Welch's t-Test

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A Welch's t-Test is a parametric statistical test that is an adaptation of the independent two-sample t-test for unequal population variances.



References

2017

When applying Welch's t-test, the calculated t-value is compared to a critical t-value which is based on the selected significance level of the test and on the number of degrees of freedom. If the calculated t-value is less than or equal to the critical value, then no evidence exists for a statistically significant difference between the two population means at the selected confidence level. The equations for the necessary calculations, including the critical t-values for common significance levels, can be found in most statistical texts and in the Unified Guidance.

2016

  • (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Welch's_t-test Retrieved:2016-9-14.
    • In statistics, Welch's t-test, or unequal variances t-test, is a two-sample location test which is used to test the hypothesis that two populations have equal means. Welch's t-test is an adaptation of Student's t-test,that is, it has been derived with the help of Student's t-test and is more reliable when the two samples have unequal variances and unequal sample sizes. These tests are often referred to as "unpaired" or "independent samples" t-tests, as they are typically applied when the statistical units underlying the two samples being compared are non-overlapping. Given that Welch's t-test has been less popular than Student's t-test and may be less familiar to readers, a more informative name is "Welch's unequal variances t-test" or "unequal variances t-test" for brevity.
(...) Welch's t-test defines the statistic t by the following formula:
[math]\displaystyle{ t \quad = \quad {\; \overline{X}_1 - \overline{X}_2 \; \over \sqrt{ \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \quad }}\, }[/math]
where [math]\displaystyle{ \overline{X}_1 }[/math], [math]\displaystyle{ s_1^2 }[/math] and [math]\displaystyle{ N_1 }[/math] are the 1st 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 [math]\displaystyle{ \nu }[/math]  associated with this variance estimate is approximated using the Welch–Satterthwaite equation:
[math]\displaystyle{ \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 }} }[/math]
Here [math]\displaystyle{ \nu_1 = N_1-1 }[/math], the degrees of freedom associated with the first variance estimate. [math]\displaystyle{ \nu_2 = N_2-1 }[/math], the degrees of freedom associated with the 2nd variance estimate.
Welch's t-test can also be calculated for ranked data and might then be named Welch's U-test.[1]
  1. Fagerland, M. W.; Sandvik, L. (2009). "Performance of five two-sample location tests for skewed distributions with unequal variances". Contemporary Clinical Trials 30: 490–496. doi:10.1016/j.cct.2009.06.007.