Kurtosis-robust estimation of the fixed effects eta-squared: An impossibility theorem for F-based estimators and a practical correction.

Abstract

Eta-squared (η2) is the most widely reported proportion-of-variance effect size in ANOVA, yet its estimators are biased and non-normality is rarely studied. We prove that, for balanced one-way fixed effects ANOVA, no function of the usual F-statistic-the class containing the estimators in routine use-is exactly unbiased for fixed effects η2, though the non-centrality and Cohen's f2 admit exact unbiased estimation; this clarifies why popular 'unbiased' estimators are only approximate. We derive the leading-order, kurtosis-induced bias of the non-centrality-based estimator and propose a kurtosis-robust correction that uses an L-moment kurtosis estimate to remove this drift. Across Monte Carlo conditions and an out-of-sample heavy-tailed family, it attains the lowest root mean square error among those studied and markedly reduces kurtosis-induced bias drift.