This thesis provides an overview of linear mixed effects models commonly used in Biostatistics for analyzing repeated measurements, which include clustered and longitudinal measurements on patients or individuals often considered in clinical studies. In this thesis, we study the properties of the maximum likelihood estimators under the assumption of skew-normal distributions for the random effects and/or random errors in linear mixed models. We find that when the "true" random effects distributions are skewed, the assumption of a skew-normal distribution for the random effects provides more robust estimators of the model parameters in terms of smaller biases and mean squared errors.
We also study the empirical levels of the likelihood ratio test for testing the significance of the skewness parameters in the skew-normal distribution. Our Simulation study suggests that the likelihood ratio test provides approximately the correct level of significance under the null hypothesis that the underlying distribution is normal.