Multivariate distributions with applications to logistic regression models for correlated categorical responses
A number of models for ordered categorical responses have appeared in the statistical literature over the past few decades. Many of these models are based on a latent variable approach in which the underlying random variable follows a continuous probability distribution. The exact value of the underlying random variable is not observed; rather, it is only known that the response occurred in one of several mutually exclusive categories. Such models are referred to as logit, or logistic regression models when an underlying logistic distribution is used;Statistical analyses with logistic regression models are straightforward when the categorical responses are assumed to be independent. Relaxation of the independence assumption allows for inclusion of a wide variety of practical applications, but appropriate models have not been thoroughly developed. Most of the research on correlated categorical responses has been confined to the development of conditional and mixed models;An alternative approach taken in this dissertation is a multivariate latent variable approach. In particular, correlated categorical responses will be considered to have arisen from the categorization of an underlying multivariate logistic random variable. This approach is a multivariate extension of the usual univariate logistic regression model; in fact, it provides a logistic regression model for each univariate margin. A major advantage of this approach is that the model parameters can be interpreted with respect to the influence of the regression variables on the univariate marginal distribution of each single response. Conditional and mixed models, on the other hand, permit only conditional interpretations of parameters;Included in the first paper is a specification of a new bivariate logistic distribution. This distribution is applied to the analysis of bivariate ordered categorical responses. A listing of the computer program developed for this analysis is given in Appendix C. In the second paper a bivariate logistic regression model is developed and illustrated and is based on the aforementioned bivariate logistic distribution. A computer listing for this model is found in Appendix D. The third paper includes a generalization of one of Gumbel's bivariate logistic distributions. A new multivariate uniform distribution is derived in the fourth paper and is used to specify a multivariate logistic distribution. It is shown that the pairwise correlations can range between zero and one and are, in general, not equal. The asymptotic normality of maximum likelihood estimators for product multinomial models is established in Appendix A. Finally, a simple expression for the Fisher information matrix arising from a product multinomial likelihood function is derived in Appendix B.