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

Abstract

Eta‐squared ( η ² ) 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 η ² , though the non‐centrality and Cohen's f ² 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.