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Model Selection and Multimodel Inference SecondEdition Springer NewYork Berlin Heidelberg Barcelona HongKong London Milan Paris Singapore Tokyo Kenneth P. Burnham David R. Anderson Model Selection and Multimodel Inference A Practical Information-Theoretic Approach Second Edition With31Illustrations 1 3 KennethP.Burnham DavidR.Anderson ColoradoCooperativeFish andWildlifeResearchUnit ColoradoStateUniversity FortCollins,CO80523-1484 USA CoverIllustration:Thecoverwasassembledfromphotosoftheyellow-belliedtoad(Bombinavarie- gata)takenbyJonasBarandumaspartofhisPh.D.programattheUniversityofZurich.Thesetoads haveindividuallyidentifiablepatternsontheirabdomenfromafewweeksfollowingmetamorphosis thatremainunchangeduntildeath.Twopairsareduplicates—butwhichtwo? CoverphotographsbyDr.JonasBarandum,St.Gallen,Switzerland.CoverdesignbyKentonAllred. LibraryofCongressCataloging-in-PublicationData Burnham,KennethP. Modelselectionandmultimodelinference:apracticalinformation-theoreticapproach /KennethP.Burnham,DavidR.Anderson.—2nded. p. cm. Rev.ed.of:Modelselectionandinference.©1998. Includesbibliographicalreferences(p. ). ISBN0-387-95364-7(alk.paper) 1.Biology—Mathematicalmodels. 2.Mathematicalstatistics. I.Burnham,KennethP. Modelselectionandinference. II.Title. QH323.5 B87 2002 570(cid:1).1(cid:1)51—dc21 2001057677 ISBN0-387-95364-7 Printedonacid-freepaper. ©2002,1998Springer-VerlagNewYork,Inc. Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(Springer-VerlagNewYork,Inc.,175FifthAvenue,NewYork,NY10010, USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orby similarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. PrintedintheUnitedStatesofAmerica. 9 8 7 6 5 4 3 2 SPIN10853081 www.springer-ny.com Springer-Verlag NewYork Berlin Heidelberg AmemberofBertelsmannSpringerScience+BusinessMediaGmbH Tomymotherandfather,LucilleR.(deceased)andJ.CalvinBurnham (deceased),andmysonanddaughter,ShawnP.andSallyA.Burnham Tomyparents,CharlesR.(deceased)andLetaM.Anderson;mywife, DaleneF.Anderson;andmydaughters,TamaraE.and AdrienneM.Anderson This page intentionally left blank Preface We wrote this book to introduce graduate students and research workers in variousscientificdisciplinestotheuseofinformation-theoreticapproachesin theanalysisofempiricaldata.Thesemethodsallowthedata-basedselection of a “best” model and a ranking and weighting of the remaining models in a pre-defined set. Traditional statistical inference can then be based on this selected best model. However, we now emphasize that information-theoretic approachesallowformalinferencetobebasedonmorethanonemodel(mul- timodel inference). Such procedures lead to more robust inferences in many cases,andweadvocatetheseapproachesthroughoutthebook. The second edition was prepared with three goals in mind. First, we have triedtoimprovethepresentationofthematerial.Boxesnowhighlightessen- tialexpressionsandpoints.Somereorganizationhasbeendonetoimprovethe flow of concepts, and a new chapter has been added. Chapters 2 and 4 have been streamlined in view of the detailed theory provided in Chapter 7. Sec- ond,conceptsrelatedtomakingformalinferencesfrommorethanonemodel (multimodel inference) have been emphasized throughout the book, but par- ticularlyinChapters4,5,and6.Third,newtechnicalmaterialhasbeenadded to Chapters 5 and 6. Well over 100 new references to the technical literature are given. These changes result primarily from our experiences while giving severalseminars,workshops,andgraduatecoursesonmaterialinthefirstedi- tion.Inaddition,wehavedonesubstantiallymorethinkingabouttheissueand reading the literature since writing the first edition, and these activities have ledtofurtherinsights. InformationtheoryincludesthecelebratedKullback–Leibler“distance”be- tween two models (actually, probability distributions), and this represents a viii Preface fundamentalquantityinscience.In1973,HirotuguAkaikederivedanestima- torofthe(relative)expectationofKullback–LeiblerdistancebasedonFisher’s maximizedlog-likelihood.Hismeasure,nowcalledAkaike’sinformationcri- terion(AIC),providedanewparadigmformodelselectionintheanalysisof empiricaldata.Hisapproach,withafundamentallinktoinformationtheory, is relatively simple and easy to use in practice, but little taught in statistics classesandfarlessunderstoodintheappliedsciencesthanshouldbethecase. Wedonotacceptthenotionthatthereisasimple“truemodel”inthebiolog- icalsciences.Instead,weviewmodelingasanexerciseintheapproximation oftheexplainableinformationintheempiricaldata,inthecontextofthedata beingasamplefromsomewell-definedpopulationorprocess.Rexstad(2001) views modeling as a fabric in the tapestry of science. Selection of a best ap- proximating model represents the inference from the data and tells us what “effects”(representedbyparameters)canbesupportedbythedata.Wefocus on Akaike’s information criterion (and various extensions) for selection of a parsimoniousmodelasabasisforstatisticalinference.Modelselectionbased on information theory represents a quite different