#StackBounty: #regression #hypothesis-testing #diagnostic $H_0$ vs $H_1$ in diagnostic testing

Bounty: 50

Consider diagnostic testing of a fitted model, e.g. testing whether regression residuals are autocorrelated (a violation of an assumption) or not (no violation). I have a feeling that the null hypothesis and the alternative hypothesis in diagnostic tests often tend to be exchanged/flipped w.r.t. what we would ideally like to have.

If are interested in persuading a sceptic that there is a (nonzero) effect, we usually take the null hypothesis to be that there is no effect, and then we try to reject it. Rejecting $H_0$ at a sufficiently low significance level produces convincing evidence that $H_0$ is incorrect, and we therefore are comfortable in concluding that there is a nonzero effect. (There are of course a bunch of other assumptions which must hold, as otherwise the rejection of $H_0$ may result from a violation of one of those assumptions rather than $H_0$ actually being incorrect. And we never have 100% confidence but only, say, 95% confidence.)

Meanwhile, in diagnostic testing of a model, we typically have $H_0$ that the model is correct and $H_1$ than there is something wrong with the model. E.g. $H_0$ is that regression residuals are not autocorrelated while $H_1$ is that they are autocorrelated. However, if we want to persuade a sceptic that our model is valid, we would have $H_0$ consistent with a violation and $H_1$ consistent with validity. Thus the usual setup in diagnostic testing seems to exchange $H_0$ with $H_1$, and so we do not get to control the probability of the relevant error.

Is this a valid concern (philosophically and/or practically)? Has it been addressed and perhaps resolved?


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