriteria ...........83
2.11.6NullHypothesisTestingIsStillImportantin
StrictExperiments ..................83
2.11.7Information-TheoreticCriteriaAreNota“Test”..84
2.11.8ExploratoryDataAnalysis .............84
2.12SomeHistoryandFurtherInsights .............85
2.12.1Entropy .......................86
2.12.2AHeuristicInterpretation ..............87
2.12.3MoreonInterpretingInformation-
TheoreticCriteria ..................87
2.12.4NonnestedModels .................88
2.12.5FurtherInsights ...................89
2.13BootstrapMethodsandModelSelectionFrequencies πi ..90
2.13.1Introduction .....................91
2.13.2TheBootstrapinModelSelection:
TheBasicIdea ....................93
2.14ReturntoFlather’sModels .................94
2.15Summary ...........................96
3BasicUseoftheInformation-TheoreticApproach98
3.1Introduction .........................98
3.2Example1:CementHardeningData ............100
3.2.1SetofCandidateModels ..............101
3.2.2SomeResultsandComparisons ...........102
3.2.3ASummary .....................106
3.3Example2:TimeDistributionofanInsecticideAddedtoa
SimulatedEcosystem ....................106
3.3.1SetofCandidateModels ..............108
3.3.2SomeResults ....................110
3.4Example3:NestlingStarlings ................111
3.4.1ExperimentalScenario ...............112
3.4.2MonteCarloData ..................113
3.4.3SetofCandidateModels ..............113
3.4.4DataAnalysisResults ................117
3.4.5FurtherInsightsintotheFirstFourteen
NestedModels ...................120
3.4.6HypothesisTestingandInformation-Theoretic
ApproachesHaveDifferent
SelectionFrequencies ................121
3.4.7FurtherInsightsFollowingFinal
ModelSelection ...................124
3.4.8WhyNotAlwaysUsetheGlobalModel
forInference? ....................125
3.5Example4:SageGrouseSurvival ..............126
3.5.1Introduction .....................126
3.5.2SetofCandidateModels ..............127
3.5.3ModelSelection ...................129
3.5.4HypothesisTestsforYear-Dependent
SurvivalProbabilities ................131
3.5.5HypothesisTestingVersusAICin
ModelSelection ...................132
3.5.6AClassofIntermediateModels ..........134
3.6Example5:ResourceUtilizationof Anolis Lizards .....137
3.6.1SetofCandidateModels ..............138
3.6.2CommentsonAnalyticMethod ...........138
3.6.3SomeTentativeResults ...............139
3.7Example6:Sakamotoetal.’s(1986)SimulatedData....141
3.8Example7:ModelsofFishGrowth .............142
3.9Summary ...........................143
4FormalInferenceFromMoreThanOneModel:
MultimodelInference(MMI)149
4.1IntroductiontoMultimodelInference ............149
4.2ModelAveraging ......................150
4.2.1Prediction ......................150
4.2.2AveragingAcrossModelParameters ........151
4.3ModelSelectionUncertainty ................153
4.3.1ConceptsofParameterEstimationand
ModelSelectionUncertainty ............155
4.3.2IncludingModelSelectionUncertaintyin
EstimatorSamplingVariance ............158
4.3.3UnconditionalConfidenceIntervals ........164
4.4EstimatingtheRelativeImportanceofVariables ......167
4.5ConfidenceSetfortheK-LBestModel ...........169
4.5.1Introduction .....................169
4.5.2 i ,ModelSelectionProbabilities,
andtheBootstrap ..................171
andtheBootstrap ..................171
4.6ModelRedundancy .....................173
4.7Recommendations ......................176
4.8CementData .........................177
4.9PineWoodData .......................183
4.10TheDurbanStormData ...................187
4.10.1ModelsConsidered .................188
4.10.2ConsiderationofModelFit .............190
4.10.3ConfidenceIntervalsonPredicted
StormProbability ..................191
4.10.4ComparisonsofEstimatorPrecision ........193
4.11FlourBeetleMortality:ALogisticRegressionExample..195
4.12PublicationofResearchResults ...............201
4.13Summary ...........................203
5MonteCarloInsightsandExtendedExamples206
5.1Introduction .........................206
5.2SurvivalModels .......................207Contentsxvii
5.2.1AChainBinomialSurvivalModel .........207
5.2.2AnExample .....................210
5.2.3AnExtendedSurvivalModel ............215
5.2.4ModelSelectionifSampleSizeIsHuge,
orTruthKnown ...................219
5.2.5AFurtherChainBinomialModel ..........221
5.3ExamplesandIdeasIllustratedwithLinearRegression...224
5.3.1All-SubsetsSelection:AGPAExample ......225
5.3.2AMonteCarloExtensionoftheGPAExample..229
5.3.3AnImprovedSetofGPAPredictionModels....235
5.3.4MoreMonteCarloResults .............238
5.3.5LinearRegressionandVariableSelection .....244
5.3.6Discussion ......................248
5.4EstimationofDensityfromLineTransectSampling....255
5.4.1DensityEstimationBackground ..........255
5.4.2LineTransectSamplingofKangaroosat
WallabyCreek ....................256
5.4.3AnalysisofWallabyCreekData ..........256
5.4.4BootstrapAnalysis .................258
5.4.5ConfidenceIntervalon D ..............258
5.4.6BootstrapSamples:1,000Versus10,000 ......260
5.4.7BootstrapVersusAkaikeWeights:ALesson
onQAICc ......................261
5.5Summary ...........................264