Distribution Theory for Glass's Estimator of Effect size and Related Estimators

TLDR

Glass’s estimator of effect size, defined as the mean difference divided by the sample standard deviation, is examined within an explicit statistical model, with attention to the impact of measurement invalidity. The authors derive the exact distribution of Glass’s estimator, demonstrate its small‑sample bias, present a minimum‑variance unbiased estimator with uniformly lower variance, provide a correction for measurement‑error attenuation, and compute optimal weighting expressions for the most precise weighted effect‑size estimate.

Abstract

Glass's estimator of effect size, the sample mean difference divided by the sample standard deviation, is studied in the context of an explicit statistical model. The exact distribution of Glass's estimator is obtained and the estimator is shown to have a small sample bias. The minimum variance unbiased estimator is obtained and shown to have uniformly smaller variance than Glass's (biased) estimator. Measurement error is shown to attenuate estimates of effect size and a correction is given. The effects of measurement invalidity are discussed. Expressions for weights that yield the most precise weighted estimate of effect size are also derived.