Welch–Satterthwaite equation

Equation to approximate pooled degrees of freedom

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 si2 (i = 1, ..., n), each respectively having νi degrees of freedom, often one computes the linear combination.

χ ′ = ∑ i = 1 n k i s i 2 . {\displaystyle \chi '=\sum _{i=1}^{n}k_{i}s_{i}^{2}.}

where k i {\displaystyle k_{i}} is a real positive number, typically k i = 1 ν i + 1 {\displaystyle 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

ν χ ′ ≈ ( ∑ i = 1 n k i s i 2 ) 2 ∑ i = 1 n ( k i s i 2 ) 2 ν i {\displaystyle \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 σi2 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.

See also

  • Pooled variance

References

  1. ^ Spellman, Frank R. (12 November 2013). Handbook of mathematics and statistics for the environment. Whiting, Nancy E. Boca Raton. ISBN 978-1-4665-8638-3. OCLC 863225343.{{cite book}}: CS1 maint: location missing publisher (link)
  2. ^ Van Emden, H. F. (Helmut Fritz) (2008). Statistics for terrified biologists. Malden, MA: Blackwell Pub. ISBN 978-1-4443-0039-0. OCLC 317778677.

Further reading

  • Satterthwaite, F. E. (1946), "An Approximate Distribution of Estimates of Variance Components.", Biometrics Bulletin, 2 (6): 110–114, doi:10.2307/3002019, JSTOR 3002019, PMID 20287815
  • Welch, B. L. (1947), "The generalization of "student's" problem when several different population variances are involved.", Biometrika, 34 (1/2): 28–35, doi:10.2307/2332510, JSTOR 2332510, PMID 20287819
  • 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. [1]