Meta-analytic entropy: information-theoretic heterogeneity beyond I-squared

Abstract

Can information-theoretic metrics detect heterogeneity features that I-squared is blind to, namely multimodality, skewness, and distributional shape? We computed Shannon entropy, Kullback-Leibler divergence, effect-precision mutual information, Fisher information, and a Normalized Entropy Index for 403 Cochrane reviews from Pairwise70. Estimation used Monte Carlo sampling with 10,000 draws per review; the index compared mixture entropy against theoretical bounds, while Silverman kernel density counted modes on precision-weighted distributions. The median Normalized Entropy Index was 0.151 (IQR 0.109–0.229; 95% bootstrap CI 0.143–0.163), and 44.7% of reviews were multimodal despite I-squared below 60%. Because such estimates grow noisier below ten studies, we stratified by k; multimodality rose from 33% (k<10) to 62% (k≥10), a real signal, not a small-sample artefact. Effect-precision mutual information was positive in every computable review and significant by permutation in 21.5%, capturing dependence consistent with, but not proving, small-study mechanisms. The Normalized Entropy Index (0–1) thus captures distributional shape that complements, rather than replaces, variance-based heterogeneity measures.