Modelling the random effects covariance matrix in longitudinal data

A 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 subject-specific 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.

[1]  Tom Leonard,et al.  Bayesian Inference for a Covariance Matrix , 1992 .

[2]  R. D. Bock,et al.  Multivariate Statistical Methods in Behavioral Research , 1978 .

[3]  R. Weiss,et al.  Diagnosing explainable heterogeneity of variance in random‐effects models , 2000 .

[4]  Xiao-Li Meng,et al.  Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage , 2000 .

[5]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[6]  R. Kass,et al.  Shrinkage Estimators for Covariance Matrices , 2001, Biometrics.

[7]  Colin O. Wu,et al.  Nonparametric Mixed Effects Models for Unequally Sampled Noisy Curves , 2001, Biometrics.

[8]  Robert E. Weiss,et al.  An Analysis of Paediatric Cd4 Counts for Acquired Immune Deficiency Syndrome Using Flexible Random Curves , 1996 .

[9]  M J Daniels,et al.  Dynamic conditionally linear mixed models for longitudinal data. , 2002, Biometrics.

[10]  M. Pourahmadi Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix , 2000 .

[11]  M. Pourahmadi,et al.  Bayesian analysis of covariance matrices and dynamic models for longitudinal data , 2002 .

[12]  M. Pourahmadi Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterisation , 1999 .

[13]  J. Raz,et al.  Linear mixed models with heterogeneous within-cluster variances. , 1997, Biometrics.

[14]  D. Hedeker,et al.  MIXREG: a computer program for mixed-effects regression analysis with autocorrelated errors. , 1996, Computer methods and programs in biomedicine.

[15]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[16]  R. Kass,et al.  Nonconjugate Bayesian Estimation of Covariance Matrices and its Use in Hierarchical Models , 1999 .

[17]  D J Kupfer,et al.  Treatment of major depression with psychotherapy or psychotherapy-pharmacotherapy combinations. , 1997, Archives of general psychiatry.

[18]  Tom Leonard,et al.  The Matrix-Logarithmic Covariance Model , 1996 .

[19]  Roderick J. A. Little,et al.  Statistical Analysis with Missing Data , 1988 .

[20]  L. Skovgaard NONLINEAR MODELS FOR REPEATED MEASUREMENT DATA. , 1996 .

[21]  P. Heagerty,et al.  Misspecified maximum likelihood estimates and generalised linear mixed models , 2001 .