Some symmetric, invariant measures of multivariate association

Cramer & Nicewander (1979) — measures of multivariate association

Result

the symmetric, linearly invariant measures of association between two sets of variables are all functions of the canonical correlations, generalize $R^2$, and are strictly ordered for any two sets.

Use

lineage of $\eta_D=(\prod\rho_i^2)^{1/d}$ as a symmetric invariant measure; van den Burg & Lewis (1988), Psychometrika 53(1):109–122, compare the $\Lambda$-based and Pillai-based indices.

Verification

abstract read (primary text, Crossref); whether the geometric mean itself is in their list is unread (paywalled, gaps.md), 5 Sep 2026.