State Space Representations of Autoregressive Linear Mixed Effects Models

Abstract

The previous chapters discussed longitudinal data analysis using linear mixed effects models and autoregressive linear mixed effects models. This chapter discusses state space representations of these models. This chapter also introduces the state space representations of time series data and the extension to multivariate longitudinal data. We use the state space representations and the Kalman filter as an alternative method to calculate the likelihoods for longitudinal data. In the autoregressive linear mixed effects models, the current response is regressed on the previous response, fixed effects, and random effects. Intermittent missing, that is the missing values in the previous response as a covariate, is an inherent problem with autoregressive models. One approach to this problem is based on the marginal form of likelihoods because they are not conditional on the previous response. State space representations with the modified Kalman filter also provide the marginal form of likelihoods without using large matrices. Calculation of likelihood usually requires matrices whose size depends on the number of observations of a subject, but this method does not. In the modified method, the regression coefficients of the fixed effects are concentrated out of the likelihood.