A Corrected Welch Satterthwaite Equation. And: What You Always Wanted to Know About Kish's Effective Sample but Were Afraid to Ask

TL;DR

This paper introduces a corrected Satterthwaite approximation for degrees of freedom in weighted variance sums, reducing bias especially with small sample sizes, and clarifies the relationship with Kish's effective sample size.

Contribution

A new bias-corrected formula for degrees of freedom that improves accuracy over the traditional Satterthwaite approximation, with connections to Kish's effective sample size.

Findings

The corrected estimator closely matches true degrees of freedom in simulations.

Original Satterthwaite estimator shows significant downward bias with small samples.

The correction improves variance estimation in multiple statistical applications.

Abstract

This article presents a corrected version of the Satterthwaite (1941, 1946) approximation for the degrees of freedom of a weighted sum of independent variance components. The original formula is known to yield biased estimates when component degrees of freedom are small. The correction, derived from exact moment matching, adjusts for the bias by incorporating a factor that accounts for the estimation of fourth moments. We show that Kish's (1965) effective sample size formula emerges as a special case when all variance components are equal, and component degrees of freedom are ignored. Simulation studies demonstrate that the corrected estimator closely matches the expected degrees of freedom even for small component sizes, while the original Satterthwaite estimator exhibits substantial downward bias. Additional applications are discussed, including jackknife variance estimation, multiple imputation total variance, and the Welch test for unequal variances.