Eigenvalue-Based Diagnostic Equity Testing: A Random Matrix Framework for Detecting Multi-Dimensional Performance Disparities in Clinical Classifiers

Evaluating whether clinical classifiers perform equitably across patient subgroups is a central requirement for the responsible deployment of machine learning in medicine. Conventional approaches test one fairness metric at a time, such as sensitivity, positive predictive value, or area under the receiver operating characteristic curve, and therefore cannot detect disparities that manifest only in the joint structure of a group-specific confusion matrix. We develop a unified hypothesis-testing framework rooted in random matrix theory that compares demographic groups through the L2 distance between their joint eigenvalue densities, yielding a scalar spectral divergence that is sensitive to every cell of the 2×2 confusion matrix simultaneously. We derive the closed-form spectral divergence for Gaussian-approximated eigenvalue densities, prove almost-sure consistency of the empirical estimator via the delta method, and construct an extreme-value (Gumbel) test statistic with family-wise error rate control. Monte Carlo experiments comprising 10,000 replications across balanced, moderately imbalanced, and severely imbalanced group-size regimes show that the spectral test keeps Type I errors close to its nominal level while achieving power exceeding 90% in complex and multi-dimensional violation scenarios, where the best single-metric competitor reaches at most 63%. Three clinical benchmark datasets from the UCI Machine Learning Repository utilised include Pima Indians Diabetes (n=768), Cleveland Heart Disease (n=303), and Heart Failure Clinical Records (n=299). Results confirm that the spectral method detects statistically significant (p<0.001) performance disparities missed by all three conventional tests. These results support eigenvalue-based divergence as a practical, model-agnostic diagnostic equity tool for clinical machine learning audits.