A likelihood-based coefficient for biomedical independence testing: the binomial-cut composite likelihood ratio

Abstract: The standard dependence summaries used in biomarker studies -- Pearson's r, Spearman's rho, Kendall's tau -- take values in [-1, 1] with 0 indicating no linear or monotone association. Zero does not distinguish independence from non-monotone dependence, so the scale cannot represent threshold effects, heteroscedasticity, and tail shifts common in biomarker practice. We formulate independence testing as a composite Bernoulli likelihood ratio: at each threshold t, comparing the Bernoulli laws of 1(Y <= t) conditionally on X versus marginally, aggregated over cut points. The resulting coefficient xi_cut lies on [0, 1] with 0 iff X and Y are independent (under continuity of Y) and 1 iff Y is a measurable function of X. Fisher weighting arises at second order from the Bernoulli likelihood, and xi_cut equals twice the threshold-averaged mutual information between X and 1(Y <= t), giving a distribution-free lower bound on I(X; Y). A second-order expansion recovers the Fisher-weighted Dette-Siburg-Stoimenov measure, which coincides under continuity with Chatterjee's rank correlation. Estimation uses a Nadaraya-Watson plug-in with a max-over-grid bandwidth; inference is by exact permutation. In biomarker-motivated simulations T_cut substantially outperforms rank-based coefficients on W-shaped non-monotone and heteroscedastic alternatives. We illustrate on the Seattle cohort (n=70, ages 21-88) of the aging plasma proteome dataset, screening all 1,305 proteins for age dependence: under Benjamini-Hochberg control at q<0.05, T_cut rejects on 70 proteins, six of which are missed by Pearson, Spearman, and Chatterjee at the same FDR level.