Modelling the random effects covariance matrix in longitudinal data

Abstract

SUMMARYA common class of models for longitudinal data are random effects (mixed) models. In these models, the random effects covariance matrix is typically assumed constant across subject. However, in many situations this matrix may differ by measured covariates. In this paper, we propose an approach to model the random effects covariance matrix by using a special Cholesky decomposition of the matrix. In particular, we will allow the parameters that result from this decomposition to depend on subjectspecific covariates and also explore ways to parsimoniously model these parameters. An advantage of this parameterization is that there is no concern about the positive definiteness of the resulting estimator of the covariance matrix. In addition, the parameters resulting from this decomposition have a sensible interpretation. We propose fully Bayesian modelling for which a simple Gibbs sampler can be implemented to sample from the posterior distribution of the parameters. We illustrate these models on data from depression studies and examine the impact of heterogeneity in the covariance matrix on estimation of both fixed and random effects.

A Reparametrization Approach for Dynamic Space-Time Models

“…Pourahmadi (1999) introduced an unconstrained parametrization procedure to model a temporal covariance matrix. Chen and and Dunson (2003) and Daniels and Zhao (2003) have also used similar parametrizations to model covariance matrices in longitudinal studies. The Cholesky decomposition of the inverse of a covariance matrix is used to associate a unique unit lower triangular and a unique diagonal matrix with each covariance matrix.…”

Bayesian Modeling of Random Effects Covariance Matrix for Generalized Linear Mixed Models

“…The GARP/IV parametrization provides parameters that can easily be modeled without the concern of the estimator being positive definite, that have a sensible interpretation, and that allow for simple computation (Daniels and Zhao, 2003;Lee et al, 2012).…”

“…where γ is a a × 1 vector of unknown dependence parameters, λ is a b × 1 vector of unknown variance parameters, design vectors w i,t j and h i,t are covariates to model the GARP/IV parameters as functions of subject-specific covariates (Pourahmadi, 2000;Pourahmadi and Daniels, 2002;Daniels and Zhao, 2003;Lee et al, 2012). Therefore, the random effects covariance matrix can be heterogeneous in the subject-specific covariates.…”

“…The positive definiteness restriction of the covariance matrix is that the IVs need to be positive (Pourahmadi, 1999(Pourahmadi, , 2000. In addition, the modified Cholesky decomposition reduces the number of parameters in the covariance matrix and is used for the estimation of the covariance matrix to analyze longitudinal Gaussian data Daniels and Zhao, 2003;Mackenzie, 2003, 2006). Lee et al (2012) proposed a GLMM with the heterogenous random effects covariance matrix depending on covariates via the modified Cholesky decomposition.…”

Bayesian statistics in medicine: a 25 year review

“…Generalized monotonic regression with random changepoints was used to model expression of a leukaemia surface antigen in relation to prognostic factors [504]. Modelling of a random effects covariance matrix was used in longitudinal studies of depression [505]. A hierarchical Bayesian birth cohort analysis was used for trends in the age of onset of Type I diabetes [506].…”