A Novel Measure of Association for Multivariate Data with Applications

Multivariate linear correlation analysis plays a crucial role in statistical research, with a variety of measures used to assess associations among variables. The adjusted squared coefficient of multiple correlation, R², is a widely used metric but depends on the choice of the response variable. Another common measure, the determinant of the correlation matrix, det(ρₓ), or its complement, 1-det(ρₓ), evaluates overall linear dependence but is highly sensitive to dimensionality. We propose a novel Frobenius norm based correlation coefficient, C², which provides a response-invariant alternative for evaluating linear association. Unlike R², C² is response‑invariant and exhibits reduced sensitivity to dimensionality compared with determinant based measures such as 1-det(ρₓ). The proposed method is not intended to replace existing approaches but rather to complement them by offering an alternative measure of linear correlation. This paper focuses exclusively on linear association, without extending to nonlinear dependence. We establish theoretical properties of C² and evaluate its point and confidence interval estimators through bootstrap simulations. Comparative analyses demonstrate that C² provides a response-invariant summary of linear association and exhibits qualitatively similar behavior to R² and 1 − det(ρₓ) under the simulation settings considered. Applications of C² include its use as a replacement for R² in the variance inflation factor for sample size adjustment, biomarker association analysis in a vaccine trial, and independence testing via C² = 0. Overall, C² provides a flexible, model‑free measure of linear association that mitigates the extreme dimensional sensitivity inherent in determinant based approaches.