# When to use the term "evidence against" when testing a directional hypotheses

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?

• This issue is an artefact of using a point null in a one-sided situation. You have found evidence strongly consistent with nonpositive slope. Commented Mar 22, 2023 at 19:10
• And no support for the hypothesis of positive slope. Commented Mar 22, 2023 at 19:13
• 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? Commented Mar 23, 2023 at 9:38