Repeated measures data describe multiple measurements taken from the same experimental unit under the different treatment conditions. In particular, researches with repeated measures data in various fields such as health and behavioral sciences, education, and psychology have an important role in applied statistics. There are many methods used to analyze the results of research designs planned with these measurements. The most important difference between these methods is the assumptions on which the models are based. One of the most important assumptions needed by classical methods is the normality assumption. Many methods are valid under the assumption of normality. However, it is not always possible to hold this assumption in applications. For this reason, in the analysis of repeated measures data, different distributions are necessary that can provide flexibility beyond the normal distribution, especially in cases where the assumption of normality does not hold. In this study, it is proposed to use Multivariate Laplace distribution (MLD) which is an alternative distribution in cases where normality assumption does not hold by examining the multivariate variance analysis model (MANOVA). Under MLD assumption, the parameter estimates for the Two-way Repeated Measures MANOVA model are carried out with the maximum likelihood (ML) estimation and ML estimates are obtained via the EM Algorithm.