High-Breakdown Inference for Mixed Linear Models

Mixed linear models are used to analyze data in many settings. These models have a multivariate normal formulation in most cases. The maximum likelihood estimator (MLE) or the residual MLE (REML) is usually chosen to estimate the parameters. However, the latter are based on the strong assumption of exact multivariate normality. Welsh and Richardson have shown that these estimators are not robust to small deviations from multivariate normality. This means that in practice a small proportion of data (even only one) can drive the value of the estimates on their own. Because the model is multivariate, we propose a high-breakdown robust estimator for very general mixed linear models that include, for example, covariates. This robust estimator belongs to the class of S-estimators, from which we can derive asymptotic properties for inference. We also use it as a diagnostic tool to detect outlying subjects. We discuss the advantages of this estimator compared with other robust estimators proposed previously and illustrate its performance with simulation studies and analysis of three datasets. We also consider robust inference for multivariate hypotheses as an alternative to the classical F-test by using a robust score-type test statistic proposed by Heritier and Ronchetti, and study its properties through simulations and analysis of real data.