A TWO-STAGE ESTIMATION PROCEDURE FOR LINEAR MIXED-EFFECTS MODELS
Let y i = (y i1,…,y in i be a vector of ni responses from subject i, where i = 1,…,m. The yi are assumed to follow the linear mixed effects model where α is a vector of p unknown fixed effects parameters; Xi is a known matrix of dimension (ni × p) linking α to yi , β i is a vector of q unobservable random effects; Zi is a known matrix of dimension (ni × q) linking β i to Yi , and ei is a vector of ni within individual random errors. We assume that the β i 's and ei 's are independent and identically distributed with means β and 0, respectively, and covariance matrices Σ and Vi , respectively. This kind of model characterizes the common structure of repeated measures, growth curve, or longitudinal data. The maximun likelihood estimator for α and the restricted maximun likelihood estimators of the variance components under the normal model are usually determined by Newton-Raphson or EM algorithms. The computations of these algorithms are not simple. In this paper, we propose a computationally simple two-stage procedure, which unlike the likelihood based procedures, does not require the normality assumptions for the β i 's and the ei 's. It is shown that the two-stage estimators for the parameters of the model are consistent estimators under the assumption that the fourth moments of the components of β i and ei exist. A simulation study also shows that the two-stage estimators are not much inferior to the likelihood based estimators under the normal model.