Efficiency Comparisons in Multivariate Multiple Regression with Missing Outcomes

We consider a follow-up study in which an outcome variable is to be measured at fixed time points and covariate values are measured prior to start of follow-up. We assume that the conditional mean of the outcome given the covariates is a linear function of the covariates and is indexed by occasion-specific regression parameters. In this paper we study the asymptotic properties of several frequently used estimators of the regression parameters, namely the ordinary least squares (OLS), the generalized least squares (GLS), and the generalized estimating equation (GEE) estimators when the complete vector of outcomes is not always observed, the missing data patterns are monotone and the data are missing completely at random (MCAR) in the sense defined by Rubin [11]. We show that when the covariance of the outcome given the covariates is constant, as opposed to the nonmissing data case: (a) the GLS estimator is more efficient than the OLS estimator, (b) the GLS estimator is inefficient, and (c) the semiparametric efficient estimator in a model that imposes linear restrictions only on the conditional mean of the last occasion regression can be less efficient than the efficient estimator in a model that imposes linear restrictions on the conditional means of all the outcomes. We provide formulae and calculations of the asymptotic relative efficiencies of the considered estimators in three important cases: (1) for the estimators of the occasion-specific means, (2) for estimators of occasion-specific mean differences, and (3) for estimators of occasion-specific dose-response model parameters.