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At least two tests are common for testing the significance of a Pearson correlation coefficient.

  1. Comparing $t=\frac{\sqrt{r^2/(1-r^2)}}{\sqrt{\frac{1}{N-2}}}$ to Student's $t$-distribution with $N-2$ degrees of freedom.
  2. Comparing $z'=\frac{\tanh^{-1}r-\tanh^{-1}\rho_0}{\sqrt{\frac{1}{N-3}}}$ to a standard normal distribution. ("Fisher's transformation.")

My current understanding is that a disadvantage of the first test is that it is only valid for $H_0:\rho=0$ (and consequently is inappropriate for constructing confidence intervals), and a disadvantage of the second test is that $z'$ is only "approximately" normally distributed (i.e. biased for small samples).

If I were deciding between these two tests (e.g. not boostrapping), I would therefore prefer the $r$-to-$t$ test when and only when I am testing $H_0:\rho=0$, and I would prefer the $r$-to-$z'$ test when and only when I am testing $H_0:\rho=\rho_0\neq0$ and my sample size is large. Is this correct?

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  • $\begingroup$ Thanks. I fixed the denominator. $\endgroup$
    – Nils Enevoldsen
    Commented Nov 20, 2019 at 15:17
  • $\begingroup$ Have you ever found yourself in a position where you actually tested against something other than $\rho=0$? $\endgroup$
    – Bernhard
    Commented Nov 20, 2019 at 15:23

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That is correct.

(The answer to my question was posted by another user as a since-deleted comment. I'm adding this comment so I can mark the question as answered.)

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