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In the sciences very often we are testing for the presence of a predicted relationship, for example, that two variables are positively correlated ($\beta > 0$). Under traditional frequentist testing, one might set the null hypothesis to be $H_0: \beta = 0$, with a compound alternative hypothesis of $H_{1,a}: \beta > 0$ and $H_{1,b}: \beta < 0$.

If one finds strong support for $H_{1,b}$ (that the slope is negative) is it correct to say that this is "evidence against $H_{1,a}$"? Typically we only state that we find evidence against the null hypothesis, but in the case of a directional test it seems intuitive that if you find strong evidence for a negative slope, then this should also count as strong evidence against a positive slope, not simply against the null of $H_0: \beta = 0$.

So is it correct to say use the term "evidence against" one of the other compound alternative hypotheses? Otherwise it seems to imply that no one can ever find evidence against a presupposed hypothesis, despite evidence to its contrary. Or is it that my null and alternative hypotheses are not formed correctly?

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  • $\begingroup$ This issue is an artefact of using a point null in a one-sided situation. You have found evidence strongly consistent with nonpositive slope. $\endgroup$ Commented Mar 22, 2023 at 19:10
  • $\begingroup$ And no support for the hypothesis of positive slope. $\endgroup$ Commented Mar 22, 2023 at 19:13
  • $\begingroup$ But what is the correct null then? If we choose $H_0: \beta > 0$ (which is what we hypothesize to be true) then this somewhat contradicts the idea of the null being "no relationship". However, if we select $H_0: \beta < 0$ then we can never "reject" $\beta > 0$ since you typically can only reject the null. And so it seems paradoxical -- how do you frame the question so that you can reject the hypothesis of a positive slope? $\endgroup$ Commented Mar 23, 2023 at 9:38

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