- 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 unequalvariance s. As such, it is an approximate solution to theBehrens–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 ith sample mean, sample variance and
sample size , respectively. Unlike in Student's t-test, the denominator is "not" based on apooled 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 ith variance estimate.
tatistical test
Once "t" and "u" have been computed, these statistics can be used with the
t-distribution to test thenull hypothesis that the two population means are equal (using atwo-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 ap-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).
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