Independent Two-Sample t-Test

An Independent Two-Sample t-Test is a student's t-test applied to two independent samples.

• AKA: Unpaired Two-Sample t-Test, Unmatched Two-Sample t-Test, Pooled Variance t-Test.
• Context:
  • It is used statistical experiments that have one test variable that is a measurement variable and a grouping variable that is a nominal variable with two categorical values.
  • It requires two independent samples drawn from two different populations.
  • It can also be generally defined as the testing of null hypothesis that states the means of the test variable (e.g mean value) are equal for the two categorical groups.
  • It is described by an Independent Two-Sample t-Test Task and solved by an Independent Two-Sample t-Test System and requires the calculation of a Independent Two-Sample t-Statistic.
  • It assumes homoscedasticity, i.e. the two independent samples are drawn two populations with equal variances.
  • It assumes that samples and population are normally distributed.
  • It is mathematically identical to a One-Way ANOVA when data is divided into two categorical groups.
  • It is replaced by a Welch's t-Test when variances are unequal, sample sizes are small and unequal.
  • It is replaced by a Mann-Whitney U Test is when data samples are non-parametric and not normally distributed.
  • It can range from being a Two-Tailed Independent Two-sample Test to being a One-Tailed Independent Two-sample Test.
• Example(s)
  • The test of the null hypothesis of whether the sample means of the VIQ among females and males are statistically different. In this case the test variable is the VIQ mean value and the grouping variable is the two categorical groups "female" and "male".
  • …
• Counter-Example(s):
  • A Welch's t-Test,
  • An One-Sample t-Test.
  • A Matched-Pair t-Test.
  • A Hotelling's Two-Sample t-Squared Test
  • a Chi-Square Test.
• See: Independent t-Test, Dependent t-Test, Student's t-Test.

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References

2017a

• (Wikipedia, 2017) ⇒ http://en.wikipedia.org/wiki/Student%27s_t-test#Independent_two-sample_t-test
  • Given two groups (1, 2), this test is only applicable when:

• the two sample sizes (that is, the number, n, of participants of each group) are equal;
• it can be assumed that the two distributions have the same variance;

Violations of these assumptions are discussed below.

The t statistic to test whether the means are different can be calculated as follows:

[math]\displaystyle{ t = \frac{\bar {X}_1 - \bar{X}_2}{s_p \sqrt{2/n}} }[/math]

where

[math]\displaystyle{ \ s_p = \sqrt{\frac{s_{X_1}^2+s_{X_2}^2}{2}} }[/math]

Here [math]\displaystyle{ s_p }[/math] is the pooled standard deviation for n=n₁=n₂ and [math]\displaystyle{ s_{X_1}^2 }[/math] and [math]\displaystyle{ s_{X_2}^2 }[/math] are the unbiased estimators of the variances of the two samples. The denominator of t is the standard error of the difference between two means.

For significance testing, the degrees of freedom for this test is 2n − 2 where n is the number of participants in each group.

2017b

• (Stattrek, 2017) ⇒ http://stattrek.com/hypothesis-test/difference-in-means.aspx?Tutorial=AP
  • This lesson explains how to conduct a hypothesis test for the difference between two means. The test procedure, called the two-sample t-test, is appropriate when the following conditions are met:
    • The sampling method for each sample is simple random sampling.
    • The samples are independent.
    • Each population is at least 20 times larger than its respective sample.
    • The sampling distribution is approximately normal, which is generally the case if any of the following conditions apply.
      • The population distribution is normal.
      • The population data are symmetric, unimodal, without outliers, and the sample size is 15 or less.
      • The population data are slightly skewed, unimodal, without outliers, and the sample size is 16 to 40.
      • The sample size is greater than 40, without outliers.

2017c

• (QCP Glossary, 2017) ⇒ https://www.quality-control-plan.com/StatGuide/ttest_unpaired.htm
  • The two-sample unpaired t test is used to test the null hypothesis that the two population means corresponding to the two random samples are equal.

Assumptions:

• Within each sample, the values are independent, and identically normally distributed (same mean and variance).
• The two samples are independent of each other.
• For the usual two-sample t test, the two different samples are assumed to come from populations with the same variance, allowing for a pooled estimate of the variance. However, if the two sample variances are clearly different, a variant test, the Welch-Satterthwaite t test, is used to test whether the means are different.

2017D

• http://www.evanmiller.org/ab-testing/t-test.html Does the average value differ across two groups?

2014

• (McDonald, 2015) ⇒ McDonald, J.H. (2014). Handbook of Biological Statistics (3rd ed.). Sparky House Publishing, Baltimore, Maryland. Pages 126-130 Retrived from: http://www.biostathandbook.com/twosamplettest.html
  • SUMMARY: Use Student's t–test for two samples when you have one measurement variable and one nominal variable, and the nominal variable has only two values. It tests whether the means of the measurement variable are different in the two groups.

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