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My question may be pedantic, but I find the syntax used to describe the "power" of a hypothesis test really annoying, and I just want to either correct my understanding of the concept, or understand why it is, seemingly, mislabeled.

My understanding of "power" is the probability that a test will reject $H_0$, when $H_a$ is true. Just from this wording, the power of a test seems to be the power in favor of $H_a$, not the power against it. In other words, if $H_a$ is true, you want your test to reject $H_a$. However, my text, Introduction to the Practice of Statistics, used as a text book at our community college, states that this probability is "the power against $H_a$".

Why do they call the probability that $H_0$ is rejected when $H_a$ is true the power against $H_0$ instead of the power in favor of $H_a$, or simply the power of $H_a$?

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    $\begingroup$ That ("power against $H_a$") appears to be a particularly unhelpful phrasing. Which page and which edition does that occur on? ... if I felt compelled to say something particularly close to that, I'd have said 'power to detect $H_a$' instead $\endgroup$
    – Glen_b
    Commented Aug 8, 2013 at 0:25
  • $\begingroup$ The statistical power of a hypothesis test is given by the probability of rejecting the null hypothesis in case it is indeed false. $\endgroup$
    – Jan Rothkegel
    Commented Apr 26, 2016 at 18:23

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I have not checked out your text, but your understanding is correct.

The alternative hypothesis (Ha) is usually stated vaguely as something like the difference between the two population means is not zero. But for the purpose of computing and interpreting power you need a definitive Ha, say that the difference between population means equals 10 (or some value). If that Ha is true, and if you accept all the assumptions of the test, power is the probability that random sampling of data from the two populations with the specified sample size will result in a P value less than alpha.

So yes, it is the power against the null hypothesis and for the alternative.

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  • $\begingroup$ I wouldn't say is the power against the null and for the alternative as it doesn't actually depend on the observations; it seems to me more a characterisation of the test itself, rather than an argument for or against either hypothesis. $\endgroup$
    – Dikran Marsupial
    Commented Aug 30, 2013 at 19:12
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    $\begingroup$ @DikranMarsupial: Right, power does not depend on the observations. But it does depend on sample size and variability. And it can't be computed without defining a clear alternative hypothesis. So power is the characteristic of a test, given a sample size & SD, and a defined alternative hypothesis. $\endgroup$
    – Harvey Motulsky
    Commented Aug 30, 2013 at 23:33
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It seems to me that it is the power of the test itself, rather than the power against either of the hypothesis. If there is very little data, the test will be unable to reject $H_0$ whether it is false or not, so the test itself has little power to reveal anything about the problem. If we have lots of data, we will be able to confidently expect to reject a false $H_0$ even if the likely effect size under $H_a$ is rather small, so the test is powerful.

Perhaps we should think of the power of a test being somewhat analogous to the (optical) power of a microscope or telescope, in that it gives an idea of how fine a distinction we can reasonably expect to resolve.

I suppose you could however argue that if we are unable to reject the null hypothesis using a test with high power, then this is "powerful" support for the idea that $H_a$ is probably false (as if $H_a$ were true, the test would be highly unlikely to fail to reject $H_0$).

See statistical power for info on the statistical meaning of "power".

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    $\begingroup$ "Power" has a very definitive meaning in statistics. Like many statistical terms, it is also an ordinary word so it is easy to get mixed up and apply non-rigorous definitions when in fact "power" in statistics is rigorously defined. $\endgroup$
    – Harvey Motulsky
    Commented Aug 29, 2013 at 15:56
  • $\begingroup$ Agreed, the (original) question appeared to be asking why this quantity was called power. $\endgroup$
    – Dikran Marsupial
    Commented Aug 30, 2013 at 18:55

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