class11 Logistic Regression

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1、BS805Class11LogisticRegressionSituationRegressionappropriate,butthedependentvariableisdichotomous.Independentvariablescanbecontinuous,categorical(forexample,intheformofdummyvariables),orordinal.Goals:(sameaslinearregression)a)Predictionofthedependentvariable(here,predi

2、ctriskofanevent)b)Associationbetweenthedependentandindependentvariablesc)ControlforconfoundersandcovariatesThelogisticmodel(simplelogisticregression)Basedontheodds,px/(1-px),andtheoddsratioploglnx=+Xnatural(odds)=ββ011-pxor,ifsamplingappropriate,β+βXe01p=xβ+βX1

3、+e011BS805Class11PropertiesofLogisticRegression•ForallvaluesofX,0

4、Xisnotconstant,andY

5、Xisnotnormallydistributed•Themodelisfitthroughmaximumlikelihood.Testingisdonethroughlikelihoodratiotestsanduseofchi-squarestatisticsratherthanFortstatistics.•Themod

6、elcomparesrisksoftheoutcomethroughtheoddsratio:p/(1-p)x1x1OR(xvsx)=12p/(1-p)x2x2Giventhemodel,β1(x1-x2)OR(x1vsx2)=e•Logisticregressionisappropriateforcase-controlstudiesaswellascross-sectionalstudies,cohortstudies,andclinicaltrials.•Estimatesandtestsareappropriateforbo

7、thretrospectiveandprospectivestudies.2BS805Class11Interpretationoflogisticregressionoutput•Testoftheoverallmodel:LikelihoodratiotestofH:β=β=...=β=0012k•Partialchi-squaretestforindividualvariables,controllingforothervariablesinthemodel22βˆχ=,1d.f.se(βˆ)•Predicte

8、dprobabilities(ifrandomsamplingwasdone)βˆ+βˆXe01pˆ=xβˆ+βˆX1+e01•OddsratiotodescribetheeffectofX(x1vsx2)βˆ(x-x)OR(xvsx)=e11212•Confidenceintervalfortheoddsratio(x1vsx2)(βˆ+ZSE(βˆ))(x-x)e1112Ifbistheslope(β)estimateforapredictorX,theoddsratioforaoneunitbdifferenceinXisex

9、p(b)=e.So,1.Ifbisnegative,theoddsratioislessthan1,andthevariablereducestheoddsoftheoutcome.2.Ifbispositive,theoddsratioisgreaterthan1,andthevariableincreasestheoddsoftheoutcome.3.Thelargertheabsolutevalueofb,thestrongertheeffectsofaoneunitchangeinthepredictorontheoddso

10、ftheoutcome.3BS805Class11PerforminglogisticregressioninSAS•Usuallythedependentvariable(Y)iscoded1foranevent,and0forno