Negative binomial regression

263 Pages·2007·1.986 MB·English

by  Joseph Hilbe

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
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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