Testing Correlation in a Three-Level Model

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Abstract 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 $${\varvec{\Sigma }} \otimes {\varvec{\Psi }}_1 \otimes {\varvec{\Psi }}_2$$ Σ ⊗ Ψ 1 ⊗ Ψ 2 , where $${\varvec{\Sigma }}$$ Σ is an arbitrary positive definite covariance matrix, and $${\varvec{\Psi }}_1$$ Ψ 1 and $${\varvec{\Psi }}_2$$ Ψ 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.

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