Welch-Satterthwaite equation

From Wikipedia, the free encyclopedia

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 sample variances.

For n samples variances s_i² (i = 1, ..., n), each having ν_i degrees of freedom, often one computes the linear combination

$\chi' = \sum_{i=1}^{n} k_{i} s_{i}^{2}.$

In general, the 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

$\nu_{\chi'} \approx \frac{(\sum_{i=1}^{n} k_{i} s_{i}^{2})^{2}} {\sum_{i=1}^{n} \frac{(k_{i} s_{i}^{2})^{2}} {\nu_{i}} }$

There is no assumption that the underlying population variances σ_i² are equal.

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

[edit] References

Satterthwaite, F. E. (1946), “An Approximate Distribution of Estimates of Variance Components.”, Biometrics Bulletin 2: 110-114
Welch, B. L. (1947), “The generalization of "student's" problem when several different population variances are involved.”, Biometrika 34: 28-35
Neter, John; John Neter, William Wasserman, Michael H. Kutner (1990). Applied Linear Statistical Models. Richard D. Irwin, Inc.. ISBN 0-256-08338-X.
'The Expression of Uncertainty and Confidence in Measurement', M3003, UKAS, December 1997