I have a colleague who is comparing several different treatments of data via canonical correlation analysis. In other words, given some time-varying signal, $a(t)$, he extracting some vector of features $v_1(t)$. He then supposes that this is a predictor for some other vector $p(t)$. To check this he computes the [first] canonical correlation coefficient, $R_1 = \text{CCA}(v_1(t),p(t))$. And then he tries some new improved feature extractor, $v_2(t)$, and compares again $R_2 = \text{CCA}(v_2(t),p(t))$.

I can find tests for the situation that $R_1$, and $R_2$ are different from zero. But what about the test that $R_1$ and $R_2$ are significantly different from each other?

Asked on behalf of a colleague, but I'd also be interested in the answer.