This project will examine new methodology for making inference about the regression parameters in the presence of missing covariate data for two commonly used classes of regression models. In particular, we examine the class of generalized linear models for general types of response data and the Cox model for survival data. The methodology addresses problems occurring frequently in clinical investigations for chronic disease, including cancer and AIDS. The specific objectives of the project are to: 1) develop and study classical and Bayesian methods of inference for the class of generalized linear models (GLM's) in the presence of missing covariate data. In particular, we will i) examine methods for estimating the regression parameters when the missing covariates are either categorical or continuous and the missing data mechanism is ignorable. Also, parametric models for the covariate distribution will be examined. The methods of estimation will focus on the Monte Carlo version of the EM algorithm (Wei and Tanner, 1990) and other related iterative algorithms. The Gibbs sampler (Gelfand and Smith, 1990) along with the adaptive rejection algorithm of Gilks and Wild (1992) will be used to sample from the conditional distribution of the missing covariates given the observed data. ii) examine estimating the regression parameters when the missing covariates are either categorical or continuous and the missing data mechanism is nonignorable. Models for the missing data mechanism will be studied. iii) develop and study Bayesian methods of inference in the presence of missing covariate data when the missing covariates are either categorical or continuous and the missing data mechanism is ignorable. Parametric prior distributions for the regression coefficients are proposed. Properties of the posterior distributions of the regression coefficients will be studied. The methodology will be implemented using Markov Chain Monte Carlo methods similar to those of Tanner and Wong (1987). iv) investigate Bayesian methods when the covariates are either categorical or continuous and the missing data mechanism is nonignorable. Multinomial models for the missing data mechanism will be studied. Dirichlet prior distributions for the multinomial parameters will be investigated. 2) develop and study classical and Bayesian methods of inference for the Cox model for survival outcomes in the presence of missing covariates. Specifically, we will i) develop and study estimation methods for the Cox model for survival outcomes in the presence of missing covariates. Methods for estimating the regression parameters when the missing covariates are either categorical or continuous will be studied. The methods of estimation will focus on an EM type algorithm similar to that of Wei and Tanner (1990). ii) study estimation of the regression parameters when the missing covariates are either categorical or continuous and the missing data mechanisms nonignorable. Models for the missing data mechanism will be studied. Bayesian methods similar to those of 1-iii) and -iv) will be investigated. Computational techniques using the Monte Carlo methods described in 1-iii) will be implemented.
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