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

TL;DR

This paper characterizes the positive definiteness of parametric block correlation matrices with constant within and between group correlations, providing practical criteria based on group means and matrix pencils.

Contribution

It establishes necessary and sufficient conditions for positive definiteness of such matrices, simplifying verification through group mean matrices and matrix pencil analysis.

Findings

Positive definiteness is equivalent to the positive definiteness of the block average matrix.

The validity depends only on the number of groups, not their sizes.

A simple characterization of positive definiteness is obtained via roots of the determinant.

Abstract

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