Afsnitsforfatter: 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.