Applied Statistics Lesson of the Day – The Independent 2-Sample t-Test with Unequal Variances (Welch’s t-Test)
March 13, 2014 Leave a comment
A common problem in statistics is determining whether or not the means of 2 populations are equal. The independent 2-sample t-test is a popular parametric method to answer this question. (In an earlier Statistics Lesson of the Day, I discussed how data collected from a completely randomized design with 1 binary factor can be analyzed by an independent 2-sample t-test. I also discussed its possible use in the discovery of argon.) I have learned 2 versions of the independent 2-sample t-test, and they differ on the variances of the 2 samples. The 2 possibilities are
- equal variances
- unequal variances
Most statistics textbooks that I have read elaborate at length about the independent 2-sample t-test with equal variances (also called Student’s t-test). However, the assumption of equal variances needs to be checked using the chi-squared test before proceeding with the Student’s t-test, yet this check does not seem to be universally done in practice. Furthermore, conducting one test based on the results of another results possible inflation of Type 1 error (Ruxton, 2006).
Some books give due attention to the independent 2-sample t-test with unequal variances (also called Welch’s t-test), but some barely mention its value, and others do not even mention it at all. I find this to be puzzling, because the assumption of equal variances is often violated in practice, and Welch’s t-test provides an easy solution to this problem. There is a seemingly intimidating but straightforward calculation to approximate the number of degrees of freedom for Welch’s t-test, and this calculation is automatically incorporated in most software, including R and SAS. Finally, Welch’s t-test removes the need to check for equal variances, and it is almost as powerful as Student’s t-test when the variances are equal (Ruxton, 2006).
For all of these reasons, I recommend Welch’s t-test when using the parametric approach to comparing the means of 2 populations.
Graeme D. Ruxton. “The unequal variance t-test is an underused alternative to Student’s t-test and the Mann–Whitney U test“. Behavioral Ecology (July/August 2006) 17 (4): 688-690 first published online May 17, 2006