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Negative binomial regression PDF

263 Pages·2007·1.986 MB·English
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NegativeBinomialRegression Atlast–abookentirelydevotedtothenegativebinomialmodelanditsmany variations.Everymodelcurrentlyofferedinacommercialstatisticalsoftwarepackage isdiscussedindetail–howeachisderived,howeachresolvesadistributional problem,andnumerousexamplesoftheirapplication.Manyofthesemodelshave neverbeforebeenthoroughlyexaminedinatextoncountresponsemodels:the canonicalnegativebinomial;theNB-Pmodel,wherethenegativebinomialexponentis itselfparameterized;andnegativebinomialmixedmodels.Writtenforpracticing researchersandstatisticianswhoneedtoupdatetheirknowledgeofPoissonand negativebinomialmodels,thebookprovidesacomprehensiveoverviewofestimating methodsandalgorithmsusedtomodelcounts,aswellasspecificmodelingguidelines, modelselectiontechniques,methodsofinterpretation,andassessmentofmodel goodnessoffit.Datasetsandmodelingcodeareprovidedonacompanionwebsite. joseph m. hilbe isanEmeritusProfessorattheUniversityofHawaiiandAdjunct ProfessorofStatisticsintheSchoolofSocialandFamilyDynamicsatArizonaState University.HehasservedasboththeSoftwareReviewsEditorandoverallAssociate EditorforTheAmericanStatisticiansince1997andiscurrentlyontheeditorialboards offiveacademicjournalsinstatistics.AnelectedFellowofboththeAmerican StatisticalAssociationandRoyalStatisticalSociety,Hilbeisauthor,withJames Hardin,ofGeneralizedEstimatingEquationsandtwoeditionsofGeneralizedLinear ModelsandExtensions. Negative Binomial Regression JOSEPH M. HILBE ArizonaStateUniversity cambridge university press Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,Sa˜oPaulo CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9780521857727 (cid:1)C J.M.Hilbe2007 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithout thewrittenpermissionofCambridgeUniversityPress. Firstpublished2007 PrintedintheUnitedKingdomattheUniversityPress,Cambridge AcatalogrecordforthispublicationisavailablefromtheBritishLibrary ISBN978-0-521-85772-7hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto inthispublication,anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. Contents Preface pageix Introduction 1 1 Overviewofcountresponsemodels 8 1.1 Varietiesofcountresponsemodel 8 1.2 Estimation 13 1.3 Fitconsiderations 15 1.4 Briefhistoryofthenegativebinomial 15 1.5 Summary 17 2 Methodsofestimation 19 2.1 DerivationoftheIRLSalgorithm 19 2.2 Newton–Raphsonalgorithms 28 2.3 Theexponentialfamily 33 2.4 Residualsforcountresponsemodels 34 2.5 Summary 36 3 Poissonregression 39 3.1 DerivationofthePoissonmodel 39 3.2 Parameterizationasaratemodel 45 3.3 Testingoverdispersion 46 3.4 Summary 48 4 Overdispersion 51 4.1 Whatisoverdispersion? 51 4.2 Handlingapparentoverdispersion 52 4.3 Methodsofhandlingrealoverdispersion 61 4.4 Summary 74 v vi Contents 5 Negativebinomialregression 77 5.1 Varietiesofnegativebinomial 77 5.2 Derivationofthenegativebinomial 79 5.3 Negativebinomialdistributions 84 5.4 Algorithms 90 5.5 Summary 96 6 Negativebinomialregression:modeling 99 6.1 Poissonversusnegativebinomial 99 6.2 Binomialversuscountmodels 103 6.3 Examples:negativebinomialregression 107 6.4 Summary 131 7 Alternativevarianceparameterizations 136 7.1 Geometricregression 137 7.2 NB-1:Thelinearconstantmodel 143 7.3 NB-H:Heterogeneousnegativebinomialregression 149 7.4 TheNB-Pmodel 151 7.5 GeneralizedPoissonregression 154 7.6 Summary 158 8 Problemswithzerocounts 160 8.1 Zero-truncatednegativebinomial 160 8.2 Negativebinomialwithendogenousstratification 164 8.3 Hurdlemodels 167 8.4 Zero-inflatedcountmodels 173 8.5 Summary 177 9 Negativebinomialwithcensoring,truncation,andsample selection 179 9.1 Censoredandtruncatedmodels–econometric parameterization 179 9.2 CensoredpoissonandNB-2models–survival parameterization 186 9.3 Sampleselectionmodels 191 9.4 Summary 195 10 Negativebinomialpanelmodels 198 10.1 Unconditionalfixed-effectsnegativebinomialmodel 199 10.2 Conditionalfixed-effectsnegativebinomialmodel 203 10.3 Random-effectsnegativebinomial 208 10.4 Generalizedestimatingequation 215 Contents vii 10.5 Multilevelnegativebinomialmodels 226 10.6 Summary 230 AppendixA:Negativebinomiallog-likelihoodfunctions 233 AppendixB:Deviancefunctions 236 AppendixC:Statanegativebinomial–MLalgorithm 237 AppendixD:Negativebinomialvariancefunctions 239 AppendixE:Datasets 240 References 242 Authorindex 247 Subjectindex 249 Preface This is the first text devoted specifically to the