Scottish and Northumbrian statisticians' meeting: 20 May 2005, University of Newcastle upon Tyne
Random-Effects Models for Multivariate Repeated Responses
G. Verbeke and S. Fieuws
Biostatistical Centre K.U.Leuven, Leuven
In the context of longitudinal data or repeated measurements, research questions are often formulated which require joint modelling of multivariate response vectors measured repeatedly within the participating subjects. Examples include longitudinal studies where many indices are measured over time and where an overall assessment of timetrends is needed or where classification of subjects based on multivariate longitudinal profiles is of interest. Other examples can be found in clustered settings, e.g. a questionnaire measuring different concepts, each by a set of items.
There are number of possible approaches to extend models for one repeatedly measured outcome to multivariate settings. We will focuss on a random-effects approach, where a mixed model is assumed for each outcome separately, and where the joint model arises from assuming a joint (multivariate) distribution for all random effects. This model is very flexible in the sense that it does not assume the outcomes to be measured the same number of times, nor at the same time points. This approach also allows combining outcomes and/or models of a different nature: Continuous and categorical outcomes; linear, generalized linear, and non-linear models.
Nevertheless, there is also a disadvantage, especially when high-dimensional outcome vectors need to be analysed, which is the case in the examples we will discuss. A new pairwise model fitting approach will be presented which can circumvent this problem, and which is applicable whatever dimensionality of the problem. In this approach, the likelihoods of all pairwise models (each model involves two outcomes) are first maximised separately instead of the likelihood of the full multivariate model. In the second step, parameter estimates are obtained by averaging over all pairs. Borrowing ideas from the pseudolikelihood framework, standard errors can be calculated and inferences become readily available. The approach will be illustrated using several examples, and the statistical properties of the estimators and associated standard errors will be evaluated using a simulation study.