Auteur de la section : Danielle J. Navarro and David R. Foxcroft

On the relationship between ANOVA and the Student t-test

There is one last thing I want to point out before finishing. It is something that a lot of people find kind of surprising, but it is worth knowing about. An ANOVA with two groups is identical to the Student t-test. No, really. It is not just that they are similar, but they are actually equivalent in every meaningful way. I will not try to prove that this is always true, but I will show you a single concrete demonstration. Suppose that, instead of running an ANOVA on our mood.gain ~ drug model, let us instead do it using therapy as the predictor. If we run this ANOVA we get an F-statistic of F(1,16) = 1.708, and a p-value = 0.210. Since we only have two groups, I did not actually need to resort to an ANOVA, I could have just decided to run a Student t-test. So let us see what happens when I do that: I get a t-statistic of t*(16) = -1.307 and a *p-value = 0.210. Curiously, the p-values are identical. Once again we obtain a value of p = 0.210. But what about the test statistic? Having run a t-test instead of an ANOVA, we get a somewhat different answer, namely t*(16) = -1.307. However, there is a fairly straightforward relationship here. If we square the *t-statistic then we get the F-statistic from before: -1.307² = 1.708.