Using the Student’s t-test with extremely small sample sizes

Abstract

Researchers occasionally have to work with an extremely small sample size, defined herein as N ≤ 5. Some methodologists have cautioned against using the t-test when the sample size is extremely small, whereas others have suggested that using the t-test is feasible in such a case. The present simulation study estimated the Type I error rate and statistical power of the one- and two-sample t-tests for normally distributed populations and for various distortions such as unequal sample sizes, unequal variances, the combination of unequal sample sizes and unequal variances, and a lognormal population distribution. Ns per group were varied between 2 and 5. Results show that the t-test provides Type I error rates close to the 5% nominal value in most of the cases, and that acceptable power (i.e., 80%) is reached only if the effect size is very large. This study also investigated the behavior of the Welch test and a rank-transformation prior to conducting the t-test (t-testR). Compared to the regular t-test, the Welch test tends to reduce statistical power and the t-testR yields false positive rates that deviate from 5%. This study further shows that a paired t-test is feasible with extremely small Ns if the within-pair correlation is high. It is concluded that there are no principal objections to using a t-test with Ns as small as 2. A final cautionary note is made on the credibility of research findings when sample sizes are small.