Interactions in Logistic Regression
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1 Interactions in Logistic Regression > # UCBAdmissions is a 3-D table: Gender by Dept by Admit > # Same data in another format: > # One col for Yes counts, another for No counts. > Berkeley = read.table(" code_n_data/berkeley2.data") > Berkeley Gender Dept Yes No 1 Male A Female A Male B Female B Male C Female C Male D Female D Male E Female E Male F Female F > # Resp var is 2 cols. Second col is Y=1 > full = glm(cbind(no,yes) ~ Dept*Gender,family=binomial,data=Berkeley) > anova(full,test='chisq') Analysis of Deviance Table Model: binomial, link: logit Response: cbind(no, Yes) Terms added sequentially (first to last) Df Deviance Resid. Df Resid. Dev Pr(>Chi) NULL Dept < 2.2e-16 *** Gender Dept:Gender ** --- Signif. codes: 0 *** ** 0.01 * Page 1 of 10
2 > # Let's see what it means. Repeating some material from an earlier analysis... > noquote(gradschool) Dept %MaleAcc %FemAcc Chisq p-value [1,] A e-05 [2,] B [3,] C [4,] D [5,] E [6,] F > Male = as.numeric(gradschool[,2]) > Female = as.numeric(gradschool[,3]) > cbind(male,female) Male Female [1,] [2,] [3,] [4,] [5,] [6,] > # On the log scale, differences are logs of odds ratios. > # Non-parallel means the odds ratio DEPENDS > logmale = log(male); logfemale = log(female) > plot(rep(1:6,2),c(logmale,logfemale), pch=" ", axes=f, + xlab="department",ylab="log Percent Acceptance") > axis(1,1:6,letters[1:6]) # X axis > axis(2) # Y axis > lines(1:6,logfemale,lty=1); lines(1:6,logmale,lty=2) > points(1:6,logmale); points(1:6,logfemale,pch=19) > legend(2,2.5,legend="female Applicants",lty=1,bty="n",pch=19) > legend(2,2.3,legend="male Applicants",lty=2,bty="n",pch=1) > title("berkeley Graduate Admissions by Department") Page 2 of 10
3 Page 3 of 10
4 > summary(full) Call: glm(formula = cbind(no, Yes) ~ Dept * Gender, family = binomial, data = Berkeley) Deviance Residuals: [1] Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) e-10 *** DeptB DeptC < 2e-16 *** DeptD e-15 *** DeptE < 2e-16 *** DeptF < 2e-16 *** GenderMale e-05 *** DeptB:GenderMale DeptC:GenderMale e-05 *** DeptD:GenderMale ** DeptE:GenderMale *** DeptF:GenderMale * --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: e+02 on 11 degrees of freedom Residual deviance: e-15 on 0 degrees of freedom AIC: Number of Fisher Scoring iterations: 3 Page 4 of 10
5 Categorical by Quantitative Interactions Parallel regression lines on the log scale mean that Log differences between groups are the same for each level of x. Odds ratios are the same for each level of x. Odds are in the same proportion at each level of x. Called a proportional odds model. Product terms represent departure from parallel lines. Translates to departure from proportional odds. To test proportional odds assumption, test regression coefficients of the product terms. Odds ratios depend on the value of x. Page 5 of 10
6 > math = read.table(" /code_n_data/mathcat.data") > math[1:5,] hsgpa hsengl hscalc course passed outcome Yes Mainstrm No Failed Yes Mainstrm Yes Passed Yes Mainstrm Yes Passed Yes Mainstrm Yes Passed Yes Mainstrm Yes Passed > attach(math) # Variable names are now available > > # Make dummy vars for course to be sure what's going on > n=length(hsgpa) > c1 = c2 = numeric(n) > c1[course=='catch-up'] = 1 > c2[course=='elite'] = 1 > # table(c1,course); table(c2,course) > c1gpa = c1*hsgpa; c2gpa = c2*hsgpa > > # Reduced model will have no interactions > redmod = glm(passed ~ hsgpa+c1+c2, family=binomial) > fullmod = glm(passed ~ hsgpa+c1+c2+c1gpa+c2gpa, family=binomial) > anova(redmod,fullmod,test='chisq') Analysis of Deviance Table Model 1: passed ~ hsgpa + c1 + c2 Model 2: passed ~ hsgpa + c1 + c2 + c1gpa + c2gpa Resid. Df Resid. Dev Df Deviance Pr(>Chi) > > # Can do it with factors > contrasts(course) = contr.treatment(3,base=3) > red = glm(passed ~ hsgpa+course, family=binomial) > full = glm(passed ~ hsgpa+course+hsgpa:course, family=binomial) Page 6 of 10
7 > anova(red,full,test='chisq') Analysis of Deviance Table Model 1: passed ~ hsgpa + course Model 2: passed ~ hsgpa + course + hsgpa:course Resid. Df Resid. Dev Df Deviance Pr(>Chi) > anova(redmod,fullmod,test='chisq') # For comparison Analysis of Deviance Table Model 1: passed ~ hsgpa + c1 + c2 Model 2: passed ~ hsgpa + c1 + c2 + c1gpa + c2gpa Resid. Df Resid. Dev Df Deviance Pr(>Chi) Consistent with proportional odds. Page 7 of 10
8 > summary(fullmod) Call: glm(formula = passed ~ hsgpa + c1 + c2 + c1gpa + c2gpa, family = binomial) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) e-11 *** hsgpa e-12 *** c c c1gpa c2gpa Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 393 degrees of freedom Residual deviance: on 388 degrees of freedom AIC: Number of Fisher Scoring iterations: 5 > summary(full) Call: glm(formula = passed ~ hsgpa + course + hsgpa:course, family = binomial) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) e-11 *** hsgpa e-12 *** course course hsgpa:course hsgpa:course Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 393 degrees of freedom Residual deviance: on 388 degrees of freedom AIC: Number of Fisher Scoring iterations: 5 Page 8 of 10
9 > betahat = redmod$coefficients; betahat (Intercept) hsgpa c1 c > > gpa = 50:100 > catchup = betahat[1]+betahat[3] + betahat[2]*gpa > elite = betahat[1]+betahat[4] + betahat[2]*gpa > mainstream = betahat[1] + betahat[2]*gpa > > GPA = rep(gpa,3); Pass = c(catchup,elite,mainstream) > plot(gpa,pass,pch=' ') > lines(gpa,catchup,lty=1) > lines(gpa,elite,lty=2) > lines(gpa,mainstream,lty=3) > title("parallel Estimated Log Odds") Page 9 of 10
10 > oddscu = exp(catchup); oddsel = exp(elite) > oddsmain = exp(mainstream) > Odds = c(oddscu,oddsel,oddsmain) > plot(gpa,odds,pch=' ') > lines(gpa,oddscu,lty=1) > lines(gpa,oddsel,lty=2) > lines(gpa,oddsmain,lty=3) > title("proportional Estimated Odds") Page 10 of 10
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