Question

I need help interpreting logistic regression results to answer the following question: Does GRE scores, undergraduate GPA and the prestige (yes or no) of their undergraduate program effect admission (yes or no) into graduate school?

Fit Group 4 Logistic Fit of ADMIT 2 By GRE 1.00 Contingency Analysis of ADMIT 2 By TOPNOTCH 2 4 Mosaic Plot Logistic Fit of ADMIT 2 By GPA 1.00 1.00 0.75 0.75 No 0.75 No No ADMIT 2 0.50 N 0.50 ADMIT 2 ADMIT 2 0.50 0.25 0.25 0.25 Yes Yes Yes 0.00 200 1 300 0.00 2.2 400 600 700 800 Yes 24 2.6 2.8 3 34 500 GRE 3.6 3.8 4 3.2 GPA Iterations 4 Whole Model Test Model -LogLikelihood Difference 6.96019 Full 243.02807 Reduced 249.98826 0.00 No TOPNOTCH 2 Contingency Table ADMIT 2 Count Yes No Total I No 97 238 335 Yes 30 35 65 Total 127 273 400 DF ChiSquare Prob>Chisa 13.92038 0.0002* Iterations 4 Whole Model Test Model -LogLikelihood Difference 6.50445 Full 243.48381 Reduced 249.98826 DF ChiSquare Prob>Chisq 13.0089 0.0003* 1 1 4 Tests N DF -Loglike RSquare (U) 400 13.5398469 0.0142 Test ChiSquare Prob>Chisq Likelihood Ratio 7.080 0.0078 7.431 0.0064* Pearson RSquare (U) 0.0278 AICC 490.086 BIC 498.039 Observations (or Sum Wgts) 400 Fit Details 4 Parameter Estimates Term Estimate Std Error ChiSquare Prob>Chisq Intercept -2.9013443 0.6060379 22.92 <.0001* GRE 0.00358221 0.000986 13.20 0.0003 For log odds of Yes/No Covariance of Estimates RSquare (U) 0.0260 AICC 490.998 BIC 498.951 Observations (or Sum Wgts) 400 Fit Details 4 Parameter Estimates Term Estimate Std Error ChiSquare Prob> ChiSq Intercept -4.3575873 1.0353175 17.72 <,0001* GPA 1.05110873 0.2988695 12.37 0.0004 For log odds of Yes/No Covariance of Estimates Fisher's Exact Test Left Right 2- Tail Prob Alternative Hypothesis 0.0057* Prob(ADMIT 2 =No) is greater for TOPNOTCH 2=No than Yes 0.9975 Prob(ADMIT 2-No) is greater for TOPNOTCH 2-Yes than No 0.0086* Prob(ADMIT 2 =No) is different across TOPNOTCH 2

Answer 1

Logistic regression, also called a logit model, is used to model dichotomous outcome variables. In the logit model the log odds of the outcome is modelled as a linear combination of the predictor variables. Basically there are three predictor variables - GRE, GPA and rank. We will treat the variables GRE and GPA as continuous. The variable rank takes on the values 1 through 4 lets say. Institutions with a rank of 1 have the highest prestige, while those with a last rank have the lowest. As expected, both GRE and GPA have a positive effect on admissions – as they go up, so do the odds of admission. Also prestige,GRE and GPA are statistically significant, as are the three terms for rank.

The logistic regression coefficients give the change in the log odds of the outcome for a one unit increase in the predictor variable. The chi-squared test statistic of 14.9, with 1 degree of freedom is associated with a p-value of 0.0001 indicating that the overall effect of rank is statistically significant. We can also exponentiate the coefficients and interpret them as odds-ratios. We can say that for a one unit increase in GPA, the odds of being admitted to graduate school increase by a factor of 1.94. The effects seen are in the directions expected, i.e. higher GRE and GPA and more prestige suggests more likely admittance i.e. lower prestige rank score means higher prestige. Now if any move up one on GPA, the odds of being admitted increase by a certain factor, i.e. more than double. If prestige rank increases by one (i.e. less prestige), the odds percentage will decrease.

After adjusting for confounding factors like gpa, gre, rank, we analyse that the odds of being admitted to graduate school versus not being admitted increase by a factor more than two times higher for candidates with higher GPA than for candidates with less scores. For logistic regression, the more general method of maximum likelihood is preferred for its robust statistical properties. Basically, the algorithm tries to find coefficients that maximize the likelihood that the probabilities are closest to 1 for people who defaulted, and close to zero for people who did not. The fitness of our model can be tested by calculating the p-value i.e. probability of a student being admitted. With a p-value of 0.0068 which is less than 0.05, we conclude that our model is statistically significant. Thus we can say that GRE scores, undergraduate GPA and the prestige of their undergraduate program effect admission into graduate school.

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