The relationship between a primary endpoint and features of longitudinal profiles of a continuous response is often 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. Methods proposed in literature require a parametric (normality) assumption on the random effects, which may be unrealistic. We propose a conditional likelihood approach which requires no assumptions on the random effects and a semiparametric full likelihood approach which requires only the assumption that the random effects have a smooth density. It is straightforward and fast to implement the conditional likelihood approach. EM algorithm is used in general for implementation of the semiparametric full likelihood approach and it involves increased computational burden. 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 some efficiency gains over the other methods and provides estimates for the underlying random effects distribution. We also illustrate the performance of the approaches by application to a study of bone mineral density and longitudinal progesterone levels in 624 women transitioning to menopause in which investigators wished to understand the association between osteopenia, characterized by bone mineral density at or below the 33rd percentile, and features of hormonal patterns over the menstrual cycle in peri-menopausal women. Data analysis results obtained from the approaches offer the analyst assurance of credible estimation of the relationship.
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