What Is REML?

• Activity Number: 301
• Type: Contributed
• Date/Time: Tuesday, July 31, 2012 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Christopher Gotwalt*+
• Companies: SAS Institute
• Keywords: REML ; Mixed Model ; Firth Adjustment ; Variance Components

Abstract

Restricted maximum likelihood (REML) estimates of linear mixed model parameters have been shown many times via simulation to have desirable statistical properties. Of particular renown is its reduced bias relative to the MLE. Existing derivations of REML have motivated it as a marginal likelihood, a likelihood of residuals, or as a maximum posteriori estimator with Jeffrey's prior on the fixed effects and a flat prior on the variance parameters. None of these derivations clearly explain its excellent bias properties. In this presentation, we show that REML is an instance of a Firth adjusted MLE. Because the Firth adjustment annihilates the first term of the asymptotic expansion of the bias of the MLE, this explains REML's excellent bias properties and establishes a unified framework for estimating variance parameters and fixed effects. Interestingly, we show that this is only the case for linear covariance structures, and suggest a new, as yet unexplored, estimator that may have superior statistical properties to REML for nonlinear correlation structures.