Welch's t-test

In statistics, Welch's t test is an adaptation of Student's t-test intended for use with two samples having possibly unequal variances. 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:

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

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:

$\nu ={{\left({s_{1}^{2} \over N_{1}}+{s_{2}^{2} \over N_{2}}\right)^{2}} \over {{s_{1}^{4} \over N_{1}^{2}\cdot \nu _{1}}+{s_{2}^{4} \over N_{2}^{2}\cdot \nu _{2}}}}={{\left({s_{1}^{2} \over N_{1}}+{s_{2}^{2} \over N_{2}}\right)^{2}} \over {{s_{1}^{4} \over N_{1}^{2}\cdot \left({N_{1}-1}\right)}+{s_{2}^{4} \over N_{2}^{2}\cdot \left({N_{2}-1}\right)}}}.\,$

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, MR19277
• [1]
• 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).