Serial correlation structures in latent linear mixed models for analysis of multivariate longitudinal ordinal responses

We propose a latent linear mixed model to analyze multivariate longitudinal data of multiple ordinal variables, which are manifestations of fewer continuous latent variables. We focus on the latent level where the effects of observed covariates on the latent variables are of interest. We incorporate serial correlation into the variance component rather than assuming independent residuals. We show that misleading inference may be drawn when misspecifying the variance component. Furthermore, we provide a graphical tool depicting latent empirical semi-variograms to detect serial correlation for latent stationary linear mixed models. We apply our proposed model to examine the treatment effect on patients having the amyotrophic lateral sclerosis disease. The result shows that the treatment can slow down progression of latent cervical and lumbar functions.