For Table 7.1, let yit = 1 when student i used substance t (t = 1, cigarettes; t = 2, alcohol; t = 3, marijuana). Fit the model, logit[P(Yit = 1)] = ui + α + βt.

a. Report and interpret the estimated fixed effects. Conduct a likelihood-ratio test of H0: β1 = β2 = β3 as a way of testing marginal homogenity.

b. Report σˆ for the random effects. In practical terms, what does (i) the large value imply? (ii) a large positive value for ui for a particular student represent?

c. Why are {βˆt} so different from {βˆt} for the marginal model in Exercise 9.1?

d. Explain how the focus differs for the random effects and marginal models than for the loglinear model (AC, AM, CM) fitted to these data in Section 7.1.7.

e. Adding race and gender to the analysis, with data as shown in Table 7.8, (i) analyze using GLMMs, (ii) compare results and interpretations to those with marginal models in Exercise 9.3.