Statistica Sinica 17(2007), 929-943

MAXIMUM LIKELIHOOD INFERENCE IN ROBUST

LINEAR MIXED-EFFECTS MODELS USING

MULTIVARIATE $t$ DISTRIBUTIONS

Peter X.-K. Song$^1$, Peng Zhang$^2$ and Annie Qu$^3$

$^1$University of Waterloo, $^2$University of Alberta and $^3$Oregon State University

Abstract: This paper focuses on the problem of maximum likelihood estimation in linear mixed-effects models where outliers or unduly large observations are present in clustered or longitudinal data. Multivariate $t$ distributions are often imposed on either random effects and/or random errors to incorporate outliers. A powerful algorithm of maximum by parts (MBP) proposed by Song, Fan and Kalbfleisch (2005) is implemented to obtain maximum likelihood estimators when the likelihood is intractable. The computational efficiency of the MBP allows us to further apply a profile-likelihood technique for the estimation of the degrees of freedom in $t$-distributions. Comparison of the Akaike information criterion (AIC) among candidate models provides an objective criterion to determine whether outliers are influential on the quality of model fit. The proposed models and methods are illustrated through both simulation studies and data analysis examples, with comparison to the existing EM-algorithm.

Key words and phrases: AIC, EM-algorithm, Gauss-Newton algorithm, longitudinal data, MBP algorithm, outlier.