Likelihood based inference for skew-normal independent linear mixed models

Abstract

Linear mixed models with normally distributed response are routinely used in longitudinal data. However, the accuracy of the assumed normal distribution is crucial for valid inference of the parameters. We present a new class of asymmetric linear mixed models that provides an efficient estimation of the parameters in the analysis of longitudinal data. We assume that, marginally, the random effects follow a multivariate skew-normal/independent distribution [M. D. Branco and D. K. Dey, J. Multivariate Anal. 79, No. 1, 99–113 (2001; Zbl 0992.62047)] and that the random errors follow a symmetric normal/independent distribution [K. Lange and J. S. Sinsheimer, J. Comput. Graph. Stat. 2, 175–198 (1993)], providing an appealing robust alternative to the usual symmetric normal distribution in linear mixed models. Specific distributions examined include the skew-normal, the skew-t, the skew-slash, and the skew-contaminated normal distribution. We present an efficient EM-type algorithm for the computation of maximum likelihood estimation of the parameters. The technique for the prediction of future responses under this class of distributions is also investigated. The methodology is illustrated through an application to the Framingham cholesterol data and a simulation study.