The robustness to non-normality of significance levels of the t and F tests

Abstract

A statistical test is called"robust" if it is insensitive to departures from the underlying assumptions, this term was introduced by Box (4).

Theoretical study made by Gayen (13), (14) and (15) showed that"Student's" t-test and the closely related F-test of analysis of variance are insensitive to departures from normality. But the F-test on the equality of two variances is very sensitive to such departures.

Empirical studies made by Norton (19) and Boneau (3) agree with Gayen's theoretical conclusions. Norton studied the effect of non-normality on the F-test of analysis of variance,and showed that the form of the sampled population had very little effect on this test. For example, for the case of three groups of sample sizes 3, for the 5% level, the percentages exceeding the theoretical limits were 7.83 and 4.77% respectively for sampling from a leptokurtic and an extremely skewed population. Such property of robustness to non-normality on the F-test of analysis of variance is also possessed by t-test. Boneau's empirical study on the effect of non-normality on the two-sample t-test showed that for two samples of size 5, the significance level is respectively 3.1 and 5.1% for the empirical distribution of t’s from the exponential and uniform distribution compare to the nominal 5% value. The discrepancy is decreased when sample size is increased.