The analysis of longitudinal polytomous data: generalized estimating equations and connections with weighted least squares

In recent years, methods have been developed for modelling repeated observations of a categorical response obtained over time on the same individual. Although situations in which the repeated response is binary or Poisson have been studied extensively, relatively little attention has been given to polytomous categorical response variable. In this paper, we extend the estimating equations initially developed for clustered discrete data by Liang and Zeger (1986, Biometrika 73, 13-22), and subsequently extended by Prentice (1988, Biometrics 44, 1033-1048), to polytomous response variables. Under certain assumptions, we illustrate that these estimating equations simplify to the weighted least squares (WLS) equations formalized by Koch et al. (1977, Biometrics 33, 133-158). This connection provides a formal framework for obtaining iterated weighted least squares model parameter estimates. Cumulative logit models are developed and applied to a representative longitudinal data set. Simulation results comparing WLS, an iterative form of WLS, and independence estimating equations using a robust estimate of the variance are presented.