Estimation of the parameters of the extended growth curve model under multivariate skew normal distribution

Abstract

The Growth Curve Model (GCM) assumes the same shape of profiles for each group, where group means are assumed to be represented by polynomials of the same degree. The model, therefore, is inappropriate when analyzing data from studies involving groups with mean growth curves represented by different shapes. We consider the Extended Growth Curve Model (EGCM), which is a model that allows the group means to follow different degrees of polynomials over time. Existing inference on EGCM assumes multivariate normal errors and produces estimators that are not optimal, when used in the analysis of data with skewed distributions. In this paper, we consider the multivariate skew normal (MSN) distribution as the underlying distribution for the EGCM and provide estimators for its mean and covariance parameters. We adopted the Restricted Expectation–Maximization (REM) algorithm, which is based on the multivariate Newton–Raphson (NR) method and Lagrangian optimization. However, the multivariate NR method and the existing REM algorithm are only applicable to vector parameters and the parameters of interest in this study are matrices. We, therefore, extended the NR approach to matrix parameters, that consequently allowed us to extend the REM algorithm to matrix estimators. The performance of the proposed estimators was examined using extensive simulations and a real data example was also considered to illustrate the application of our proposed estimators.