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.

翻译：本文提出了针对独立方差分量加权和的自由度近似方法的修正版本，该近似方法源于Satterthwaite（1941，1946）的研究。当分量自由度较小时，原始公式会产生有偏估计。通过精确矩匹配推导的修正项，通过引入考虑四阶矩估计的调整因子来校正偏差。我们证明当所有方差分量相等且忽略分量自由度时，Kish（1965）的有效样本量公式可作为特例推导得出。模拟研究表明，即使在分量规模较小的情况下，修正后的估计量仍能紧密匹配期望自由度，而原始Satterthwaite估计量则表现出显著的下偏偏差。本文还讨论了其他应用场景，包括刀切法方差估计、多重插补总方差以及针对异方差的韦尔奇检验。