An alternative parameterization of the general linear mixture model for longitudinal data with non-ignorable drop-outs

This paper considers the mixture model methodology for handling non-ignorable drop-outs in longitudinal studies with continuous outcomes. Recently, Hogan and Laird have developed a mixture model for non-ignorable drop-outs which is a standard linear mixed effects model except that the parameters which characterize change over time depend also upon time of drop-out. That is, the mean response is linear in time, other covariates and drop-out time, and their interactions. One of the key attractions of the mixture modelling approach to drop-outs is that it is relatively easy to explore the sensitivity of results to model specification. However, the main drawback of mixture models is that the parameters that are ordinarily of interest are not immediately available, but require marginalization of the distribution of outcome over drop-out times. Furthermore, although a linear model is assumed for the conditional mean of the outcome vector given time of drop out, after marginalization, the unconditional mean of the outcome vector is not, in general, linear in the regression parameters. As a result, it is not possible to parsimoniously describe the effects of covariates on the marginal distribution of the outcome in terms of regression coefficients. The need to explicitly average over the distribution of the drop-out times and the absence of regression coefficients that describe the effects of covariates on the outcome are two unappealing features of the mixture modelling approach. In this paper we describe a particular parameterization of the general linear mixture model that circumvents both of these problems.