Estimation for Generalized Linear Models When Covariates Are Subject-specific Parameters in a Mixed Model for Longitudinal Measurements

Abstract

In many studies, a primary endpoint and longitudinal measures of a continuous response are collected for each participant along with other covariates, and the association between the primary endpoint and features of the longitudinal profiles is of interest. One challenge is that the features of the longitudinal profiles are observed only through the longitudinal measurements, which are subject to measurement error and other variation.

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 subject-specific effects from individual regression fits yields biased inference, and several methods for reducing this bias have been proposed, which include regression calibration, pseudo-expected estimating equations, and refined regression calibration (for logistic primary model only). However, these methods require a parametric (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 for the generalized linear model parameters require no assumptions on the random effects and yield consistent inference regardless of the true distribution of random effects. This approach is straightforward and fast to implement. However, this approach can not give insight into the features of the random effects and inference on the longitudinal data process is not available as it conditions away the random effects. Furthermore, conditional estimation approach might be less efficient relative to full likelihood approach.

In order to estimate the distribution of random effects and to improve efficiency, 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 very mild assumption that the random effects have a smooth but unspecified density. When the primary endpoint is normally distributed given the random effects, implementation of this approach is straightforward via optimization techniques. EM algorithm is used for implementation of this approach when the primary endpoint does not follow a normal model given the random effects and it involves increased computational burden.

The performance of both approaches is demonstrated via simulation. Results show that, in contrast to methods predicated on a parametric (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 full likelihood approach shows the potential of efficiency gains over other methods. We also illustrate the performance of the approaches by application to a study of bone mineral density and longitudinal progesterone levels in peri-menopausal women. Data analysis results obtained from the approaches offer the analyst assurance of credible estimation of the relationship.