Robustness of correlation test to non-normality I'm trying to reconcile two seemingly opposite statements about robustness to non-normality of the Pearson's correlation test statistic (where the null means "no correlation").
This CV answer says:

Very non-robust.

This biostat handbook says:

[...] numerous simulation studies have shown that linear regression and
  correlation are not sensitive to non-normality; one or both
  measurement variables can be very non-normal, and the probability of a
  false positive (P<0.05, when the null hypothesis is true) is still
  about 0.05 (Edgell and Noon 1984, and references therein).

What am I missing?
 A: Since whuber has given a comprehensive analysis of the behavior of the distributions of p-values under a null of zero-correlation, I'll focus my comments elsewhere.


*

*Robustness in relation to hypothesis tests doesn't only mean level-robustness (getting close to the desired significance level). Besides only looking at one level and only at two-sided tests, the study appears to have ignored impact on power. There's no much point saying that you're keeping close to a 5% rejection rate under the null if you also end up with a 5% rejection rate* for large deviations from the null. 
* (or maybe worse, if the test ends  up biased under the non-normal distributions for some alternatives) 
Investigating power is considerably more involved. For a start, with these distributions you'd have to be looking at specifying some copula or copulas, presumably with close to a linear relationship in the untransformed variables, and certainly with close to some specified value for the population correlation coefficient. You'll have to look at several effect sizes (at least), and possibly both negative and positive dependence. 
Nevertheless, if one is to understand the properties of inference with the test in these situations, one cannot ignore the potential impact on power.

*It would seem odd to discuss that particular test of the Pearson correlation without examining alternative tests - for example, permutation tests of the Pearson correlation, rank tests like Kendall's tau and Spearman's rho (which not only have good performance when the normal assumptions hold, but which also have direct relevance to the issue with copulas needed for a power study that I mentioned before), perhaps robustified versions of the correlation coefficient, possibly also bootstrap tests.
