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I have seen several texts that assert that for left-tailed and right-tailed tests, $H_0$ should be specified only in the form of, for example, $H_0: \mu = x$. The rationale for this seems to be that it is superfluous to say $H_0: \mu \leq x$ or $H_0: \mu \geq x$ because the definition of the $H_A$ (using $<$ or $>$) and the resulting directionality (left-tailed or right-tailed) of the test already implies whether we mean $\leq$ or $\geq$ when we specify $H_0$.

Is this strictly true or just a matter of style? Are there any cases, either with testing $\mu$ or in other areas, where this is not true? I think I've also seen texts that say just that $H_0$ must use an equals sign of some kind (i.e $=$, $\leq$, or $\geq$) but I'm not sure.

TooTone
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sparc_spread
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2 Answers2

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It's not simply a matter of style - and if it were, I'd actually advocate for including the inequality.

What makes the equality both relevant and important is two fold:

(1) often, one is really only interested in the equality case or the alternative - for example, when the inequality one the other side from the alternative would be extremely unlikely (e.g. the likely way a new drug works is fairly well understood and has no mechanism by which cholesterol could be worse - it either helps lower cholesterol or it doesn't).

(2) the way the null distribution (the distribution of the test statistic under the null) is generated in a one-tailed test is to deal only with the equality case; while the $<$ part of $\leq$ may be logically possible, it's ignored in figuring out significance levels (like $P(T\geq t_c|H_0)$ for some statistic $T$) and p-values ($P(T\geq T_\text{obs}|H_0)$).

(That said, in practice many people - though far from everyone - do include the inequality in stating the null, rather than just the equality. Different books take different stances on it.)

Glen_b
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  • You wrote "while the < part of ≤ may be logically possible", do you know an example of a NHST where the Null is an inequality? I was under the impression that to do such a thing one would need to assume a probability distribution for the parameter (i.e. do Bayesian statistics). – Michael Mar 28 '19 at 15:23
  • Many people hold that the negation of the alternative is the correct null, and the logic/mathematics for the $\leq$ one-sided test has a fairly long history. As an outline of a situation where it makes sense, consider a proposed intervention or treatment that is a bit more expensive per subject (after accounting for economies of scale) than the current one. You could only be interested in it if it was actually better than the current one, and all the worse cases would be no more of interest than the equality case. This kind of situation crops up all the time with one tailed tests... ctd
    • – Glen_b Mar 29 '19 at 00:41
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    ctd... (On the other hand there are cases where you really don't anticipate any possible case "worse than" no effect and then writing it only as an equality makes perfect sense). In between there are situations where proponents of either way of writing the null in a one-sided test can happily argue their points. 2. There's no need to consider any kind of prior on the parameter; you take the significance level as the largest type I error rate in the region of the null. This is standard. (Though you'd really prefer that to occur on the boundary if possible) – Glen_b Mar 29 '19 at 00:41
  • Just to make sure I understood your last comment: the "$p$-value" under a null hypothesis of the form $\mu \ge \mu_0$ would then be defined as $\max_{\nu \ge \mu_0}P(T\leq t|\mu = \nu)$ and not as $P(T\leq t|\mu \ge \mu_0)$? Here $T$ is the test statistic an $t$ its observed value. – Michael Apr 02 '19 at 10:07
  • Yes; it will typically occur at $\mu_0$ – Glen_b Apr 03 '19 at 00:10