Abstract: We proposed a new approach to model longitudinal data consisting of transitional frequencies classified according to an ordered categorical response variable. Following an approach of Kalbfleisch and Lawless (1985), the responses are assumed to be sampled from an underlying continuous-time finite-state-space Markov chain, with the further assumption that direct transitions are strictly between adjacent states, owing to the ordered categorical nature of the response variable. The model admits a parsimonious parameterization in terms of the transition probability rates (intensity parameters) between adjacent states over an infinitesimal period. It is assumed that after a suitable transformation (link function), the intensity parameters 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 misspecification, the likelihood goodness-of-fit test is asymptotically equivalent to the Pearson Chi-square goodness-of-fit test, with the usual limiting Chi-square distribution. We illustrate the new approach with two data sets.
Key words and phrases: Asymptotic normality, continuous-time finite-state-space Markov chain, goodness-of-fit test, maximum likelihood estimation.