It seems to me that, after having a predetermined level of significance, the Neyman-Pearson approach only requires to see if the observed $x$ lies in the rejection region or not. In this case, how can one relate it to the $p$-value? Or is it an alternative way to judge if $x$ is rejected or not by using $p$-values? After reading some references about Neyman-Pearson and Fisher's treatments, I am confused.
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2It may help you to read my answer here: When to use Fisher and Neyman-Pearson framework? As for how the rejection region relates to p-values, the RR is defined as that region where $p<\alpha$. – gung - Reinstate Monica Oct 22 '13 at 01:51
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It is an excellent explaniation. So under Neyman's setting, does it requre any property of T(x), so that p<α is equivalent to x belongs to the rejection region? – user28363 Oct 22 '13 at 01:58
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I'm not sure I totally follow your question, but there is no property of T(x) being in the rejection region except that p<a. – gung - Reinstate Monica Oct 22 '13 at 02:04
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My question is that, how does p<alpha guarantee that x is rejected. Because Neyman-Pearson does not necessrily require p-value in the procedure, so I think this p-value has to be defined first. – user28363 Oct 22 '13 at 03:55
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1Yes, p<a guarantees that x is rejected. The N-P approach is based on maintaining a maximum type I error rate, so alpha comes 1st, you figure out what T(x) is when p=a, & that defines the rejection region. – gung - Reinstate Monica Oct 22 '13 at 04:15
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I have a pretty lengthy post with relevant citations about the Fisher-Neyman-Pearson NHST framework and its origins. As a reminder of some definitions which answer some of your questions (by making these two pieces more explicit):
- The $p$-value is the probability of obtaining an estimate equal to or more extreme than the observed value if the null hypothesis is true. Note that we can only tentatively reject the null, and can neither accept the null hypothesis $H_0$ nor can we accept the alternative hypothesis $H_1$ given this information.
- The $\alpha$ value is the probability of rejecting the null hypothesis when it is actually true. You may notice that this makes $\alpha$ and $p$ different things. The $\alpha$ value is a pre-determined cutoff or a "zone of danger" with which we approximate $p$ values potentially making wrong decisions. We can have $p =[.01,.02,.03,.04,.05]$ but the $\alpha$ here is always set at $.05$. We cannot ever know with certainty whether we have avoided a false rejection, but setting $\alpha$ low enough should, in theory, reduce the number of false rejections by a lot.
On to your actual questions, the Neyman-Pearson perspective was the first to adopt this very idea of $\alpha$ level cutoffs. While in their original paper which proposed this never discussed an explicit cutoff like $.05$, they present a very lengthy discussion of why this may be important (given the reasons noted above along with others). So unlike Fisher, N-P proposed a decision-making framework which explicitly built in $\alpha$ as a way to make hardline decisions about a given $p$-value observed in the data.
Shawn Hemelstrand
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