Nonparametric Estimation of Random Effect Variance with Partial Information from the Clusters

This work is a new proposal for estimating the variance of the random effects in case the knowledge of the internal variability of the clusters is (or might be) assumed to be known. Here by clusters we mean, for instance, second-level units in multi-level models (schools, hospitals etc.), or subjects in repeated measure experiments. The proposed approach is useful whenever the variability of the response in a linear model can be viewed as the sum of two independent sources of variability, one that is common to all clusters and it is unknown, and another which is assumed to be available and it is clusterspecific. The responses here have to be thought as functions of the first-level observations, whose variability is known to depend only on the cluster's specificities. These settings include linear mixed models (LMM) when the estimators of the effects of interest are obtained conditionally on each cluster. The model may account for additional informations on the clusters, such as covariates, or contrast vectors. An estimator of the common source of variability is obtained from the residual deviance of the model, opportunely re-scaled, through the moment method. An iterative procedure is then suggested (whose initial step depends on the available information), that turns out to be a special case of the EM-algorithm.