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萨特思韦特（1941, 1946）近似法的修正版本。已知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，引入了一个考虑四阶矩估计的校正因子以调整偏差。我们证明当所有方差分量相等且忽略分量自由度时，基什（1965）有效样本量公式可作为本修正的特例出现。模拟研究表明，即使在分量样本量较小时，修正后的估计量仍能紧密匹配期望自由度，而原始萨特思韦特估计量则表现出显著的下偏。本文进一步探讨了其在刀切法方差估计、多重插补总方差以及韦尔奇异方差检验等领域的应用。