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:

$t \quad = \quad {\; \overline{X}_1 - \overline{X}_2 \; \over \sqrt{ \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \quad }}\,$

where $\overline{X}_{i}$, $s_{i}^{2}$ and $N_{i}$ are the $i$^thsample 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:

$\nu \quad = \quad {{\left( \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \; \right)^2 } \over { \quad {s_1^4 \over N_1^2 \cdot \nu_1} \; + \; {s_2^4 \over N_2^2 \cdot \nu_2 } \quad }} \quad = \quad {{\left( \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \; \right)^2 } \over { \quad { s_1^4 \over N_1^2 \cdot \left({N_1-1}\right)} \; + \; {s_2^4 \over N_2^2 \cdot \left({N_2-1}\right) } \quad }} \,$

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

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.