Negative Binomial Regression

573 Pages·2011·3.319 MB·English

by  Joseph M. Hilbe

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NegativeBinomialRegression
SecondEdition
ThissecondeditionofNegativeBinomialRegressionprovidesacomprehensive
discussionofcountmodelsandtheproblemofoverdispersion,focusingattentionon
themanyvarietiesofnegativebinomalregression.Asubstantialenhancementfromthe
firstedition,thetextprovidesthetheoreticalbackgroundaswellasfullyworkedout
examplesusingStataandRformosteverymodelhavingcommercialandRsoftware
support.ExamplesusingSASandLIMDEParegivenaswell.Thisneweditionisan
idealhandbookforanyresearcherneedingadviceontheselection,construction,
interpretation,andcomparativeevaluationofcountmodelsingeneral,andofnegative
binomialmodelsinparticular.
Followinganoverviewofthenatureofriskandriskratioandthenatureofthe
estimatingalgorithmsusedinthemodelingofcountdata,thebookprovidesan
exhaustiveanalysisofthebasicPoissonmodel,followedbyathoroughanalysisofthe
meaningsandscopeofoverdispersion.SimulationsandrealdatausingbothStataand
Rareprovidedthroughoutthetextinordertoclarifytheessentialsofthemodelsbeing
discussed.Thenegativebinomialdistributionanditsvariousparameterizationsand
modelsarethenexaminedwiththeaimofexplaininghoweachtypeofmodel
addressesextra-dispersion.Newtothiseditionarechaptersondealingwithendogeny
andlatentclassmodels,finitemixtureandquantilecountmodels,andafullchapter
onBayesiannegativebinomialmodels.Thisneweditionisclearlythemost
comprehensiveappliedtextoncountmodelsavailable.
JOSEPH M. HILBE isaSolarSystemAmbassadorwithNASA’sJetPropulsion
LaboratoryattheCaliforniaInstituteofTechnology,anAdjunctProfessorofstatistics
atArizonaStateUniversity,andanEmeritusProfessorattheUniversityofHawaii.
ProfessorHilbeisanelectedFellowoftheAmericanStatisticalAssociationand
electedMemberoftheInternationalStatisticalInstitute(ISI),forwhichheisthe
foundingChairoftheISIastrostatisticscommitteeandNetwork.Heistheauthorof
LogisticRegressionModels,aleadingtextonthesubject,co-authorofRforStata
Users(withR.Muenchen),andofbothGeneralizedEstimatingEquationsand
GeneralizedLinearModelsandExtensions(withJ.Hardin).

