Estimation for generalized linear models when covariates are subject-specific parameters in a mixed model for longitudinal measurements

In many studies, a primary endpoint and longitudinal measures of a continuous response are collected for each participant, and the association between the primary endpoint and features of the longitudinal profiles is of interest. A relevant framework assumes that the longitudinal data follow a linear mixed model whose random effects are covariates in a generalized linear model for the primary endpoint. Naive implementation by imputing random effects from individual regression fits yields biased inference, and several methods for reducing this bias have been proposed. However, these methods require normality assumption on the random effects, which may be unrealistic. Adapting a strategy of Stefanski and Carroll (1987 Biometrika 74:703--716), we propose a conditional estimation approach in which estimators require no assumptions on the random effects and yield consistent inference regardless of the true random effects distribution. This approach is straightforward and fast to implement. However, it can not give insight into the features of the random effects and might be less efficient relative to full likelihood approach. We further propose a semiparametric full likelihood approach which approximates the random effects distribution by the seminonparametric density representation of Gallant and Nychka (1987 Econometrica 55: 363--390). This approach requires only the assumption that the random effects have a smooth but unspecified density. When the primary endpoint is normally distributed given the random effects, implementation is straightforward via optimization techniques. EM algorithm is used for implementation when the primary endpoint is not normal given the random effects and it involves increased computational burden. The performance of both approaches is demonstrated via simulation and by application to a study of bone mineral density in peri-menopausal women. Results show that, in contrast to methods predicated on normality assumption for the random effects, the approaches yield valid inferences under departures from this assumption and are competitive when the assumption holds. The semiparametric likelihood approach shows potential of improved efficiency.