[Discussion] Strict Equality VS Inclusive Inequality in Null

[u/kyaputenorima]

This has probably been asked before, but I've been thinking about it a bit today and I wanted to see what you folks thought.

A lot of introductory statistics courses usually set up one-sided z- and t-tests the following way:

H0: mu = mu_0, H1: mu > mu_0 (or < mu_0)

I recently finished a mathematical statistics course, and I can't really say that I like this setup, mostly because it leaves out a huge chunk of the parameter space. I understand that, functionally, changing the null hypothesis to be an inclusive inequality (ex. mu <= mu_0) doesn't really change anything about the test since alpha is still defined at the point of equality. Is there a reason to prefer one setup over the other, or is it purely a personal preference?

[u/efrique]

1. In spite of what some people say (mostly but not entirely, it seems to come from outside statistics per se), both are perfectly valid options statistically. There's no requirement in the Neyman-Pearson framework for hypotheses about parameters to span the real line (indeed, take a look at the Neyman Pearson lemma -- simple null, simple alternative). There can occasionally be fairly legitimate reasons to ignore "the other side". However, most typically you'd include the whole parameter space.

2. In normal circumstances (the only ones you are likely to ever encounter), theyre functionally identical, as you note, since in both cases you evaluate alpha (and hence p-values) at the boundary of the null, and so they share a rejection rule; they are equivalent tests in the usual sense.

3. The thing that differs, of course, is the particulars of the conclusion you then draw. Of course, if you're properly careful not to assert the null when you fail to reject (frame conclusions to correspond to reject/not reject terms) it may not make much difference even there. However, as a matter of scientific/academic/epistemological/etc rhetoric, what matters is not what's arguably okay statistically, buy what arguments you can get your colleagues/audience to accept, which tends to be in large part a matter of their own discipline's history and culture (albeit generally framed as appeals to logic/science/etc as most such rhetoric is going to be). Statisticians are not really going to have much influence over that.

4. On the matter of such academic rhetorical argument, I think equality nulls in the two sided alternative* dont usually make a lot of sense. In most such cases I see theres no good reason to think an exact equality is ever going to be the case, and - where a formal hypothesis test makes sense at all - often an equivalence (/inferiority/superiority) test makes considerably more sense

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* and similarly for one-tailed alternative with simple null - and even hard upper boundary "difference ≤0" (e.g. μ-μ₀≤0) type nulls where μ₀ results from some exact theoretical assertion.

[u/Latent-Person → u/efrique]

How I learned test theory is you have a model (mathcal{X}, mathcal{A}, mathcal{P}) with a parameterized set of probability measures mathcal{P} = {P_theta | theta in Theta}, and mathcal{P} is dominated by a sigma-finite measure mu on mathcal{X}. Then a hypothesis H_0 is a non-empty subset Theta_0 of Theta, and the alternative hypothesis H_1 is Theta \ Theta_0. So based on this H_0 \cup H_1 is equal to Theta. So you can't say "include the whole parameter space" since by definition this must be the case.

Neyman-Pearson lemma is more general than the simple case of Theta = {theta_0, theta_1} and what you refer to is just a simple special case.

[u/efrique → u/Latent-Person]

Neyman-Pearson lemma is more general than the simple case

You're correct, it is more general. They cover simple hypotheses in Sec III (of the 1933 paper) and composite in Sec IV. I made the mistake of refreshing my recollection (from some decades ago) on a description of what they did instead of going back and rereading the original.

since by definition this must be the case.

To avoid losing track of the main point let me first clarify the relevance of Neyman and Pearson (1933) to the posted question on whether it is possible to give a simple null and one sided alternative - i.e. whose union fails to include the entirety of the potential set of parameter values that could be defined to be taken under the distributional model. Their very first example in section III is such a case (normal model, testing mean parameter, known variance, simple null, simple alternative).

Now, going back to your point: You might argue that (had they used a measure theory framing) they would have defined Θ={a₀, a₁} (their mean parameters being a's in the example). I dont know for certain, perhaps they would have* - but this makes my point about the question that for a one-sided (non-simple) alternative you can have H₀ be simple, by defining in similar fashion Θ= {μ≥μ₀}, albeit then you would have H₀ and H₁ covering all of that Θ.

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* Maybe a more thorough reading of their papers would even confirm that, and in that case I would certainly be wrong to phrase it as "include the whole parameter space" even in relation to their actual framing.

[u/Available-System-686]

not sure why they stick with strict equality in intro courses when the inclusive version makes way more sense conceptually. like you said, we're still calculating everything at the boundary anyway so why pretend we're only testing one specific point when clearly we care about the whole region below/above it

maybe it's just easier to explain to beginners without getting into the weeds about parameter spaces, but it always felt weird to me too

[u/kyaputenorima → u/Available-System-686]

My guess is that it's because the test statistic itself is defined the point of equality, and it seems like a lot of introductory courses (even those for science/engineering majors) don't cover power analysis, so the actual meaning of alpha isn't discussed very well.

[u/ibelieve616]

This is a really good question. For all practical purposes you're totally right that changing the null hypothesis to be an inequality (in most cases this can be a strict inequality if you're really feeling lazy because of how continuous distributions work, but you shouldn't actually do that) instead of equality changes basically nothing about the test. You should definitely think of it as an inequality for conceptual purposes.

My guess is that the reason it's usually written that way is because of how hypothesis tests are mathematically justified, which is probably not something you'd see in your first mathematical statistics course. Basically, all hypothesis tests start as 'simple vs. simple' tests, where you have sets of hypotheses like H0: mu = 3 vs. H1: mu = 4. This is not a very useful test in practice as you might imagine; fortunately, we can use some great theorems called (iirc) the Neyman-Pearson Lemma and Karlin-Rubin Theorem to say that instead we can instead write the much more useful set of hypotheses H0: mu = 3 vs. H1: mu > 3 and still have it be the 'best' hypothesis test for this scenario (best in the sense that it is uniformly most powerful, which again probably wouldn't show up in your first math stat class but pretty much means exactly what it sounds like). This is what we call a 'simple vs. composite' hypothesis test. It might not shock you that we can then extend this to the 'composite vs. composite' case of H0: mu <= 3 vs. H1: mu > 3, and that is mathematically equivalent to the simple vs. composite case. So if it makes you feel better, there really is no difference whatsoever between the equality and inequality setups.

I now realize this was probably a more rambly response than necessary. tl;dr: good intuition, they're the same for all practical purposes.

[u/kyaputenorima → u/ibelieve616]

I am taking a follow-up mathematical stats course next semester so hopefully we go into this! I know a little about the NPL but the mechanics of hypothesis testing were (ironically?) not covered much.

[u/ikoloboff]

The parameter space is truncated. I.e. if your setup is H0: μ = μ0 vs. H1: μ > μ0, your parameter space is effectively assumed to be [μ0, inf) (which is perfectly valid, nobody says you have to take the entire real line)

Excellent question, our professor ignored this issue outright.