Generalized log-linear models with random effects, with application to smoothing contingency tables

Abstract

We define a class of generalized log-linear models with random effects. For a vector of Poisson or multinomial means m and matrices of constants C and A, the model has the form C log A μ = X β + Zu, where β are fixed effects and u are random effects. The model contains most standard models currently used for categorical data analysis. We suggest some new models that are special cases of this model and are useful for applications such as smoothing large contingency tables and modeling heterogeneity in odds ratios. We present examples of its use for such applications. In many cases, maximum likelihood model fitting can be handled with existing methods and software. We outline extensions of model fitting methods for other cases. We also summarize several challenges for future research, such as fitting the model in its most general form and deriving properties of estimates used in smoothing contingency tables.