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Another question has gotten me thinking about testing more than just correlated/not correlated. Let's say that I have some data and find the correlation to be $0.15$. However, I then realize that correlation doesn't really interest me unless it's at least $0.10$. Moreover, a correlation of $-0.10$ is equally interesting, and I don't care about the sign, just if the variables have a sufficiently strong relationship for me to care.

I have gotten as far as thinking that I would want to test my observed $r^2$ against a null of $\rho^2\le 0.1^2 = 0.01$. I cannot think of a good way to test this. Certainly if I want to test against $\rho^2=0$ then I can just do a permutation test.

What about if I want to test the more general $\rho^2\le \eta?$

Thanks!

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    $\begingroup$ Why can’t you simulate the empirical distribution of your r2 in repeated samples and take an empirical quantile? $\endgroup$
    – Fr1
    Commented Aug 20, 2019 at 13:58
  • $\begingroup$ @Fr1 So take bootstrap samples of my bivariate observations, calculate $r^2$ and then hit it with ecdf(R2)(eta)? $\endgroup$
    – Dave
    Commented Aug 20, 2019 at 14:11
  • $\begingroup$ yes: If I understood correctly (as I believe) ecdf(R2)(eta).. then yes, this is what I was referring to $\endgroup$
    – Fr1
    Commented Aug 20, 2019 at 14:13
  • $\begingroup$ Upvoted the question because this is the kind of stuff in statistics that I terribly like.. the non parametric tests and empirical distributions are something that has an incredibly wide spectrum of applications, and many times are incredibly useful (at least when you can bootstrap in reasonable time! Which sometimes is difficult!! Or requires huge computational power, so huge costs, direct infrastructure or cloud or a mix of the two) $\endgroup$
    – Fr1
    Commented Aug 20, 2019 at 14:16
  • $\begingroup$ @pglpm $H_0: \theta \le \theta_0$ is a totally acceptable hypothesis test. I'm allowed to test $\theta = \mu_1 - \mu_0 \le 0.25$, so why not $\theta = \rho^2 \le 0.25$? $\endgroup$
    – Dave
    Commented Aug 20, 2019 at 18:39

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