Simultaneous modeling of marginal distributions and log-multiplicative associations for ordered multivariate categorical data.

There is a great deal of literature on modeling (separately) either the univariate or joint distribution of a two-way or multi-way contingency table. For example, in the GEE methodology, the univariate margins are of interest, and the bivariate associations are treated as a nuisance. Conversely, in standard log-linear methodology, the univariate margins are often fixed, and the bivariate and higher-order interactions are of interest. Neither of these approaches is suitable when both the univariate and bivariate associations are of direct interest. A methodology that maximizes the Poisson log-likelihood subject to the constraints implied by a generalized log-linear model has been developed. Although this methodology has been used to simultaneously model both the joint and marginal distributions, there are many instances in multivariate response settings where the interest lies mainly in the marginal relationships among the responses and not the joint distribution. Furthermore, this methodology is limited to models which are linear in the model parameters. We extend this methodology by introducing models that simultaneously model the univariate marginal logits with linear models and the associations that characterize the bivariate marginal distributions with log-multiplicative models. Log-multiplicative models have useful interpretations for ordinally scaled variables, and, in particular, they permit the estimation of category scores for the categories. This facilitates comparisons of response scales for several response variables, the assessment of category distinguishability, and the investigation of the effects of collapsing or combining categories. Our development of generalized log-non-linear modeling is in terms of the $R\times C$ contingency table, with special emphasis on tables where there is a one-to-one correspondence between the row and column categories. In addition, we consider models for the special case of the $I\times(I+1)$ table that arises when a single response variable is measured at two time points, but at the second time point an additional level of the outcome is observed. Situations in which an $I\times(I+1)$ table results include cases where death occurs or there are missing data, such as dropouts from clinical or panel studies, at the second time point.PhDBiological SciencesBiostatisticsPure SciencesStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/130548/2/9732147.pd