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I was reading a social science paper that tried to explain the correlation between two variables. In that reference there was mention of its standard-error and p-value and that got me to thinking whether the author was using an unknown formula (to me) involving both the explanatory and response variables. My hunch is that he was measuring the standard-error and p-value for the response variable in question.

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  • $\begingroup$ Your question lacks detail. Which paper? Why did you think that the author was using a formula "involving both the explanatory and response variables"? $\endgroup$
    – Glen_b
    Jan 1, 2014 at 5:50
  • $\begingroup$ Probably it is about the standard error of the correlation coefficient and a p value for testing whether the two variables are truely correlated? $\endgroup$
    – Michael M
    Jan 1, 2014 at 10:50

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Let's forget "explanatory" and "response" for a moment and just consider two random vectors $\mathbf{x}$ and $\mathbf{y}$. You're interested in a the sampling distribution of the scalar quantity $$r = f(\mathbf{x}, \mathbf{y}),$$ where $f$ is the function for the sample correlation.

If you want to learn about the general problem, you could spend weeks/months/years studying "transformations of random variables." If you wanted a brute force solution, you could run simulations. Or you could search for derived results, like in this JSTOR link. For instance, when the population correlation $\rho = 0$, $$\frac{1+r}{1-r} \sim F(n-2, n-2).$$

You could derive confidence intervals off of that. But for a more general expression of the sampling distribution of $r$, see this blog article. It's a pretty nasty expression.

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