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 $u$ associated with this variance estimate is approximated using the Welch-Satterthwaite equation:

:$u\; =$left( {s_1^2 over N_1} + {s_2^2 over N_2} ight)^2 } over s_1^4 over N_1^2 cdot u_1}+{s_2^4 over N_2^2 cdot u_2}.,

Here $u\_i$ is $N\_i-1$, the degrees of freedom associated with the $i$^th variance estimate.

tatistical test

Once "t" and "$u$" 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

*
* [http://biol09.biol.umontreal.ca/BIO2041e/Correction_Welch.pdf]
*Sawilowsky, Shlomo S. (2002). [http://tbf.coe.wayne.edu/jmasm/sawilowsky_behrens_fisher.pdf Fermat, Schubert, Einstein, and Behrens–Fisher: The Probable Difference Between Two Means When σ₁ ≠ σ₂] "Journal of Modern Applied Statistical Methods", 1(2).