5.7 From the same survey referred to in Problem 4.16, Table 5.11 cross-classifies whether a person smokes frequently with the four scales of the MBTI personality test. SAS reports model −2 log likelihood values of 1130.23 with only an intercept term, 1124.86 with also the main effect predictors, 1119.87 with also all the two-factor interactions, and 1116.47 with also all the three-factor interactions.

a. Write the model for each case, and show that the numbers of parameters are 1, 5, 11, and 15.

b. According to AIC, which of these four models is preferable?

c. When a classification table for the model containing the four main effect terms uses the sample proportion of frequent smokers of 0.23 as the cutoff, sensitivity = 0.48 and specificity = 0.55. The area under the ROC curve is c = 0.55. Does knowledge of personality type help you predict well whether someone is a frequent smoker? Explain.