A TWO-STAGE ESTIMATION PROCEDURE FOR LINEAR MIXED-EFFECTS MODELS

Abstract

Let y _i = (y _i1,…,y _{in i} be a vector of n_i responses from subject i, where i = 1,…,m. The y_i are assumed to follow the linear mixed effects model

where α is a vector of p unknown fixed effects parameters; X_i is a known matrix of dimension (n_i × p) linking α to y_i , β _i is a vector of q unobservable random effects; Z_i is a known matrix of dimension (n_i × q) linking β _i to Y_i , and e_i is a vector of n_i within individual random errors. We assume that the β _i 's and e_i 's are independent and identically distributed with means β and 0, respectively, and covariance matrices Σ and V_i , 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 e_i '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 e_i exist. A simulation study also shows that the two-stage estimators are not much inferior to the likelihood based estimators under the normal model.

Keywords:

• Consistent
• Method of moments
• Repeated measures
• Unbalanced data

Acknowledgments