Welch–Satterthwaite equation

In statistics and uncertainty analysis, the Welch–Satterthwaite equation is used to calculate an approximation to the effective degrees of freedom of a linear combination of independent sample variances, also known as the pooled degrees of freedom,^[1]^[2] corresponding to the pooled variance.

For $n$ sample variances $s i 2 (i = 1, ..., n)$ , each respectively having $ν i$ degrees of freedom, often one computes the linear combination

\chi '=\sum _{{i=1}}^{n}k_{i}s_{i}^{2}.

where $k_{i}$ is a real positive number, typically $k_{i}={\frac {1}{\nu _{i}+1}}$ . In general, the probability distribution of $χ'$ cannot be expressed analytically. However, its distribution can be approximated by another chi-squared distribution, whose effective degrees of freedom are given by the Welch–Satterthwaite equation

\nu _{{\chi '}}\approx {\frac {\displaystyle \left(\sum _{{i=1}}^{n}k_{i}s_{i}^{2}\right)^{2}}{\displaystyle \sum _{{i=1}}^{n}{\frac {(k_{i}s_{i}^{2})^{2}}{\nu _{i}}}}}

There is no assumption that the underlying population variances $σ i 2$ are equal. This is known as the Behrens–Fisher problem.

The result can be used to perform approximate statistical inference tests. The simplest application of this equation is in performing Welch's t test.

References

Satterthwaite, F. E. (1946), "An Approximate Distribution of Estimates of Variance Components.", Biometrics Bulletin, 2: 110&ndash, 114, doi:10.2307/3002019
Welch, B. L. (1947), "The generalization of "student's" problem when several different population variances are involved.", Biometrika, 34: 28&ndash, 35, doi:10.2307/2332510
Neter, John; John Neter; William Wasserman; Michael H. Kutner (1990). Applied Linear Statistical Models. Richard D. Irwin, Inc. ISBN 0-256-08338-X.
Michael Allwood (2008) "The Satterthwaite Formula for Degrees of Freedom in the Two-Sample t-Test", AP Statistics, Advanced Placement Program, The College Board.

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Welch–Satterthwaite equation

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