Assumptions of correlation test vs regression slope test (significance testing)

If my understanding is correct, then

1. the test on a regression slope in a simple bivariate regression - i.e. the test of $\mathcal{H}_0$: $b = 0$ in $Y' = a + bX$ and
2. the test of a correlation, i.e. $\mathcal{H}_0$ : $\rho=0$

appear to involve different assumptions. The first assumes normality of errors (conditional distribution of Y given X) while the second is reported to assume bivariate normality of X and Y. Yet the tests produce identical p-values in every case I've ever seen and several trustworthy sources (e.g. David Howell's Psych Stats Text, van Belle et al's Biostats text) assert that these are the same test.

Now, bivariate normality (as far as I can deduce) implies that the conditional distribution of Y given X is Normal with a constant variance (equal to $var(Y)\times(1-\rho^2 )$), which is the stated assumption in the regression slope test. So is it the case that bivariate normality is not truly required in the test of the correlation - that the more narrow assumption regarding the conditional distribution is the only required assumption?

• Common (F or t) testing in simple linear regression Y ~ X coefficient assumes error normality and homoscedasticity for Y. But linear correlation testing implies that same thing in opposite regression X ~ Y too. But that, if I'm not mistaken, holds only when the distribution is bivariate normal. So, when you are testing $r$ by the same approach as you do in regression, the normality is the assumption. (But you could do $r$ testing some other alternative ways, not requiring the normality: permutation/montecarlo/bootstrap.) – ttnphns Nov 10 '15 at 13:59
• First comment was too long - I'll try again. Thanks ttnphns. I'm confused, however, as to why linear correlation testing would be described as having assumptions on errors in both directions (I agree, that is my interpretation as well) when the hypothesis test computationally is the same. How could the computations be valid for some cases (regression use when X non-normally distributed) but not others (correlation use when X non-normally distributed) when they are exactly the same? Or are the tests somehow telling me different things, even though people describe them as the same? – J Taylor Nov 10 '15 at 14:26
• @JTaylor see stats.stackexchange.com/questions/2125/… or stats.stackexchange.com/questions/32464/… or it should clear things up a little bit. – Tim Nov 10 '15 at 14:29
• Testing of correlation coefficient for significance is more problematic than testing of regressional beta (athough beta of simple regression = correlation, numerically as statistics). Yes, because we have to account for the fact that errors involve both Y and X in this case. If you are ready to assume both-way error normality (and hence bivariate normality) you may use for $r$ the same F test approach/formula (to test against $\rho$=0) as for the beta. But generally, testing $r$ and beta are two different tasks. – ttnphns Nov 10 '15 at 14:44
• @Tim - thanks for the pointers. I think you were trying to clear up confusion on what regression is vs correlation, and how Y~X is different from X~Y. I think I have a handle on that (though I could be mistaken). I don't see how the links address my question though. If I say that I want to test H0:rho=0, how would you know which assumption to check (bivariate normality or conditional normality)? But why would that matter if the test is exactly the same? – J Taylor Nov 10 '15 at 14:54