An Improved Satterthwaite (1941, 1946) Effective df Approximation

Abstract

This study introduces a correction to the approximation of effective df as proposed by Satterthwaite, specifically addressing scenarios where component df are small. The correction is grounded in analytical results concerning the moments of standard normal random variables. This modification is applicable to complex variance estimates that involve both small and large df, offering an enhanced approximation of the higher moments required by Satterthwaite’s framework. Additionally, this correction extends and partially validates the empirically derived adjustment by Johnson and Rust, as it is based on theoretical foundations rather than simulations used to derive empirical transformation constants. Finally, the proposed adjustment also provides a correction to the estimate of the total variance in cases missing data have been replaced by multiple imputations such as in the case of plausible values in national and international large scale assessments.