Linear mixed models and penalized least squares.pdfVIP

Linear mixed models and penalized least squares Douglas M. Bates Department of Statistics, University of Wisconsin–Madison Saikat DebRoy Department of Biostatistics, Harvard School of Public Health Abstract Linear mixed-effects models are an important class of statistical models that are not only used directly in many fields of applications but also used as iterative steps in fitting other types of mixed-effects models, such as generalized linear mixed models. The parameters in these models are typically estimated by maximum likelihood (ML) or restricted maximum likelihood (REML). In general there is no closed form solution for these estimates and they must be determined by iterative algorithms such as EM iterations or general nonlinear optimization. Many of the intermediate calculations for such iterations have been expressed as generalized least squares problems. We show that an alternative representation as a penalized least squares problem has many advantageous computational properties including the ability to evaluate explicitly a profiled log-likelihood or log-restricted likelihood, the gradient and Hessian of this profiled objective, and an ECME update to refine this objective. Key words: REML, gradient, Hessian, EM algorithm, ECME algorithm, maximum likelihood, profile likelihood, multilevel models 1 Introduction We will establish some results for the penalized least squares representation of a general form of a linear mixed-effects model, then show how these results specialize for particular models. The general form we consider is 2 2 −1 y = Xβ + Zb + ∼ N (0, σ I ), b ∼ N (0, σ Ω ), ⊥ b (1) where y is the n-dimensional response vector, X is a