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 sample variances.

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

:$chi\text{'}\; =\; sum\_\{i=1\}^\{n\}\; k\_\{i\}\; s\_\{i\}^\{2\}.$

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"'

:$u\_\{chi\text{'}\}\; approx\; frac\{(sum\_\{i=1\}^\{n\}\; k\_\{i\}\; s\_\{i\}^\{2\})^\{2\; \{sum\_\{i=1\}^\{n\}\; frac\{(k\_\{i\}\; s\_\{i\}^\{2\})^\{2\; \{\; u\_\{i\; \}$

There is "no" assumption that the underlying population variances "σ_i"² 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.

References

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* cite book
last = Neter
first = John
coauthors = John Neter, William Wasserman, Michael H. Kutner
title = Applied Linear Statistical Models
publisher = Richard D. Irwin, Inc.
date = 1990
isbn = 0-256-08338-X