Propriety of the Posterior Distribution and Existence of the MLE for Regression Models With Covariates Missing at Random

Abstract

Characterizing model identifiability in the presence of missing covariate data is a very important issue in missing data problems. In this article, we characterize the propriety of the posterior distribution of the regression coefficients for some general classes of regression models, including the class of generalized linear models (GLM's) and parametric survival models with right-censored data. Toward this goal, we derive some very general and easy-to-check conditions for the matrix of covariates. We also derive sufficient conditions for the existence of the maximum likelihood estimates and establish novel results for checking propriety of the posterior when the sample size is large. Several theorems are given to establish propriety of the posterior and the existence of the maximum likelihood estimator. The conditions reduce to solving a system of linear equations, which can be carried out using software such as MAPLE, IMSL, or SAS. We assume that the missing covariates are missing at random and assume an improper uniform prior for the regression coefficients. In addition, we establish these results assuming a very general form for the covariate distribution, allowing for both missing categorical and/or continuous covariates. A small dataset is used to illustrate that the posterior can be improper based on complete cases while proper when all of the cases are used in the analysis. Two real datasets are presented to demonstrate verification of posterior propriety for GLM's and parametric survival models, and also to illustrate propriety for large datasets.

KEY WORDS :

• Binary regression model
• Covariate distribution
• Design matrix
• Generalized linear model
• Maple
• Missing at random
• Necessary and sufficient conditions
• Noninformative prior
• Parametric survival model
• Poisson regression model