This is somewhat related to another question that I asked. The question I have is, when doing hypothesis testing, when the alternative hypothesis is a range, the null hypothesis is still a point value.
As an example, when testing whether a correlation coefficient is greater than 0.5, the null hypothesis is "correlation = 0.5" instead of "correlation <= 0.5". Why is this the case? (or have I got it wrong?)

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    $\begingroup$ This question is essentially the same as stats.stackexchange.com/q/7853/919 . $\endgroup$ – whuber Mar 12 '11 at 20:42
  • $\begingroup$ You're essentially right. I somehow missed that Q completely, which is why I didn't spot this when commenting on this questioner's previous Q. $\endgroup$ – onestop Mar 13 '11 at 9:13

First, it is not always the case. There might be a composite null.

Most standard tests have a simple null because in the framework of Neyman and Pearson the aim is to provide a decision rule that permits you to control the error of rejecting the null when it is true. To control this error you need to specify one distribution for the null.

When you have a composite hypothesis there are many possibilities. In this case, there are two natural types of strategies, either a Bayesian one (i.e. put weights on the different null distribution) or a minimax one (where you want to construct a test that has a controlled error in the worst case.

In the Bayesian setting, using the posterior, you are rapidly back to the case of a simple null. In the minimax setting, if the null is something like corre $\leq$ 0.5 it might be that the problem is equivalent to using the simple null corre = 0.5. Hence to avoid talking about minimax people directly take the simple null that is the 'extreme point' of the composite setting. In the general case it is often possible to transform the composite minimax null into a simple null... hence treating rigorously the case of a composite null is to my knowledge mostly done by going back somehow to a simple null.

  • $\begingroup$ In fact, this applies to the alternative hypothesis as well. $\endgroup$ – charles.y.zheng Mar 12 '11 at 15:10
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    $\begingroup$ I don't think you need to specify a simple null hypothesis in the N-P framework. The explanation appears in my reply at stats.stackexchange.com/questions/7853/… $\endgroup$ – whuber Mar 12 '11 at 20:45
  • $\begingroup$ @whuber Sorry but what is N-P framework ? $\endgroup$ – robin girard Mar 13 '11 at 6:30
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    $\begingroup$ N-P = Neyman-Pearson $\endgroup$ – onestop Mar 13 '11 at 9:09
  • $\begingroup$ @whuber you're right ! but somehow the case you point is covered by the second part of my answer ? the composite alternative is handled within a minimax framework and the composite minimax is transformed into a simple null $\endgroup$ – robin girard Aug 21 '12 at 12:09

I don't think that the null hypothesis should always be something like correlation=0.5. At least in the problems which I have come across that wasn't the case. For example in information theoretic statistics the following problem is considered. Suppose that $X_1, X_2, \cdots, X_n$ are coming from an unknown distribution $Q$. In the simplest case we want to test between two distributions $P_1$ and $P_2$. So the hypotheses are $H_1:Q=P_1$ and $H_2:Q=P_2$.

  • $\begingroup$ ah, didn't see robin's post! $\endgroup$ – Ashok Mar 12 '11 at 10:29

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