9.12 For the insomnia study summarized in Table 9.6, model (9.2) compared treatments while controlling for initial response of time to fall asleep.

a. Add an interaction term to model (9.2). Summarize how the estimated treatment effect varies according to the initial responses by showing that the estimated treatment log odds ratio changes from 0.00 to 1.41 as the initial response score goes from 10 to 75.

b. Now fit the model without interaction by treating initial response as qualitative, using dummy variables. Show that the estimated treatment log odds ratio is 0.911 (SE = 0.249), and interpret.

c. Now fit the model with interaction terms by treating initial response as qualitative. Explain why the results suggest that the active treatment seems relatively more successful at the two highest initial response levels.