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In hypothesis testing for the value of the population mean, when computing type II error $\beta$, if the alternative hypothesis includes many possibilities, is it always the case that $\beta$ is approximated? In other words, when we compute a numerical value for $\beta$, based on the mean from the sample $\bar{X}$, shouldn't we write $\beta \approx P[\text{do not reject } H_0 \mid \mu = \bar{X} ]$?

If there is no other information available, of course, it's the only way to compute $\beta$. It's the best we can do. But since $\bar{X}$ may very well not be $\mu$, shouldn't it be concerned that it's $\approx$ and not =?

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    $\begingroup$ Gene, welcome to the site. You should migrate questions, not ask them on multiple sites. You need to find a way to close the original math.SE question if you expect to get an answer here. Your question per se is unclear though. The alternative hypotheses are formulated as $\mu=\mu_1$ or $\mu \in \Omega_1$, while $\bar X$ may be a part of the test statistic, but not the hypothesis. $\endgroup$
    – StasK
    Feb 17, 2013 at 14:37
  • $\begingroup$ Thank you... I asked for the question to be migrated. As for the hypotheses, in this case, $H_0$ is of the form $\mu=\mu_0$ and $H_1$ is $\mu\not=\mu_0$. Indeed $\bar{X}$ is not part of the hypothesis; it is apart of the test statistic. And this is what's getting me: the sample statistics are affecting $H_1$... @drknexus Actually, I understand how $\beta = P[\text{do not reject } H_0|\mu\not=\mu_0]$, but not $\beta = P[\text{do not reject } H_0|\mu=\mu_1]$ where $\mu_1$ is taken to be the value found in the sample. And also... just the sample equiv. of $\beta$ would make sense... hence approx. $\endgroup$ Feb 17, 2013 at 17:55

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Well, given that $\beta$ uses the greek symbol, as a matter of nomenclature we'd expect it to be a population value. As a population value the number is considered to be certain (=), not an estimate ($\approx$). Moreover, as alluded to by @StasK, the hypothesis are about the population parameters, not the sample statistics.

One problem here is with notation. $\beta$ and $1-\beta$ (power) are relevant only in worlds where $\mu_{control} \neq \mu_{treatment}$. Therefore, your question should be "shouldn't we write $\beta \approx P[\text{do not reject }H_0 | \mu_{control} \neq \mu_{treatment}]$".

I think some of the confusion here comes from a priori vs posthoc power analysis. In the case of a priori analyses we make assumptions about the population parameters and thereby yield the associated $\beta$ given those assumptions.

Although it is true that in the case of posthoc analyses we are using sample statistics to calculate $\beta$ we are implicitly considering them to be population parameters. Therefore, although it is true that $\beta \approx P[\text{retain }H_0 | \mu = \bar{X}]$ it assumed that what we really are saying is that $\beta = P[\text{do not reject }H_0 | \mu_{control} = \mu_{treatment}]$. Therefore, in the formulation including $\bar{X}$ the quantity being estimated should not have the $\beta$ notation, but the sample equivelant. If you are willing to make the (common) assumption (for this type of analysis) that $\bar{X} = \mu_{treatment}$ then you are good to use $=$ rather than $\neq$.

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  • $\begingroup$ Thank you for your answer... Since computing "sample" $\beta$ on different sample values can change it drastically, it shows how writing $\beta \approx ...$ is wrong. It is indeed a "sample" $\beta$ that I want to refer to. Then $\beta (\bar{X})$ is an appropriate notation for it? $\endgroup$ Feb 17, 2013 at 18:18
  • $\begingroup$ I am not aware of an appropriate notation to report "sample" $\beta$. I think that may be because the notion of a sample $\beta$ is incoherent. $\beta$ and power ($1-\beta$) refer either to the proportion of samples that will have a given outcome given $\alpha$ or refer to the proportion of the time we expect each outcome. At any rate, both frameworks indicate that $\beta$ is telling us something about the sampling distribution of outcomes where any given $\bar{X}$ is telling us about a specific sample. $\endgroup$ Feb 18, 2013 at 2:01
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    $\begingroup$ As an aside, I have an inkling that you could create confidence intervals around $\bar{X}$ and $S$ and calculate $\beta$ under 5% and 95% case scenarios (or maybe 9.75% and 90.25%) and thereby create a confidence interval for $\beta$ that would appropriately reflect the uncertainty of the sample estimates of the population parameters. $\endgroup$ Feb 18, 2013 at 2:04
  • $\begingroup$ ... if I were to guess at a notation it would simply be lower-case 'b' in the same way we use lowercase 's' to stand for the sample version of $\sigma$. $\endgroup$ Feb 18, 2013 at 2:07
  • $\begingroup$ I was starting to think that $\beta$ should stay in a form as a function of $\bar{X}$, but I'm curious about the confidence interval idea... $\endgroup$ Feb 18, 2013 at 11:45

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