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We consider the set Bp of parametric block correlation matrices with p blocksof various (and possibly different) sizes, whose diagonal blocks are compoundsymmetry (CS) correlation matrices and off-diagonal blocks are constantmatrices. Such matrices appear in probabilistic models on categorical data,when the levels are partitioned in p groups, assuming a constant correlationwithin a group and a constant correlation for each pair of groups. We obtaintwo necessary and sufficient conditions for positive definiteness of elementsof Bp. Firstly we consider the block average map $\phi$, consisting inreplacing a block by its mean value. We prove that for any A $\in$ Bp , A ispositive definite if and only if $\phi$(A) is positive definite. Hence it isequivalent to check the validity of the covariance matrix of group means, whichonly depends on the number of groups and not on their sizes. This theorem canbe extended to a wider set of block matrices. Secondly, we consider the subsetof Bp for which the between group correlation is the same for all pairs ofgroups. Positive definiteness then comes down to find the positive definiteinterval of a matrix pencil on Sp. We obtain a simple characterization bylocalizing the roots of the determinant with within group correlation values.

On the validity of parametric block correlation matrices with constant within and between group correlations