Logistic Regression and Odds Ratio：Logistic回归分析和比值比

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1、LogisticRegressionandOddsRatioOddsRatioReviewOutcome(BrainTumor)RiskFactorYesNoTotal(Benzene)(Case)(Control)Yes502070No100130230Total150150300OutcomeYesNoRiskFactorTotal(Disease)(NoDisease)Yesaba+bRiskfactorcanalwaysbe(Exposed)arrangedaseithercolumnorNocdc+drowvaria

2、ble,butthedefinition(Unexposed)Totala+cb+dnofoddsratioisstillthesame.Letp1betheprobabilityofsuccessinrow1(probabilityofBrainTumorinrow1)1-p1istheprobabilityofnotsuccessinrow1(probabilityofnoBrainTumorinrow1)Oddofgettingdiseaseforthepeoplewhowereexposedtotheriskfacto

3、r:(pˆ1isanestimateofp1)aO+=P[disease

4、exposed]=P[disease

5、exposed]p1pˆ1a+ba501-P[disease

6、exposed]P[nodisease

7、exposed]=»=b===2.51-p11-pˆ1b20a+bLetp0betheprobabilityofsuccessinrow2(probabilityofBrainTumorinrow2)1-p0istheprobabilityofnotsuccessinrow2(probabilityofnoBrain

8、Tumorinrow2)Oddofgettingdiseaseforthepeoplewhowerenotexposedtotheriskfactor:(pˆ0isanestimateofp0)P[disease

9、unexposed]P[disease

10、unexposed]cO-==1-P[disease

11、unexposed]P[nodisease

12、unexposed]p0pˆ0c+dc100=»====.771-p01-pˆ0dd130c+dTheOddsRatioofhavingbraintumorforpeoplewho

13、wereexposedtotheriskfactorversusnotexposed:p1pˆ1aOR=q=O+=1-p1»qˆ=1-pˆ1=[a/(a+b)]/[b/(a+b)]=b=ad=50´130=3.25O-p0pˆ0[c/(c+d)]/[d/(c+d)]cbc20´1001-p01-pˆ0dInterpretation:Theoddsofhavingbraintumorare3.25timeshigherforthosewhoexposedtobenzenethanthosewhowerenotexposedtob

14、enzene.Ifq>1,thentheoddsofsuccessarehigherforcolumn1(riskfactorpresent)thancolumn2(riskfactornotpresent).Ifq<1,thentheoddsofsuccessarelowerforcolumn1(riskfactorpresent)thancolumn2(riskfactornotpresent).Ifq=1,thentheoddsofsuccessareequalforcolumn1(riskfactorpresent)a

15、ndcolumn2(riskfactornotpresent).A.Chang1LogisticRegressionandOddsRatioConfidenceintervalforoddsratio:Forlargesample,thelogofoddsratio,,followsasymptoticallyanormaldistribution.ln(qˆ)The(1–a)100%confidenceintervalestimatefortheLogOddsRatioisln(qˆ)±z×s*a/2The(1–a)100%

16、confidenceintervalestimatefortheOddsRatiois(eln(qˆ)-za/2×s*,eln(qˆ)+za/2×s*)qˆ=ad1111where,standarderrorofln(qˆ)iss*»,and+++a,b,canddshoul