An Improved Satterthwaite Effective Degrees of Freedom Correction for Weighted Syntheses of Variance

Abstract

This article presents an improved approximation for the effective degrees of freedom in the Satterthwaite (1941, 1946) method which estimates the distribution of a weighted combination of variance components The standard Satterthwaite approximation assumes a scaled chisquare distribution for the composite variance estimator but is known to be biased downward when component degrees of freedom are small. Building on recent work by von Davier (2025), we propose an adjusted estimator that corrects this bias by modifying both the numerator and denominator of the traditional formula. The new approximation incorporates a weighted average of component degrees of freedom and a scaling factor that ensures consistency as the number of components or their degrees of freedom increases. We demonstrate the utility of this adjustment in practical settings, including Rubin's (1987) total variance estimation in multiple imputations, where weighted variance combinations are common. The proposed estimator generalizes and further improves von Davier's (2025) unweighted case and more accurately approximates synthetic variance estimators with arbitrary weights.