A monte-carlo Simulation study of the properties of Residual Maximum Likelihood (REML) estimators for the linear Gaussian mixed model

Abstract

The linear Gaussian mixed model is a tool box for analyzing experimental as well as non experimental designs in a flexible way elaborately. It is the model that contains mixtures of fixed effects as well as random effects. There are several ways to estimate fixed effects and variance components of the random effects. The most commonly used methods are Iterative Generalized Least Squares (IGLS) Estimation, Maximum Likelihood (ML) Estimation and Residual Maximum Likelihood (REML) Estimation. Of these methods many researchers prefer the REML method. This method is an iterative method thus its properties cannot be studied analytically. In the past simulation studies have been used only to study the properties of unbiasedness and efficiency of these REML estimators. These simulation studies have been of a small scale and usually have examined estimation of only either fixed or random effects but not both. Also the affect of varying sample size on the properties of the estimators have not been studied. Therefore the aim of this paper is to study the major desirable properties of estimators, namely, unbiasedness, consistency, sufficiency and efficiency for the REML method of estimation for both fixed and random effects for varying sample sizes and for varying ratios of variance of random effect to error variance. This was achieved by using an extensive Monte Carlo Simulation study. Code for this simulation study was developed using Java programming language. The results indicate that the Residual Maximum Likelihood estimation (REML) method holds all the desired properties for fixed effects. However for variance components of random effects and errors it does not hold the property of sufficiency and also though when the ratio of variance of random effects to error variance is small it holds the property of efficiency it is not so efficient when this ratio is large