多水平模型(英文原著) chap7

《多水平模型(英文原著) chap7》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1、Chapter7Discreteresponsedata7.1ModelsfordiscreteresponsedataAllthemodelsofpreviouschaptershaveassumedthattheresponsevariableiscontinuouslydistributed.Wenowlookatdatawheretheresponseisessentiallyacountofevents.Thiscountmaybethenumberoftimesaneventoccursoutofafixednumberof‘trials’i

2、nwhichcaseweusuallydealwiththeresultingproportionasresponse:anexampleistheproportionofdeathsinapopulation,classifiedbyage.Wemayhaveavectorofcountsrepresentingthenumbersofeventsofdifferentkindswhichoccuroutofatotalnumberofevents:anexampleisgiveninchapter3wherewestudiedthenumberofr

3、esponsestoeach,ordered,categoryofaquestiononabortionattitudes.Statisticalmodelsforsuchdataarereferredtoas‘generalisedlinearmodels’(McCullaghandNelder,1989).A2-levelmodelcanbewritteninthegeneralform(7.1)whereistheexpectedvalueoftheresponsefortheij-thlevel1unitandfisanonlinearfunct

4、ionofthe‘linearpredictor’.Notethatweallowrandomcoefficientsatlevel2.Themodeliscompletedbyspecifyingadistributionfortheobservedresponse.WheretheresponseisaproportionthisistypicallytakentobebinomialandwheretheresponseisacounttakentobePoisson.Equation(7.1)isaspecialcaseofthenonlinea

5、rmodelstudiedinchapter5andweshallbeusingtheresultsgiventhere.Itremainsforustospecifythenonlinear‘link’functionf.Table7.1listssomeofthestandardchoices,withlogarithmschosentobasee.Inadditiontothesewecanalsohavethe‘identity’function,butthiscancreatedifficultiessinceitallows,inprinci

6、ple,predictedcountsorproportionswhicharerespectivelylessthanzerooroutsidetherange(0,1).Nevertheless,inmanycases,usingtheidentityfunctionproducesacceptableresultswhichmaydifferlittlefromthoseobtainedwiththenonlinearfunctions.Inthefollowingsectionsweconsidereachcommontypeofmodelint

7、urnwithexamples.Table7.1Somenonlinearlinkfunctions.ResponseNameProportionlogitProportioncomplementaryloglogVectorofproportionsmultivariatelogitCountlog147.2ProportionsasresponsesConsiderthe2-levelvariancecomponentsmodelwithasingleexplanatoryvariablewheretheexpectedproportionismod

8、elledusingalogitlinkfunction(7.2)Theobse