Sufficient dimension reduction has been extensively explored in the context of independent and identically distributed data. In this article we generalize sufficient dimension reduction to longitudinal data and propose an estimating equation approach to estimating the central mean subspace. The proposed method accounts for the covariance structure within each subject and improves estimation efficiency when the covariance structure is correctly specified. Even if the covariance structure is misspecified, our estimator remains consistent. In addition, our method relaxes distributional assumptions on the covariates and is doubly robust. To determine the structural dimension of the central mean subspace, we propose a Bayesian-type information criterion. We show that the estimated structural dimension is consistent and that the estimated basis directions are root-[Formula: see text] consistent, asymptotically normal and locally efficient. Simulations and an analysis of the Framingham Heart Study data confirm the effectiveness of our approach.
Keywords: Central mean subspace; Dimension reduction; Estimating equation; Longitudinal data; Semiparametric efficiency; Sliced inverse regression.