approach in the statistical sciences,andtheresultingselectedmodelmaydiffersubstantiallyfrommodel selectionbasedonsomeformofstatisticalnullhypothesistesting. Werecommendtheinformation-theoreticapproachfortheanalysisofdata fromobservationalstudies.Inthisbroadclassofstudies,wefindthatallthevar- ious hypothesis-testing approaches have no theoretical justification and may often perform poorly. For classic experiments (control–treatment, with ran- domization and replication) we generally support the traditional approaches (e.g.,analysisofvariance);thereisaverylargeliteratureonthisclassicsubject. However,forcomplexexperimentswesuggestconsiderationoffittingexplana- tory models, hence on estimation of the size and precision of the treatment effects and on parsimony, with far less emphasis on “tests” of null hypothe- ses,leadingtothearbitraryclassification“significant”versus“notsignificant.” Instead,astrengthofevidenceapproachisadvocated. Wedonotclaimthattheinformation-theoreticmethodsarealwaysthevery bestforaparticularsituation.Theydorepresentaunifiedandrigoroustheory, an extension of likelihood theory, an important application of information theory,andtheyareobjectiveandpracticaltoemployacrossaverywideclassof empiricalproblems.Inferencefrommultiplemodels,ortheselectionofasingle “best”model,bymethodsbasedontheKullback–Leiblerdistancearealmost certainlybetterthanothermethodscommonlyinusenow(e.g.,nullhypothesis testingofvarioussorts,theuseofR2,ormerelytheuseofjustoneavailable model).Inparticular,subjectivedatadredgingleadstooverfittedmodelsand theattendantproblemsininference,andistobestronglydiscouraged,atleast inmoreconfirmatorystudies. Parameter estimation has been viewed as an optimization problem for at leasteightdecades(e.g.,maximizethelog-likelihoodorminimizetheresidual sum of squared deviations). Akaike viewed his AIC and model selection as “... anaturalextensionoftheclassicalmaximumlikelihoodprinciple.”This Preface ix extension brings model selection and parameter estimation under a common framework—optimization.However,theparadigmdescribedinthisbookgoes beyondmerelythecomputationandinterpretationofAICtoselectaparsimo- niousmodelforinferencefromempiricaldata;itrefocusesincreasedattention onavarietyofconsiderationsandmodelingpriortotheactualanalysisofdata. Modelselection,undertheinformation-theoreticapproachpresentedhere,at- tempts to identify the (likely) best model, orders the models from best to worst,andproducesaweightofevidencethateachmodelisreallythebestas aninference. Severalmethodsaregiventhatallowmodelselectionuncertaintytobeincor- poratedintoestimatesofprecision(i.e.,multimodelinference).Ourintention istopresentandillustrateaconsistentmethodologythattreatsmodelformu- lation,modelselection,estimationofmodelparametersandtheiruncertainty inaunifiedmanner,underacompellingcommonframework.Wereviewand explain other information criteria (e.g., AIC , QAIC , and TIC) and present c c several examples to illustrate various technical issues, including some com- parisons with BIC, a type of dimension consistent criterion. In addition, we provide many references to the technical literature for those wishing to read furtheronthesetopics. This is an applied book written primarily for biologists and statisticians using models for making inferences from empirical data. This is primarily a sciencebook;wesayrelativelylittleaboutdecisionmakinginmanagementor managementscience.Researchbiologistsworkingeitherinthefieldorinthe laboratorywillfindsimplemethodsthatarelikelytobeusefulintheirinvesti- gations.Researchersinotherlifesciences,econometrics,thesocialsciences, and medicine might also find the material useful but will have to deal with examplesthathavebeentakenlargelyfromecologicalstudiesoffree-ranging vertebrates,astheseareourinterests.Appliedstatisticiansmightconsiderthe information-theoreticmethodspresentedherequiteusefulandasuperioralter- nativetothenullhypothesistestingapproachthathasbecomesotortuousand uninformative.Wehopematerialsuchasthiswillfinditswayintoclassrooms whereapplieddataanalysisandassociatedsciencephilosophyaretaught.This bookmightbeusefulasatextforacourseforstudentswithsubstantialexpe- rienceandeducationinstatisticsandapplieddataanalysis.Asecondprimary audience includes honors or graduate students in the biological, medical, or statisticalsciences.Thoseinterestedintheempiricalscienceswillfindthisma- terialusefulbecauseitoffersaneffectivealternativeto(1)thewidelytaught, yetoftenbothcomplexanduninformative,nullhypothesistestingapproaches and(2)thefarlesstaught,butpotentiallyveryuseful,Bayesianapproaches. Readersshouldideallyhavesomematurityinthequantitativesciencesand experienceindataanalysis.Severalcoursesincontemporarystatisticaltheory andmethodsaswellassomephilosophyofsciencewouldbeparticularlyuse- fulinunderstandingthematerial.Someexposuretolikelihoodtheoryisnearly essential,butthosewithexperienceonlyinleastsquaresregressionmodeling will gain some useful insights. Biologists working in a team situation with
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