Multivariate distributions with correlation matrices for nonlinear repeated measurements

The polynomial growth curve model based on the multivariate normal distribution has dominated the analysis of continuous longitudinal repeated measurements for the last 50 years. The main reasons include the ease of modelling dependence because of the availability of the correlation matrix and the linearity of the regression coefficients. However, a variety of other useful distributions also involve a correlation matrix: the multivariate Student's t, multivariate power exponential, and multivariate skew Laplace distributions as well as Gaussian copulas with arbitrarily chosen marginal distributions. With modern computing power and software, nonlinear regression functions can be fitted as easily as linear ones. By a number of examples, we show that these distributions, combined with nonlinear regression functions, generally yield an improved fit, as compared to the standard polynomial growth curve model, and can provide different conclusions.