Mailund on the Internet

I just saw a great quote that reminds me of the post on p-values I wrote a few days ago.

To get it, you just need a little back ground (in case you didn't read the earlier post).

With a classical hypothesis test, you have a null model that gives you a distribution of outcomes, and to test the hypothesis you make an observation, say \(\hat{x}\), and you then consider how likely it is to get a value as high or higher as \(\hat{x}\) under this null distribution.

The p-value of \(\hat{x}\) is the probability, under the null distribution, of observing anything higher than \(\hat{x}\) and if this probability is lower than some pre-chosen threshold \(\alpha\), you reject the null hypothesis.  Not that you ever observe any outcome higher than \(\hat{x}\), you just reject the hypothesis if it is unlikely to observe anything higher than \(\hat{x}\).

The hypothesis is rejected based on how much probability it puts on values larger than \(\hat{x}\), not on any values actually observed, higher than \(\hat{x}\).

What the use of P implies, therefore, is that a hypothesis that may be true may be rejected because it has not predicted observable results that have not occurred.

Jeffreys

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See here.

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