A Partial Correlation Coefficient and Coefficient of Determination for Multivariate Normal Repeated Measures Data

In a regression setting, the partial correlation coefficient is often used as a measure of ‘standardized’ partial association between the outcome y and each of the covariates in x′= (x1,...,xK). In a linear regression model estimated by using ordinary least squares, with y as the response, the estimated partial correlation coefficient between y and xk can be shown to be a monotone function of the z-statistic for testing whether the regression coefficient of xk is 0; it can also be shown to be a monotone function of the likelihood ratio statistic. When y is normal but the data are clustered so that y and x are obtained from each member of a cluster, maximum likelihood using a multivariate normal distribution is often used to estimate the regression parameters of the model for y given x. For clustered data, we propose two measures of partial association: the exact same monotone function of the repeated measures’z-statistic that is used to form the partial correlation coefficient in the normal linear regression setting and the exact same monotone function of the likelihood ratio statistic that is used to form the partial correlation coefficient in the normal linear regression setting. We also propose an R2-like measure which can be used to measure the overall predictive ability of the model. To illustrate the method, we use a longitudinal study concerning the stress on the wall of the heart from chemotherapy in children with leukaemia.