Christopher Gotwalt, Director of Statistical Research and Development, SAS Institute

What is REML? Why Does it Work? And How Do We Extend it to the Generalized Linear Mixed Model?

Restricted Maximum Likelihood (REML) estimates of the linear mixed model parameters is one of the most commonly used methods in statistics because REML estimates of variance components have lower bias than Maximum Likelihood estimates.  Outside of balanced cases, the reason why it works so well has been in many ways a mystery.  We present a new derivation for the REML method for the estimating the parameters of the linear mixed model.  We demonstrate that in the case of variance models the REML estimator is an instance of a Firth adjusted estimator, a technique that reduces the bias of maximum likelihood estimators.  The new derivation is advantageous because it suggests a new method for reducing the small sample bias of variance estimators in the generalized linear mixed model (GLMM). It is well known that in small samples the standard methods of estimating variance parameters in GLMM’s are biased downward, leading  to unacceptably high Type I error rates in tests of the fixed effects in these models.  We apply the Firth technique to the one and two treatment logistic regression model with simple random effects.  We demonstrate via simulation that, relative to estimators using pseudo likelihood and quadrature based maximum likelihood, the Firth estimates are both less biased and the Type I error rate of tests on the fixed effects is superior.

Bio

Chris studied mathematics at the University of Florida, graduating in 1997 and in 2003 received his Ph.D. in Statistics at NCSU under the guidance of his advisor, Dennis Boos.  After graduation he joined the JMP development team as a statistical software developer, focusing primarily on statistical computing.  While working as a software developer for JMP, Chris developed numerical algorithms for a wide variety of statistical applications including linear mixed models, optimal design of experiments, parametric time to event modeling, and neural networks.  Chris took on management responsibilities in 2008 and is now the Director of Statistical Research and Development for JMP.   Chris lives in Raleigh, NC, with his wife Jessica. 

2015 QPRC