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I was recently posed a homework question. It will be a few days before I can get in-person help from faculty, so I wanted to ask here. The question was:

A medical resident doing research wants to know if the mean heart rate after a particular type of trauma differs from the healthy population rate of 72 beats/min. They consider a mean difference of 6 beats/min to be clinically meaningful. They also choose 9.1 beats/min as the variation based on a previously published study. How many patients will be needed to carry out the study at 0.0005% significance and 82% power?

I've been struggling to find an answer for the "power" portion of the question, because every reference (between our two textbooks, internet searches, and CV) has had power calculations relative to a specific alternative hypothesis; and as far as I've seen, this is how power is defined.

I've since handed in the homework, and have emailed the professor and a TA. The professor directed me to some lecture notes that supposedly lead to a formula for sample size based on a specified significance, power, and rejection criteria:

$$n = \left[\frac{\sigma(z_\alpha + z_\beta)}{u_1-u_0}\right]^2$$

with $$z_\alpha = \frac{c - \mu_0}{\sigma / \sqrt{n}},\quad z_\beta = \frac{c-\mu_1}{\sigma/\sqrt{n}}$$

and $c$ is (presumably, the notes are not very clear) the cutoff value for the critical region.

But as far as I can tell, these still rely on a simple (not compound) alternative hypothesis $\mu_1$.

The TA's response was:

There’s actually a alternative hypothesis. There’s maybe some wording problem with this question, it actually means the null hypothesis is that a mean difference of 0 is clinically significant. So the alternative hypothesis would be a mean difference of 6 is clinically significant. Then you can use the formula.

So perhaps I'm just misreading the problem; but I still can't make sense of what the "mean difference of 6" is supposed to be, if not a compound alternative hypothesis (i.e. that $H_1: Y \leq 66 \text{ or } Y \geq 78$).

The questions I'd like answered are thus:

  1. Is this question bunk?
  2. If not, what is the alternative hypothesis? Should I be using the difference of two means (i.e., a paired t test or similar) rather than a test about one mean?
  3. Is it true that the power of a statistical test is always with respect to a particular (i.e., simple; non-compound) alternative hypothesis?
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  • $\begingroup$ Please add the self-study tag. $\endgroup$ – Michael R. Chernick Dec 2 '19 at 1:58
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The TA's response doesn't quite make sense to me (I wonder whether there's a typing mistake or if it's an issue of understanding), but the original question does make sense.

With a composite alternative hypothesis, power is different at different points in the alternative space. If you want a value for power you need to pick one.

The question supplies one obvious point of interest under the alternative (the minimum clinically relevant mean difference), and hints at no others, so which specific alternative it is intended that you use seems obvious enough.

With that, I think you have all the ingredients you need to use the formula.

With a two-tailed test, a "mean difference of 6" merely indicates that $|\mu-\mu_0|=6$. You don't need to worry about whether the difference is positive or negative; focus on using a test statistic of $|z|$. Whether that strictly counts as compound or not depends on your point of view.

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  • $\begingroup$ Got it: so (presumably) the questions was asking for the power at 6; not saying that $\pm$ 6 was the rejection criteria. I think I know where to go from here... $\endgroup$ – Peter Dragos Dec 2 '19 at 15:50
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    $\begingroup$ Correct. Power is "if the difference was 6, what's the chance you reject?" ... in this case you then want to look at "How big does $n$ need to be to make that chance 82%?" $\endgroup$ – Glen_b -Reinstate Monica Dec 2 '19 at 22:47

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