Dynamic Models for Longitudinal Ordinal Non-stationary Categorical Data

Abstract

When a nominal categorical response along with a multidimensional covariate are repeatedly collected over a small period of time from a large number of independent individuals, it is standard to use the so-called multinomial logits to model the marginal means or probabilities of such responses in terms of time dependent covariates. However, because of the difficulties in modeling the correlations of the repeated multinomial responses, some researchers over the last two decades have analyzed this type of repeated multinomial data by using certain ‘working’ odds ratio, or ‘working’ correlation structures. But, in the context of longitudinal binary data analysis, these ‘working’ correlations based approaches were shown to produce inefficient regression estimates as compared to simpler moment and/or quasi-likelihood approaches. The situation can be worse for longitudinal nominal multinomial/categorical data. In this paper, following the recent correlation models proposed by Sutradhar (Longitudinal Categorical Data Analysis. Springer, New York, 2014, Chap. 3) for the stationary nominal and ordinal categorical data, we discuss similar correlation models but for non-stationary ordinal categorical data. More specifically we exploit the multinomial dynamic logits (MDL) in two different ways to develop correlation models for ordinal categorical data. Under model 1, the ordinal responses are first treated to be nominal and a multinominal dynamic logits model is written for correlations between repeated nominal categorical responses. The regression and correlation parameters involved in the dynamic logits are, however, computed from an updated cumulative dynamic logits model which accommodates the actual ordinal nature of the data. Under model 2, a direct dynamic logits relationship is developed linking the cumulative multinomial responses over time. A lag 1 conditional likelihood is then exploited to estimate the desired regression and correlation parameters. Some asymptotic properties of the estimators under both models are also discussed.