State Space Representation of an Autoregressive Linear Mixed Effects Model for the Analysis of Longitudinal Data

Recently, we proposed an autoregressive linear mixed effects model for the analysis of longitudinal data in which the current response is regressed on the previous response, fixed effects, and random effects (Funatogawa et al., Statist. Med. 2007; 26:2113-2130). The model represents profiles approaching random equilibriums. Because intermittent missing is an inherent problem of the autoregressive (conditional) model, we provided the marginal (unconditional) representation of the model and the likelihood. In this study, we further provide a state space form of the model for calculating the likelihood without using large matrices. The proposed state space form corresponds to the marginal form of the likelihood instead of the conditional one. We modified the method proposed by Jones (1993) for a state space form of a usual linear mixed effects model. Following Jones (1993), the regression coefficients of the fixed effects are concentrated out of the likelihood.