Negative Binomial Regression
Second Edition
JOSEPH M. HILBE
JetPropulsionLaboratory,
CaliforniaInstituteofTechnologyand
ArizonaStateUniversity
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,Tokyo,MexicoCity
CambridgeUniversityPress
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PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521198158
(cid:1)C J.M.Hilbe2007,2011
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2007
Reprintedwithcorrections2008
Secondedition2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Hilbe,Joseph.
Negativebinomialregression/JosephM.Hilbe.–2nded.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-521-19815-8(hardback)
1.Negativebinomialdistribution. 2.Poissonalgebras. I.Title.
QA161.B5H55 2011
519.2(cid:2)4–dc22 2010051121
ISBN978-0521-19815-8Hardback
Additionalresourcesforthispublicationatwww.statistics.com/hilbe/nbr.php
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
Contents
Prefacetothesecondedition pagexi
1 Introduction 1
1.1 Whatisanegativebinomialmodel? 1
1.2 Abriefhistoryofthenegativebinomial 5
1.3 Overviewofthebook 11
2 Theconceptofrisk 15
2.1 Riskand2×2tables 15
2.2 Riskand2×ktables 18
2.3 Riskratioconfidenceintervals 20
2.4 Riskdifference 24
2.5 Therelationshipofrisktooddsratios 25
2.6 Marginalprobabilities:jointandconditional 27
3 Overviewofcountresponsemodels 30
3.1 Varietiesofcountresponsemodel 30
3.2 Estimation 38
3.3 Fitconsiderations 41
4 Methodsofestimation 43
4.1 DerivationoftheIRLSalgorithm 43
4.1.1 Solvingfor∂LorU–thegradient 48
4.1.2 Solvingfor∂2L 49
4.1.3 TheIRLSfittingalgorithm 51
4.2 Newton–Raphsonalgorithms 53
4.2.1 DerivationoftheNewton–Raphson 54
4.2.2 GLMwithOIM 57
v
vi Contents
4.2.3 Parameterizingfromµtox(cid:2)β 57
4.2.4 Maximumlikelihoodestimators 59
5 Assessmentofcountmodels 61
5.1 Residualsforcountresponsemodels 61
5.2 Modelfittests 64
5.2.1 Traditionalfittests 64
5.2.2 Informationcriteriafittests 68
5.3 Validationmodels 75
6 Poissonregression 77
6.1 DerivationofthePoissonmodel 77
6.1.1 DerivationofthePoissonfromthebinomial
distribution 78
6.1.2 DerivationofthePoissonmodel 79
6.2 SyntheticPoissonmodels 85
6.2.1 Constructionofsyntheticmodels 85
6.2.2 Changingresponseandpredictorvalues 94
6.2.3 Changingmultivariablepredictorvalues 97
6.3 Example:Poissonmodel 100
6.3.1 Coefficientparameterization 100
6.3.2 Incidencerateratioparameterization 109
6.4 Predictedcounts 116
6.5 Effectsplots 122
6.6 Marginaleffects,elasticities,anddiscretechange 125
6.6.1 MarginaleffectsforPoissonandnegativebinomial
effectsmodels 125
6.6.2 DiscretechangeforPoissonandnegative
binomialmodels 131
6.7 Parameterizationasaratemodel 134
6.7.1 Exposureintimeandarea 134
6.7.2 SyntheticPoissonwithoffset 136
6.7.3 Example 138
7 Overdispersion 141
7.1 Whatisoverdispersion? 141
7.2 Handlingapparentoverdispersion 142
7.2.1 CreationofasimulatedbasePoissonmodel 142
7.2.2 Deleteapredictor 145
7.2.3 Outliersindata 145
7.2.4 Creationofinteraction 149
Contents vii
7.2.5 Testingthepredictorscale 150
7.2.6 Testingthelink 152
7.3 Methodsofhandlingrealoverdispersion 157
7.3.1 Scalingofstandarderrors/quasi-Poisson 158
7.3.2 Quasi-likelihoodvariancemultipliers 163
7.3.3 Robustvarianceestimators 168
7.3.4 Bootstrappedandjackknifedstandarderrors 171
7.4 Testsofoverdispersion 174
7.4.1 ScoreandLagrangemultipliertests 175
7.4.2 Boundarylikelihoodratiotest 177
7.4.3 R2 andR2 testsforPoissonandnegative
p pd
binomialmodels 179
7.5 Negativebinomialoverdispersion 180
8 Negativebinomialregression 185
8.1 Varietiesofnegativebinomial 185
8.2 Derivationofthenegativebinomial 187
8.2.1 Poisson–gammamixturemodel 188
8.2.2 DerivationoftheGLMnegativebinomial 193
8.3 Negativebinomialdistributions 199
8.4 Negativebinomialalgorithms 207
8.4.1 NB-C:canonicalnegativebinomial 208
8.4.2 NB2:expectedinformationmatrix 210
8.4.3 NB2:observedinformationmatrix 215
8.4.4 NB2:Rmaximumlikelihoodfunction 218
9 Negativebinomialregression:modeling 221
9.1 Poissonversusnegativebinomial 221
9.2 Syntheticnegativebinomial 225
9.3 Marginaleffectsanddiscretechange 236
9.4 Binomialversuscountmodels 239
9.5 Examples:negativebinomialregression 248
Example1:Modelingnumberofmaritalaffairs 248
Example2:Heartprocedures 259
Example3:Titanicsurvivaldata 263
Example4:Healthreformdata 269
10 Alternativevarianceparameterizations 284
10.1 Geometricregression:NBα=1 285
10.1.1 Derivationofthegeometric 285
10.1.2 Syntheticgeometricmodels 286
viii Contents
10.1.3 Usingthegeometricmodel 290
10.1.4 Thecanonicalgeometricmodel 294
10.2 NB1:Thelinearnegativebinomialmodel 298
10.2.1 NB1asQL-Poisson 298
10.2.2 DerivationofNB1 301
10.2.3 ModelingwithNB1 304
10.2.4 NB1:Rmaximumlikelihoodfunction 306
10.3 NB-C:Canonicalnegativebinomialregression 308
10.3.1 NB-Coverviewandformulae 308
10.3.2 SyntheticNB-Cmodels 311
10.3.3 NB-Cmodels 315
10.4 NB-H:Heterogeneousnegativebinomialregression 319
10.5 TheNB-Pmodel:generalizednegativebinomial 323
10.6 GeneralizedWaringregression 328
10.7 Bivariatenegativebinomial 333
10.8 GeneralizedPoissonregression 337
10.9 PoissoninverseGaussianregression(PIG) 341
10.10 Othercountmodels 343
11 Problemswithzerocounts 346
11.1 Zero-truncatedcountmodels 346
11.2 Hurdlemodels 354
11.2.1 Theoryandformulaeforhurdlemodels 356
11.2.2 Synthetichurdlemodels 357
11.2.3 Applications 359
11.2.4 Marginaleffects 369
11.3 Zero-inflatednegativebinomialmodels 370
11.3.1 OverviewofZIP/ZINBmodels 370
11.3.2 ZINBalgorithms 371
11.3.3 Applications 374
11.3.4 Zero-alterednegativebinomial 376
11.3.5 Testsofcomparativefit 377
11.3.6 ZINBmarginaleffects 379
11.4 Comparisonofmodels 382
12 Censoredandtruncatedcountmodels 387
12.1 Censoredandtruncatedmodels–econometric
parameterization 387
12.1.1 Truncation 388
12.1.2 Censoredmodels 395