Testing Correlation in a Three-Level Model

In this paper, we present a statistical approach to evaluate the relationship between variables observed in a two-factors experiment. We consider a three-level model with covariance structure \({\varvec{\Sigma }} \otimes {\varvec{\Psi }}_1 \otimes {\varvec{\Psi }}_2\), where \({\varvec{\Sigma }}\) is an arbitrary positive definite covariance matrix, and \({\varvec{\Psi }}_1\) and \({\varvec{\Psi }}_2\) are both correlation matrices with a compound symmetric structure corresponding to two different factors. The Rao’s score test is used to test the hypotheses that observations grouped by one or two factors are uncorrelated. We analyze a fermentation process to illustrate the results. Supplementary materials accompanying this paper appear online.