Categorical Data Analysis by Example

215 Pages·2017·5.365 MB·English

by  Graham J. G. Upton

CATEGORICAL
DATA ANALYSIS
BY EXAMPLE
GRAHAM J. G. UPTON
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CATEGORICAL DATA
ANALYSIS BY EXAMPLE
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CATEGORICAL DATA
ANALYSIS BY EXAMPLE
GRAHAM J. G. UPTON
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LibraryofCongressCataloging-in-PublicationData
Names:Upton,GrahamJ.G.,author.
Title:Categoricaldataanalysisbyexample/GrahamJ.G.Upton.
Description:Hoboken,NewJersey:JohnWiley&Sons,2016.|Includesindex.
Identifiers:LCCN2016031847(print)|LCCN2016045176(ebook)|ISBN9781119307860(cloth)|
ISBN9781119307914(pdf)|ISBN9781119307938(epub)
Subjects:LCSH:Multivariateanalysis.|Log-linearmodels.
Classification:LCCQA278.U682016(print)|LCCQA278(ebook)|DDC519.5/35–dc23
LCrecordavailableathttps://lccn.loc.gov/2016031847
PrintedintheUnitedStatesofAmerica
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CONTENTS
PREFACE XI
ACKNOWLEDGMENTS XIII
1 INTRODUCTION 1
1.1 WhatareCategoricalData? 1
1.2 ATypicalDataSet 2
1.3 VisualizationandCross-Tabulation 3
1.4 Samples,Populations,andRandomVariation 4
1.5 Proportion,Probability,andConditionalProbability 5
1.6 ProbabilityDistributions 6
1.6.1 TheBinomialDistribution 6
1.6.2 TheMultinomialDistribution 7
1.6.3 ThePoissonDistribution 7
1.6.4 TheNormalDistribution 7
1.6.5 TheChi-Squared(𝜒2)Distribution 8
1.7 *TheLikelihood 9
2 ESTIMATION AND INFERENCE FOR CATEGORICAL
DATA 11
2.1 GoodnessofFit 11
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vi CONTENTS
2.1.1 Pearson’sX2 Goodness-of-FitStatistic 11
2.1.2 *TheLinkBetweenX2 andthePoissonand
𝜒2-Distributions 12
2.1.3 TheLikelihood-RatioGoodness-of-FitStatistic,
G2 13
2.1.4 *WhytheG2 andX2 StatisticsUsuallyHaveSimilar
Values 14
2.2 HypothesisTestsforaBinomialProportion(Large
Sample) 14
2.2.1 TheNormalScoreTest 15
2.2.2 *LinktoPearson’sX2 Goodness-of-FitTest 15
2.2.3 G2 foraBinomialProportion 15
2.3 HypothesisTestsforABinomialProportion(Small
Sample) 16
2.3.1 One-TailedHypothesisTest 16
2.3.2 Two-TailedHypothesisTests 18
2.4 IntervalEstimatesforABinomialProportion 18
2.4.1 Laplace’sMethod 19
2.4.2 Wilson’sMethod 19
2.4.3 TheAgresti–CoullMethod 20
2.4.4 SmallSamplesandExactCalculations 20
References 22
3 THE 2 × 2 CONTINGENCY TABLE 25
3.1 Introduction 25
3.2 Fisher’sExactTest(ForIndependence) 27
3.2.1 *DerivationoftheExactTestFormula 28
3.3 TestingIndependencewithLargeCellFrequencies 29
3.3.1 UsingPearson’sGoodness-of-FitTest 30
3.3.2 TheYatesCorrection 30
3.4 The2×2TableinaMedicalContext 32
3.5 MeasuringLackofIndependence(Comparing
Proportions) 34
3.5.1 DifferenceofProportions 35
3.5.2 RelativeRisk 36
3.5.3 Odds-Ratio 37
References 40
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CONTENTS vii
4 THE I × J CONTINGENCY TABLE 41
4.1 Notation 41
4.2 IndependenceintheI×J ContingencyTable 42
4.2.1 EstimationandDegreesofFreedom 42
4.2.2 Odds-RatiosandIndependence 43
4.2.3 GoodnessofFitandLackofFitofthe
IndependenceModel 43
4.3 Partitioning 46
4.3.1 *AdditivityofG2 46
4.3.2 RulesforPartitioning 49
4.4 GraphicalDisplays 49
4.4.1 MosaicPlots 49
4.4.2 CobwebDiagrams 50
4.5 TestingIndependencewithOrdinalVariables 52
References 54
5 THE EXPONENTIAL FAMILY 55
5.1 Introduction 55
5.2 TheExponentialFamily 56
5.2.1 TheExponentialDispersionFamily 57
5.3 ComponentsofaGeneralLinearModel 57
5.4 Estimation 58
References 59
6 A MODEL TAXONOMY 61
6.1 UnderlyingQuestions 61
6.1.1 WhichVariablesareofInterest? 61
6.1.2 WhatCategoriesShouldbeUsed? 61
6.1.3 WhatistheTypeofEachVariable? 62
6.1.4 WhatistheNatureofEachVariable? 62
6.2 IdentifyingtheTypeofModel 63
7 THE 2× J CONTINGENCY TABLE 65
7.1 AProblemwithX2 (andG2) 65
7.2 UsingtheLogit 66
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viii CONTENTS
7.2.1 EstimationoftheLogit 67
7.2.2 TheNullModel 68
7.3 IndividualDataandGroupedData 69
7.4 Precision,ConfidenceIntervals,andPredictionIntervals 73
7.4.1 PredictionIntervals 74
7.5 LogisticRegressionwithaCategoricalExplanatory
Variable 76
7.5.1 ParameterEstimateswithCategoricalVariables
(J > 2) 78
7.5.2 TheDummyVariableRepresentationofa
CategoricalVariable 79
References 80
8 LOGISTIC REGRESSION WITH SEVERAL
EXPLANATORY VARIABLES 81
8.1 DegreesofFreedomwhentherearenoInteractions 81
8.2 GettingaFeelfortheData 83
8.3 Modelswithtwo-VariableInteractions 85
8.3.1 LinktotheTestingofIndependencebetweenTwo
Variables 87
9 MODEL SELECTION AND DIAGNOSTICS 89
9.1 Introduction 89
9.1.1 Ockham’sRazor 90
9.2 NotationforInteractionsandforModels 91
9.3 StepwiseMethodsforModelSelectionUsingG2 92
9.3.1 ForwardSelection 94
9.3.2 BackwardElimination 96
9.3.3 CompleteStepwise 98
9.4 AICandRelatedMeasures 98
9.5 TheProblemCausedbyRareCombinationsofEvents 100
9.5.1 TacklingtheProblem 101
9.6 SimplicityversusAccuracy 103
9.7 DFBETAS 105
References 107