Welch's t test

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In statistics, Welch's t test is an adaptation of Student's t-test intended for use with two samples having possibly unequal variances.^[1] As such, it is an approximate solution to the Behrens–Fisher problem.

Formulas

Welch's t-test defines the statistic t by the following formula:

$t\quad =\quad {\;\overline {X}_{1}-\overline {X}_{2}\; \over {\sqrt {\;{s_{1}^{2} \over N_{1}}\;+\;{s_{2}^{2} \over N_{2}}\quad }}}\,$

where $\overline {X}_{{i}}$ , $s_{{i}}^{{2}}$ and $N_{{i}}$ are the $i$ ^th sample mean, sample variance and sample size, respectively. Unlike in Student's t-test, the denominator is not based on a pooled variance estimate.

The degrees of freedom $\nu$ associated with this variance estimate is approximated using the Welch–Satterthwaite equation:

$\nu \quad \approx \quad {{\left(\;{s_{1}^{2} \over N_{1}}\;+\;{s_{2}^{2} \over N_{2}}\;\right)^{2}} \over {\quad {s_{1}^{4} \over N_{1}^{2}\nu _{1}}\;+\;{s_{2}^{4} \over N_{2}^{2}\nu _{2}}\quad }}$

Here $\nu _{i}$ = $N_{i}-1$ , the degrees of freedom associated with the $i$ ^th variance estimate.

Statistical test

Once t and $\nu$ have been computed, these statistics can be used with the t-distribution to test the null hypothesis that the two population means are equal (using a two-tailed test), or the null hypothesis that one of the population means is greater than or equal to the other (using a one-tailed test). In particular, the test will yield a p-value which might or might not give evidence sufficient to reject the null hypothesis.

References

↑ Welch, B. L. (1947). "The generalization of "Student's" problem when several different population variances are involved". Biometrika 34 (1–2): 28–35. doi:10.1093/biomet/34.1-2.28. MR 19277.

Further reading

Daniel Borcard, Lecture Note Appendix: t-test with Welch correction, excerpt from Legendre, P. and D. Borcard. Statistical comparison of univariate tests of homogeneity of variances.
Sawilowsky, Shlomo S. (2002). "Fermat, Schubert, Einstein, and Behrens–Fisher: The Probable Difference Between Two Means When σ₁ ≠ σ₂". Journal of Modern Applied Statistical Methods 1 (2): 461–472.

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