# On the specific difference between RM ANOVA and paired sample t-test

Paired sample t-test and Repeated Measures ANOVA with a dichotomous predictor are two tests that answer the same question, and output the same p-value as far as I can tell.

This question has been asked before, but it was focused on the assumption of both tests. What strikes me more here is that the test statistics are different: one is based on the $$t$$ distribution while the other on the $$F$$ distribution. Is there any theoretical way to explain why these two distributions should be close to each other in the case of a dichotomous predictor?

If $$X\sim t(\nu)$$, i.e. $$X$$ has a $$t$$-distribution with $$\nu$$ degrees of freedom, then $$X^{2}\sim F(1, \nu_{2} = \nu)$$. It follows that in the case of two repeated measurements, the squared $$t$$ value of the paired $$t$$-test is equal to the $$F$$ value of the repeated measures ANOVA.
Here is an example using SPSS. There are two timepoints with repeated measurements. First the paired $$t$$-test: The $$t$$ value is $$-0.151$$ with $$173$$ degrees of freedom. Now the repeated measures ANOVA: The $$F$$ value is $$0.023$$ which is equal to $$(-0.151)^{2}$$, as expected. The degrees of freedom are $$1$$ and $$173$$. The $$p$$-values are identical.