*Statistical Modelling* **3** (2003), 251–271

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

Brent A. Coull

Department of Biostatistics,

Harvard School of Public Health,

655 Huntington Ave.,

Boston, MA 02115,

USA.

eMail: bcoull@hsph.harvard.edu

Alan Agresti

Department of Statistics,

University of Florida,

Gainesville,

USA

### 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** = **Xb** + **Zu**, where **b**
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.
#### Keywords:

Association model; generalized linear mixed model; marginal model; mixture model; multinomial;
logit model; ordinal data; overdispersion; Poisson.

#### Downloads:

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SAS code
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