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1、6Classi cationandRegressionTreesClassi cationandregressiontrees(CARTs)areanapproachtodiscoveringrelationshipsamongalargenumberofindependent(predictor)variablesandacategoricalorcontinuoustrait.Classi cationtreesareappliedtocategoricaloutcomes,whileregressiontreesapplytocontinuou
2、straits.Bothinvolvetheapplicationofarecursivealgorithmthataimstopartitionindividualsintogroupsinawaythatminimizesthewithin-groupheterogeneity.CARTwasoriginallydescribedbyBreimanetal.(1993)andhasgainedpopularityinrecentyearsasamethodforidentifyingstructureinhigh-dimensionaldatas
3、ettings.Inthefollowingsections,webeginbydescribingmethodsforcon-structingatree.Thisinvolvesde ningameasureofheterogeneity,orwhatiscommonlyreferredtoasnodeimpurity,aswellasdetermininghowpredictorvariablesareinputintothemodel.Bothofthesecomponentswillimpacttheresultingtreeandneed
4、tobeconsideredandde nedcarefullytore
ectthescienti cquestionsathand.Wethendescribemethodsforre ningthistreetoarriveata nalreproduciblemodel.FurtherdiscussionsofCARTmethodscanbefoundinBreimanetal.(1993)andZhangandSinger(1999).InChap-ter7,wedescribeextensionsoftheCARTmodel,includ
5、ingrandomforestsandlogicregressiontreesthato ersomeadditionaladvantages.6.1Buildingatree6.1.1RecursivepartitioningSupposeourdataconsistofatraity=(y1;:::;yn)andppotentialpredictorvariablesgivenbyx;:::;x,wherex=(x;:::;x)Tandi=1;:::;n1pj1jnjindicatesindividual.Ingeneral,interestli
6、esindiscoveringtherelationshipbetweenX=[x1;:::;xp]andy.Atreeisconstructedby rstdeterminingthevariablexj,amongthesetofallpotentialpredictors,thatismostpredic-tiveofthetraity.Thephrasemostpredictive"isusedlooselyhereandisde nedexplicitlyinSection6.1.2below.Individualsinoursample
7、arethenA.S.Foulkes,AppliedStatisticalGeneticswithR:ForPopulation-basedAssociation157Studies,UseR,DOI:10.1007/978-0-387-89554-36,cSpringerScience+BusinessMediaLLC20091586Classi cationandRegressionTreesdividedintotwogroupsbasedontheircorrespondingvalueofxj.Suppose,forexample,that
8、xjisabinaryindicatorforthepresenceofthevariantalleleat
