February 21: Douglas Bates, University of Wisconsin

UNIVERSITY OF MINNESOTA

SEMINAR

School of Statistics
College of Liberal Arts

Computing REML or ML Estimates for Linear or Nonlinear Mixed-Effects Models

Douglas Bates
Department of Statistics
University of Wisconsin

Monday, February 21, 2000
4:00-5:00 PM, Room 150 Ford Hall
Social at 3:30 PM in Room 300 Ford Hall

Abstract

Statistical models for the analysis of data from designed experiments often incorporate terms for both fixed effects, which characterize population behaviour, and random effects, which characterize differences between individual subjects. Traditional analysis of variance approaches to estimating parameters in such models are difficult to implement (and to understand) and are fragile with respect to missing data. Modern, direct approaches to such models use maximum likelihood (ML) or restricted maximum likelihood (REML) estimation. Optimizing the estimation criterion with respect to the parameters can be a formidable task, especially when there is a large amount of data.

We describe several methods of streamlining the optimization of the REML and/or the maximum-likelihood parameter estimation criteria for linear mixed-effects models. First, we condition on the ratio of the variance components or, in general, the relative precision factors. Conditional on these values, the criteria can be easily calculated after a penalized least squares problem, which is a non-iterative calculation. When optimizing this profiled log-likelihood or restricted log-likelihood with respect to the parameters in the relative precision factor, the dimension of the optimization problem is considerably reduced. By taking advantage of the special structure of the penalized least squares problem the amount of information that must be stored and the amount of computation to be performed can be substantially reduced. Second, we use the matrix-logarithm parameterization for the relative precision factors to provide a more stable optimization problem. Third, we derive both the EM and the Newton-Raphson updates so the optimization method can begin with the EM iterations then switch to the Newton-Raphson updates when near the optimum.

All of these techniques can also be used in the optimization of the estimation criteria in nonlinear mixed-effects models. These methods are implemented in the 3.1 version of the NLME library for the S-PLUS and R statistical software packages.

(Joint work with Jose Pinheiro, Bell Labs, Lucent Technologies)