Autoregressive Linear Mixed Effects Models

Abstract

In the previous chapter, longitudinal data analysis using linear mixed effects models was discussed. This chapter discusses autoregressive linear mixed effects models in which the current response is regressed on the previous response, fixed effects, and random effects. These are an extension of linear mixed effects models and autoregressive models. Autoregressive models regressed on the response variable itself have two remarkable properties: approaching asymptotes and state-dependence. Asymptotes can be modeled by fixed effects and random effects. The current response depends on current covariates and past covariate history. Three vector representations of autoregressive linear mixed effects models are provided: an autoregressive form, response changes with asymptotes, and a marginal form which is unconditional on previous responses. The marginal interpretation is the same with subject specific interpretation as well as linear mixed effects models. Variance covariance structures corresponding to AR(1) errors, measurement errors, and random effects in the baseline and asymptote are presented. Likelihood of marginal and autoregressive forms for maximum likelihood estimation are also provided. The marginal form can be used even if there are intermittent missing values. We discuss the difference between autoregressive models of the response itself which focused in this book and models with autoregressive error terms.