Regression analysis when covariates are regression parameters of a random effects model for observed longitudinal measurements

We consider regression analysis when covariate variables are the underlying regression coefficients of another linear mixed model. A naive approach is to use each subject's repeated measurements, which are assumed to follow a linear mixed model, and obtain subject-specific estimated coefficients to replace the covariate variables. However, directly replacing the unobserved covariates in the primary regression by these estimated coefficients may r esult in a significantly biased estimator. The aforementioned problem can b e evaluated as a generalization of the classical additive error model where repeated measures are considered as replicates. To correct for these biase s, we investigate a pseudo-expected estimating equation (EEE) estimator, a regression calibration (RC) estimator, and a refined version of the RC esti mator. For linear regression, the first two estimators are identical under certain conditions. However, when the primary regression model is a nonline ar model, the RC estimator is usually biased. We thus consider a refined re gression calibration estimator whose performance is close to that of the ps eudo-FEE estimator but does not require numerical integration. The RC estim ator is also extended to the proportional hazards regression model. In addi tion to the distribution theory, we evaluate the methods through simulation studies. The methods are applied to analyze a real dataset from a child gr owth study.