How do you choose which side of the tail to use in an hypothesis test? Disclaimer: even recommending a book on it is enough for me, but I've searched on 10 so far and none teach how to choose which side of the tail in a general sense.
My test statistic T is:
$$\cfrac{(\hat{\beta}_{1} + \hat{\beta}_{2} - (\beta_{1}+\beta_{2}))^2}{\hat{\sigma}^2_{\hat{\beta}_{1}} + \hat{\sigma}^2_{\hat{\beta}_{2}} + \hat{\sigma}^2_{\hat{\beta}_{1},\hat{\beta}_{2}}} $$
My null hypothesis is $$H_{0} : \beta_{1}+\beta_{2} = k$$
My alternative is $$H_{1} : \beta_{1}+\beta_{2} \geq k$$
The test statistic clearly has an $F_{1,n-k}$ distribution, and I want my $\alpha$ to be 0.05. Most books recommend to use the tail in the direction of the null hypothesis, but why is it the case?
My logic is failing me completely, and I can't understand why I would use a right tail rather than the left tail. Thanks for your answers!
 A: I highly recommend Bickel & Docksum for understanding hypothesis testing. They are very formal with what they do.
When you do a hypothesis test, besides setting the null hypothesis, you have flexibility of setting the direction of alternative hypothesis, namely, if it is one-sided or two sided, as well as its direction. The former means that you would test for any violation of null hypothesis, regardless of direction. The latter means that you would search for violation of null hypothesis in only one direction, and you then have to choose which direction that is. After you have chosen a direction, you will test if the tail of the probability distribution under null hypothesis is large or small integrated after the point of the true value. The logic is as follows: The null hypothesis is violated if the true result in the chosen direction is more extreme than a result in that direction that could have happened randomly.
For example, you observe a patient's weight for a year, then administer weight loss pills and observe for another week. You are interested if weight loss due to pills was significant under null hypothesis that pills don't work. Patient lost 1kg in a week. You will test if the patient could have potentially lost more than (or equal) than 1kg in a random week by chance. If no, then pills work with given confidence level, if yes, then test failed inconclusively.
Generally you can apply a two-sided test to a one-sided problem, and it will work, but you will lose a factor of 2 in your statistical power, because you have to test both directions.
A: Usually the claim complementary to the one you hope to establish is posed as the null, so you can say that the alternative is consistent with the data when/if you reject. You don't provide enough context to determine which is the case in your situation.
However, you do need to cover the parameter space with your hypotheses (assuming that values less than $k$ are possible). Your hypotheses do not tell you what to do when $\beta_{1}+\beta_{2}<k$, so they are incomplete.
For a strong superiority test, for example, you should have something like this:
$$H_{0} : \delta = \beta_{1}+\beta_{2} \leq k $$
$$H_{1} : \delta = \beta_{1}+\beta_{2} > k$$
The null here is a composite one: it’s an interval rather than a single point you get with a two-sided null hypothesis. But it's hard to calculate a probability of seeing a sum under the null where there's an infinite number of points at which to evaluate that probability. What we do instead is to calculate the probability at the most extreme point of the null hypothesis, closest to alternative parameter space, which is at $\delta = k$. This means that the p-value is exact only for $\delta = k$. If $\delta < k$, then our p-value is just a conservative bound on the type I error rate. This is also the reason why statistics packages will express the one-sided null as $\delta = k$ rather than $\delta \leq k$. It is technically correct, but very confusing notation.
In this strong superiority example, seeing a $\delta$ way out in the left tail will not lead to a rejection of the null. Only values far out in the right tail count as evidence against it.
