Evaluation of the method R procedure for one-way random effects models

Abstract

Method R (MR) is procedure (Reverter et al., 1994) for variance component estimation. It is best applied in situations where the computation of the preferred methods of ML and REML is infeasible or impossible to implement. Computationally, MR requires only the empirical best linear unbiased predictors (EBLUP's) for the random effects from the mixed model equations (MME). A single MR estimate for a variance component is the value for which the slope of the linear regression of the random effects EBLUP's from all the data, on the random effects EBLUP's from a sub-sample of the data, equals its expected value. Typically, the median of MR values obtained from repeated sub-sampling of the data is used. To date, properties of the MR estimator are still poorly understood. Our investigation of balanced and unbalanced one-way random effects models reveals MR estimators to be conditional REML estimates based on the whole and sub-sample means in the balanced case, but not for the unbalanced case. Simulations of MR for the one-way balanced random effects model demonstrate the robustness of the median estimator of MR estimates for multiple sub-samples. The large sample variance of the MR estimator for the balanced one-way random effects model is decomposed into two components, the true MR variance and variance due to sub-sampling. From the derived expressions, it is shown that for the intra-class correlation coefficient, the asymptotic efficiency MR relative to the MLE is 1. Our large sample results also demonstrate that the sub-sampling variance for 50% sub-sampling, is not a suitable estimator for the true MR variance in general, contrary to the assumption used in Mallinckrodt et.al. (1997). The Bivariate MR procedure proposed by Reverter (1994) is shown to be equivalent to performing a multivariate regression of EBLUP's from all the data on EBLUP's from a sub-sample of the data. For the bivariate one-way random effects model, the MR procedure provides a conditional REML estimator for matrix of regression coefficients.