negative binomial regression model. Important to researchers desiring to model count response data, the procedurehasonlyrecentlybeenaddedtothecapabilitiesofleadingcommercial statisticalsoftware.However,itisnowoneofthemostcommonmethodsused by statisticians to accommodate extra correlation – or overdispersion – when modeling counts. Since most real count data modeling situations appear to involve overdispersion, the negative binomial has been finding increased use amongstatisticians,econometricians,andresearcherswhocommonlyanalyze countresponsedata. Thisvolumewillexploreboththetheoryandvarietiesofthenegativebino- mial.Itwillalsoprovidethereaderwithexamplesusingeachtypeofmajorvari- ationithasundergone.However,ofprimeimportance,thetextwillalsoattempt toclarifydiscrepanciesregardingthenegativebinomialthatoftenappearinthe statisticalliterature.Whatexactlyisanegativebinomialmodel?Howdoesit relatetoothermodels?Howisitsvariancefunctiontobedefined?Isitamem- ber of the family of generalized linear models? What is the most appropriate mannerbywhichtoestimateparameters?Howareparameterstobeinterpreted, andevaluatedastotheirworth?Whatarethelimitsofitsapplicability?How hasitbeenextendedtoformmorecomplexmodels?Theseareimportantques- tions that have at times found differing answers depending on the author. By examininghowthenegativebinomialmodelarisesfromthenegativebinomial probabilitymassfunction,andbyconsideringhowmajorestimatingmethods relate to the estimation of its parameters, we should be able to clearly define eachvarietyofnegativebinomialaswellasthelogicunderlyingtherespective extensions. Thegoalofthistextistoserveasahandbookofnegativebinomialregression models, providing the reader with guidelines of how to best implement the modelintotheirresearch.Althoughweshallprovidethemathematicsofhow ix x Preface thevarietiesofnegativebinomialmodelarederived,theemphasiswillbeon clarityandapplication.Thetexthasbeenwrittentobeunderstandabletoanyone having a general background in maximum likelihood theory and generalized linearmodels.Togainfullbenefitofthetheoreticalaspectsofthediscussion, thereadershouldalsohaveaworkingknowledgeofelementarycalculus. TheStatastatisticalpackage(http://www.stata.com)isusedthroughoutthe texttodisplayexamplemodeloutput.Althoughmanyofthestatisticalmodels discussedinthetextareofferedasastandardpartofthecommercialpackage,I havewrittenanumberofmoreadvancednegativebinomialmodelsusingStata’s proprietaryhigherprogramminglanguage.Theseprograms,calledadofilesby Stata, display results that appear identical to official Stata procedures. Some 25oftheseStataprogramshavebeenpostedtotheBostonCollegeSchoolof EconomicsSSCarchive,accessedat:http://ideas.repec.org/s/boc/bocode.html. Programsareorderedbyyear,withthemostrecentpostedatthebottomofthe respectiveyearofsubmission.Moststatisticalprocedureswrittenforthistext canbefoundinthe2004files. LIMDEP software (http://www.limdep.com) is used to display output for examples related to negative binomial mixed models, the NB-P model, neg- ative binomial selection models, and certain types of truncated and censored models.TheseprogramsweredevelopedbyProf.WilliamGreeneofNewYork University,authoroftheLIMDEPpackage.StataandLIMDEPstatisticalsoft- warecontainmoreproceduresrelatedtonegativebinomialregressionthanall other packges combined. I recommend that either of these two packages be obtainedifthereaderintendstoduplicatetextexamplesattheirsite.Abasic NB-2modelinRisprovidedaspartoftheMASSpackage,basedinVenables andRipley(2002).NegativebinomialmodelsinRarelimitedasofthiswriting, butmoreadvancedmodelsaresuretofollowinthenearfuture. All data sets and Stata ado files related to models used in the text can be downloadedfrom:www.cambridge.org/XXXXX.Eachadofilewillhaveadate oforiginassociatedwithit.Occasionallyupdatesoradditionswillbemadeto thissite;itisrecommendedthatyoucheckitfromtimetotime,updatingtothe mostrecentiterationoftheprocedureofinterest.Ialsointendtopostadditional materialsrelatedtonegativebinomialmodelingatthissite. Ishallusethefollowingcitationandreferenceconventionsthroughoutthe text. Program commands, variable and data set names, as well as statistical output,arealldisplayedinCourierNewtypewriterfont.Datasetsandcommand namesareinbold,e.g.medpar, glm.Ishallfollowstandardconventionswith respecttomathematicalexpressions. Thismonographisbasedonseminarsandclassesrelatedtocountresponse modelsthatIhavetaughtoverthepast20years.Inparticular,thepresentationof

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