1. For multivariate binary data there are several possible parameterizations of the model that have been proposed for regression analysis of the correlated outcomes. All models for multivariate categorical data are special cases of the canonical log-linear model. The difference between models is the way that the joint probabilities are structured in terms of the parameters of interest. (a) Log-linear model (DHLZ 8.2.1) Consider a data set with four binary measurements taken on each of the subject. Define the saturated log-linear model. Give an interpretation for the parameters On, and 0iz ; for a given j = 1, 2, 3, 4 and k = j (where i indexes the subject). (b) What is E(Y ) in terms of the log-linear model parameters? (c) Bahadur model DHLZ 8.2.2 presents a "fully marginal" model for binary data known as the Bahadur model. For the data above write down the likelihood function based on the Bahadur model with three-way and four-way correlations assumed to be zero. Given an interpretation for the parameters , and p for a given j and k = j. (d) Consider a single pair of binary response variables (Y,, Y,,). Derive the correlation Pink = Cor(Y,,Y ) such that it is a function of E(Y, | Y = 1) - E(Y;|Y = 0). What's pijk when E(Y,;) = E(Y) Give a simple interpretation for pig in this case. (c) Consider an exchangeable correlation matrix for binary Y;,j = 1, -.. , n; with a single scalar predictor X. = X. Please derive the score function of GEE and discuss its relationship to quasi-likelihood using a beta-binomail variance model for (EY,n,). (Note: see DHLZ pages 60-61 for R-! [the notation for R in DHLZ is (f) Partially marginal model: Fitzmaurice and Laid (1993) considered the QEF model Pr(Y. ) = exp { On + >,Y,,+) ick and the transformation (0, w) - (#, w). Derive the score equations (of (8, a)) for