Section author: Danielle J. Navarro and David R. Foxcroft

On the relationship between ANOVA and the Student t-test

There’s one last thing I want to point out before finishing. It’s something that a lot of people find kind of surprising, but it’s worth knowing about. An ANOVA with two groups is identical to the Student t-test. No, really. It’s not just that they are similar, but they are actually equivalent in every meaningful way. I won’t 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’s instead do it using therapy as the predictor. If we run this ANOVA we get an F-statistic of F(1,16) = 1.71, and a p-value = 0.210. Since we only have two groups, I didn’t actually need to resort to an ANOVA, I could have just decided to run a Student t-test. So let’s see what happens when I do that: I get a t-statistic of t(16) = -1.3068 and a p-value = 0.21. 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) = -1.3068. However, there is a fairly straightforward relationship here. If we square the t-statistic then we get the F-statistic from before: -1.3068² = 1.7077.