Welch's t-Test

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.

• AKA: Welch's Unequal Variances t-Test, Unequal Variance t-Test, Separate Variances t-Test, Welch's Two-Sample t-Test, Welch's Unpaired t-Test, Aspin-Welch's t-Test, Aspin-Welch's-Satterthwaite t-Test, Welch-Scatterthwaite's t-Test.
• Context:
  • It is described by Welch's t-Test Task and solve by a Welch's t-Test System.
  • It requires the calculation of Welch's t-Statistic.
  • It used instead of the independent two-sample t-test when population variances are different.
• Example(s):
  • An Welch's t-test task for testing whether the sample means of the mile run time (test variable) among athletes (group 1, nominal variable) and non-athletes (group 2, nominal variable) are statistically different.
  • …
• Counter-Example(s):
  • An Independent Two-Sample t-Test.
  • a Mann–Whitney U test.
  • a Wilcoxon Signed-rank Test.
• See: Paired Difference Test.

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References

2017

• (itrcweb,2017) ⇒ Retrieved on 2017-04-02 from http://www.itrcweb.org/gsmc-1/Content/GW%20Stats/5%20Methods%20in%20indiv%20Topics/5%2011%20Two%20Sample%20Tests.htm
  • Welch’s t-test assumes that each population is normally distributed and requires that no temporal trends exist in the data, no spatial variability is present, and samples are statistically independent. One advantage of Welch’s t-test is that it does not require you to assume that population variances are equal. Another advantage is that while Welch’s t-test provides statistical power comparable to other two-sample tests, it is much simpler to use than other similar tests. The only calculations required are computing the mean, standard deviation, variance, t-statistic, and degrees of freedom. Many statistical software packages offer Welch’s t-test, but most do not determine if the requirements and assumptions are met.

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.

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