Random effects (mixed) models are a class of models used frequently to model longitudinal data. They offer many advantages including the ability to handle different observation times across subjects and to allow non-stationary covariance structures. In these models, little time is typically spent on modelling the random effects covariance matrix. In particular, examining whether this matrix is the same for all subjects or whether it differs depending on subject-specific characteristics is often neglected in the modelling. For discrete longitudinal data modelled using generalized linear mixed models, ignoring this heterogeneity can result in biased estimates of the fixed effects [1
]. For continuous longitudinal data, which will be our focus here, the standard errors for the fixed and random effects, and consequently, inferences, will be incorrect, the random effects will not be shrunk correctly, and prediction of subject-specific trajectories can be poor. In addition, incorrectly modelling the covariance structure in the presence of missing data can result in biased estimates of fixed effects.
Accounting for heterogeneity in covariance matrices has recently been discussed by several authors. In marginal models, Chiu et al.
] model the covariance matrix using a log matrix parameterization and obtain estimates using estimating equations. In non-linear mixed models, heterogeneous covariance structures are frequently used, but typically with variance a function of the mean and constant correlations across subjects [3
]. Pourahmadi and Daniels [4
] develop a class of models they call dynamic conditionally linear mixed models in which the marginal covariance matrix is allowed to vary across individuals, but they consider the random effects covariance matrix to be constant across subjects. In the context of linear mixed models, Lin et al.
] examined heterogeneity in the within-individual variances in linear mixed models and Zhang and Weiss [6
] discussed heterogeneity in the random effects covariance matrix but mainly consider models that allow the entire matrix to differ by a multiplicative factor. Little work has been done on modelling the entire random effects covariance matrix.
Here, we propose an approach that allows all the parameters of the random effects covariance matrix to be modelled, not just the variances. Specifically, we will model the parameters that result from a modified Cholesky decomposition of the covariance matrix (this decomposition has also been called a modified Gaussian [7
]). This decomposition has been used previously to model the marginal covariance matrix of longitudinal observations on a subject [4
]. This decomposition results in parameters that can be easily modelled without concern of the resulting estimator being positive definite, have nice interpretations, and allow for relatively easy model fitting (sampling from the posterior distribution). Some discussion of interpretation in the context of modelling the random effects covariance matrix will be given in Section 2.
We reanalyse a data set previously discussed in Pourahmadi and Daniels [4
]. The data was from a series of five depression studies conducted in Pittsburgh from 1982–1992 [11
]. Patients were assigned one of two active treatments and measured at baseline and then weekly for 16-weeks. Here, we examine the rate of improvement of, and dependence in, weekly depression scores over this 16-week period for the 549 subjects with no missing baseline covariates. Previous work investigated the time to recovery from depression [11
] and examined the rate of improvement in depression scores with a different class of models than those proposed here [4
Preliminary analyses suggested that the random effects covariance matrix was not the same for subjects with different combinations of treatment (two levels: drug and psychotherapy versus psychotherapy only) and initial severity of depression (two levels: high and low). This motivated our modelling framework, in which we will model the parameters resulting from the modified Cholesky decomposition as a function of both drug (treatment) and severity.
In Section 2, we discuss the modified Cholesky decomposition of the covariance matrix, the interpretation of the parameters in this setting, and its use in modelling the random effects covariance matrix. We build models for the random effects covariance matrix on the depression data and discuss our results in Section 3. Conclusions and discussion comprise Section 4.