[6gvONxR4sf7o]

>Define for me in an objective way what "at least as extreme" is.

Come up with some one dimensional test statistic T whose distribution D you know under your null hypothesis. Define a one sided p value for data x as p(t <= x).

It sounds like your statistic is 0 if the sequence is always "HHTHT" and 1 otherwise? In this case your p value is 1 unless every attempt is "HHTHT" in which case it's zero, so the test statistic is 0 with probability 1/32^k for k attempts. The more attempts you do, the smaller p gets if the null is false. It's working as intended. For this test, a threshold of p=0.05 would be dumb, but it's always dumb.

It's not an awful test assuming you came up with your test statistic and "HHTHT" before collecting your data. It meshes with the intuition of betting your friend "Hey I bet if you flip this coin you'll get HHTHT." If they proceed to flip it and see HHTHT, they are reasonable to think maybe you know something they don't.

If you come up with your test statistic after the fact, there's theory around p hacking to formalize the intuition of why it's not convincing to watch your friend flip some sequence of coins and then tell them "dude, I totally knew it was going to be that" after the fact.

[cygaril]

A more general method is to use the likelihood ratio, ie the ratio of the likelihood of an outcome under the alternative hypothesis to its likelihood under the null hypothesis. And then pick the outcomes which for which this ratio is highest as the ones which will cause you to reject the null hypothesis. Equivalently, the p-value is the probability under the null hypothesis that the likelihood ratio would be at least this large.

This works in the discrete case too, and gives p=1/32 in the original coin flip case.

[6gvONxR4sf7o]

Is the likelihood ratio test more general? I thought that one of the benefits of the usual NHST framework was that you only need the distribution of your stat under the null. With LRT don't you need the distribution under both the null and the alternative? How do you frame a null of mu = 0 against an alternative of mu != 0 with x ~ D_mu in this way?

[cygaril]

You don't necessarily need the distribution under the alternative to determine the values for which the likelihood ratio will be highest. In your example, the tails will be the areas of maximum likelihood for any (symmetric) alternative.

[6gvONxR4sf7o]

Huh, TIL. Thanks :)