Welch's t-Test Task

A Welch's t-Test Task is a statistical hypothesis testing task used to describe an Welch's t-test.

• AKA: .
• Context:
  • It can be solved by an Welch's t-Test System.
  • Task Input:
    • Input Data :
      • X =[math]\displaystyle{ \{x_1,x_2,\cdots,x_n\} }[/math] , first sample dataset drawn from a statistical population of interest. This is the first categorical group of the test variable.
      • Y =[math]\displaystyle{ \{y_1,y_2,\cdots,y_n\} }[/math], second sample dataset drawn from a second statistical population of interest. This is the first categorical group of the test variable.
    • Input Parameters:
      • [math]\displaystyle{ \alpha_0 }[/math] = a significance level value or a confidence level (a percentage).
  • output:
    • Welch's t-test statistic value,
    • P-value or Region of Acceptance
    • Region of Rejection (optional)
    • Decision Errors (optional).
  • Task Requirements
    • Specific Requirement(s): [math]\displaystyle{ \sigma(X) \neq \sigma(Y) }[/math], Unequal Population variances. It can include a Bartlett's Test or a Levene's Test.
    • Hypotheses Statement: a null hypothesis and an alternative hypothesis defined according to one-tailed Welch's t-test or two-tailed Welch's t-test.
    • Test Statistic Calculation: It is solved by Welch's t-statistic calculating system.
    • P-value and/or Region of acceptance computation: these require a t-distribution calculator or t-table.
    • Decision Rule: Null hypothesis is reject if P-value is less than [math]\displaystyle{ \alpha_0 }[/math] or if the t-test statistic value follows outside region of acceptance.
• Example(s):
  • Let's consider a Welch's t-test for comparing the mean mile run time (test variable/measurement variable) for Athletes (grouping variable, first categorical variable) and Non-athletes (grouping variable, second categorical value). The task can be solved by Welch's t-Test System based on the scipy statistical function stats.ttest_ind() [1] where the variable equal_var = False.
    • Task Input:
      • Input data: dataset retrieved from http://libguides.library.kent.edu/SPSS/IndependentTTest, dataset corresponding to test variable MileMinDur (Mile run time in minutes) and the grouping variable: Athlete (categorical value = 1) and Non-Athletes (categorical value = 0).
      • Hypothesis statement:

Null hypothesis: "the difference of the means is equal to zero" ⇒ [math]\displaystyle{ H_0:\; \mu_{nonathlete} - \mu_{athlete} = 0 }[/math]

Alternative hypothesis: "the difference of the means is not equal to zero" ⇒ [math]\displaystyle{ H_1:\; \mu_{nonathlete} - \mu_{athlete} \neq 0 }[/math]

• Task output:
• Levene test statistic is 102.563129443 and p-value is 1.4800514645e-21. Since p-value is too small, variances are unequal.
• Welch's t-statistic is -15.0486789157 and p-value is 5.82457889026e-39. P-value is too small, running time between 'Athletes' and 'Non-Athletes' is very different.

• • …
• Counter-Example(s):
  • Matched-Pair t-Test Task.
  • Linear Regression t-Test.
  • Independent Two-Sample t-Test Task.
  • One-Sample t-Test Task.
• See: Independent Two-Sample t-Test System, Statistical Significance, Sample Average, Sample Variance.

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References

2017a

• (Wikipedia, 2017) ⇒ http://en.wikipedia.org/wiki/Welch's_t-test
  • 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.

2017

• (Kent State University Libraries, 2017) ⇒ Retrived from SPSS Tutorials: Independent Samples t Test http://libguides.library.kent.edu/SPSS/IndependentTTest on 2017-04-30
  • (...) 7. Homogeneity of variances (i.e., variances approximately equal across groups). When this assumption is violated and the sample sizes for each group differ, the p value is not trustworthy. However, the Independent Samples t Test output also includes an approximate t statistic that is not based on assuming equal population variances; this alternative statistic, called the Welch t Test statistic, may be used when equal variances among populations cannot be assumed. The Welch t Test is also known an Unequal Variance T Test or Separate Variances T Test.

2014

• (McDonald, 2014) ⇒ McDonald, J.H., (2014). Handbook of Biological Statistics (3rd ed.). Sparky House Publishing, Baltimore, Maryland. Retrieved from http://www.biostathandbook.com/twosamplettest.html which contains handbook's content of pages 126-130.
  • The two-sample t-test also assumes homoscedasticity (equal variances in the two groups). If you have a balanced design (equal sample sizes in the two groups), the test is not very sensitive to heteroscedasticity unless the sample size is very small (less than 10 or so); the standard deviations in one group can be several times as big as in the other group, and you'll get P<0.05 about 5% of the time if the null hypothesis is true. With an unbalanced design, heteroscedasticity is a bigger problem; if the group with the smaller sample size has a bigger standard deviation, the two-sample t-test can give you false positives much too often. If your two groups have standard deviations that are substantially different (such as one standard deviation is twice as big as the other), and your sample sizes are small (less than 10) or unequal, you should use Welch's t–test instead.

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