A GENERALIZED LINEAR MODEL FOR REPEATED ORDERED CATEGORICAL RESPONSE DATA

Abstract

We proposed a new approach to model longitudinal data consisting of transitional frequencies classied according to an ordered categorical response vari- able. Following an approach of Kalbeisc h and Lawless (1985), the responses are as- sumed to be sampled from an underlying continuous-time nite-state-space Markov chain, with the further assumption that direct transitions are strictly between ad- jacent states, owing to the ordered categorical nature of the response variable. The model admits a parsimonious parameterization in terms of the transition probabil- ity rates (intensity parameters) between adjacent states over an innitesimal period. It is assumed that after a suitable transformation (link function), the intensity pa- rameters are linear functions of some (possibly time-dependent) covariates. We show that under very mild regularity conditions including a full-rank condition on the \design matrix, the maximum likelihood (ML) estimators are consistent and asymptotically normal. We also show that, under the same set of regularity conditions and under the null hypothesis of no model misspecication, the likeli- hood goodness-of-t test is asymptotically equivalent to the Pearson Chi-square goodness-of-t test, with the usual limiting Chi-square distribution. We illustrate the new approach with two data sets.