Newton—Raphson and EM Algorithms for Linear Mixed-Effects Models for Repeated-Measures Data

Abstract

We develop an efficient and effective implementation of the Newton—Raphson (NR) algorithm for estimating the parameters in mixed-effects models for repeated-measures data. We formulate the derivatives for both maximum likelihood and restricted maximum likelihood estimation and propose improvements to the algorithm discussed by Jennrich and Schlüchter (1986) to speed convergence and ensure a positive-definite covariance matrix for the random effects at each iteration. We use matrix decompositions to develop efficient and computationally stable implementations of both the NR algorithm and an EM algorithm (Laird and Ware 1982) for this model. We compare the two methods (EM vs. NR) in terms of computational order and performance on two sample data sets and conclude that in most situations a well-implemented NR algorithm is preferable to the EM algorithm or EM algorithm with Aitken's acceleration. The term repeated measures refers to experimental designs where there are several individuals and several measurements taken on each individual. In the mixed-effects model each individual's vector of responses is modeled as a parametric function, where some of the parameters or “effects” are random variables with a multivariate normal distribution. This model has been successful because it can handle unbalanced data (different designs for different individuals), missing data (observations on all individuals are taken at the same design points, but some individuals have missing data), and jointly dependent random effects. The price for this flexibility is that the parameter estimates may be difficult to compute. We propose some new methods for implementing the EM and NR algorithms and draw conclusions about their performance. We also discuss extensions of the mixed-effects model to incoporate nonindependent conditional error structure and nested-type designs.

Key Words:

• Growth curve
• Longitudinal data
• Random effects