Efficient algorithms for robust estimation in linear mixed-effects models using the multivariate t distribution

Linear mixed-effects models are frequently used to analyze repeated measures data. because they model flexibly the within-subject correlation often present in this type of data. The most popular linear mixed-effects model for a continuous response assumes normal distributions for the random effects and the within-subject errors, making it sensitive to outliers. Such outliers are more problematic for mixed-effects models than for fixed-effects models, because they may occur in the random effects, in the within-subject errors, or in both, making them harder to be detected in practice. Motivated by a real dataset from an orthodontic study, we propose a robust hierarchical linear mixed-effects model in which the random effects and the within-subject errors have multivariate t-distributions, with known or unknown degrees-of-freedom, which are allowed to vary with groups of subjects. By using a gamma-normal hierarchical structure, our model allows the identification and classification of both types of outliers, comparing favorably to other multivariate t models for robust estimation in mixed-effects models previously described in the literature, which use only the marginal distribution of the responses. Allowing for unknown degrees-of-freedom, which are estimated from the data. our model provides a balance between robustness and efficiency, leading to reliable results for valid inference. We describe and compare efficient EM-type algorithms, including ECM, ECME, and PX-EM, for maximum likelihood estimation in the robust multivariate t model. We compare the performance of the Gaussian and the multivariate t models under different patterns of outliers. Simulation results indicate that the multivariate t substantially outperforms the Gaussian model when outliers are present in the data, even in moderate amounts.