A New Look at the Models for Ordinal Categorical Data Analysis

The multinomial/categorical responses, whether they are nominal or ordinal, are recorded in counts under all categories/cells involved. The analysis of this type of multinomial data is traditionally done by exploiting the marginal cell probabilities-based likelihood function. As opposed to the nominal setup, the computation of the marginal probabilities is not easy in the ordinal setup. However, as the ordinal responses in practice are interpreted by collapsing multiple categories either to binary data using a single cut-point or to tri-nomial data using two cut-points, most of the studies over the last four decades, first modeled the associated cumulative probabilities using a suitable such as logit, probit, log-log, or complementary log-log link function. Next the marginal cell probabilities were computed by subtraction, in order to construct the desired estimating function such as moment or likelihood function. In this paper we take a new look at this ordinal categorical data analysis problem. As opposed to the existing studies, we first model the ordinal categories using a multinomial logistic marginal approach by pretending that the adjacent categories are nominal, and then construct the cumulative probabilities to develop the final model for ordinal responses. For inferences, we develop the cut points based likelihood or generalized quasi-likelihood (GQL) estimating functions for the purpose of the estimation of the underlying regression parameters. The new GQL estimation approach is developed in details by utilizing both tri-nomial and binary (or binomial) collapsed structures. The likelihood analysis is also discussed. A data example is given to illustrate the proposed models and the estimation methodologies. Furthermore, we also examine the asymptotic properties of the likelihood and GQL estimators for the regression parameters for both tri-nomial and binary types of cumulative response based models.