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\noindent Statistica Sinica {\bf 13}(2003), 207-226
\vspace{3pc} \normalsize

\centerline{\large\bf A GENERALIZED LINEAR MODEL FOR REPEATED}
\vspace{2pt}
\centerline{\large\bf ORDERED CATEGORICAL RESPONSE DATA}
\vspace{.25cm}
\centerline{K. S. Chan and B. Munoz-Hernandez}
\vspace{.4cm}
\centerline{\it University of Iowa and University of Costa Rica}
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\begin{quotation}
\noindent {\it 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.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Asymptotic normality, continuous-time finite-state-space Markov chain,
goodness-of-fit test, maximum likelihood estimation. \par
\end{quotation}
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