Proportional likelihood ratio mixed model for discrete longitudinal data.

Rathouz and Gao [2] and Luo and Tsai [3] proposed valuable extensions to the generalized linear model for modeling a nonlinear monotonic relationship between the mean response and a set of covariates. In their extensions for discrete data the baseline response distribution is unspecified and is estimated from the data. We propose to extend this model for the analysis of longitudinal data by incorporating random effects into the linear predictor, and using maximum likelihood for estimation and inference. Motivated in particular by longitudinal studies of clinical scale outcomes, we developed an estimation procedure for a finite-support response using a generalized expectation-maximization algorithm where Gauss-Hermite quadrature is employed to approximate the integrals in the E step of the algorithm. Upon convergence, the observed information matrix is estimated through second-order numerical differentiation of the log-likelihood function. Asymptotic properties of the maximum likelihood estimates follow under the usual regularity conditions. Simulation studies are conducted to assess its finite-sample properties and compare the proposed model to the generalized linear mixed model. The proposed method is illustrated in an analysis of data from a longitudinal study of Huntington's disease.