Timeline for $p$-value: Fisherian vs. contemporary frequentist definitions
Current License: CC BY-SA 4.0
14 events
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| Dec 2, 2023 at 16:24 | comment | added | Richard Hardy | Thank you for the elaboration! I have spent an hour thinking about this but am feeling slow today... | |
| Dec 2, 2023 at 14:08 | comment | added | Sextus Empiricus | Possibly the use of densities is used in defining cut-off regions in a two-sided Fisher exact test when the distribution is non-symmetric and Fisher may have used that approach as well? | |
| Dec 2, 2023 at 14:04 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Dec 2, 2023 at 13:18 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Dec 2, 2023 at 13:10 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Dec 2, 2023 at 12:56 | comment | added | Sextus Empiricus | @RichardHardy I have added an example for the observation of a pair of variables that follow $x,y \sim Laplace(\mu,1/2)$ and the consideration of the null test $H_0: \mu = 0$. The scatterplot shows the null joint distribution of x, y. The curve shows the null distribution of the chi-squared statistic. The regions of lowest density do not correspond for the two representations. | |
| Dec 2, 2023 at 12:51 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Dec 2, 2023 at 12:45 | history | edited | Sextus Empiricus | CC BY-SA 4.0 |
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| Dec 2, 2023 at 11:31 | comment | added | Richard Hardy | The part you explain in 2) is clear to me, but what is your $H_0$ and what are the test statistics you are comparing (explicitly, as functions of the sample)? And what are their distributions under $H_0$? | |
| Dec 2, 2023 at 11:27 | comment | added | Sextus Empiricus | This quote “First, F tests and χ2 tests are typically rejected only for large values of the test statistic. Clearly, in Fisherian testing, that is inappropriate” makes little sense to me. 1) I am unaware of Fisherman testing using the density of the statistic to decide about 'extreme' values and the computation of a p-value. 2) The chi-squared statistic might have a low density for small values, but this contrasts with the joint distribution of the sample having a high value. The statistic that is used changes the density and makes this density based p-value ambiguous. | |
| Dec 2, 2023 at 11:25 | comment | added | Richard Hardy | I wrote (and meant) falling, not failing. I do not quite follow your argument; I think a bit more detail would help bridge the gap between your examples and how they can be used in hypothesis testing where we have a $H_0$, a test statistic and its distribution under $H_0$. | |
| Dec 2, 2023 at 11:21 | comment | added | Sextus Empiricus | @RichardHardy it is not really considered failing. The p-values can be correct. But the issue is that there are no uniquely defined p-values when you consider the highest density region. If you consider two sided chi-squared test, then the p-value will differ if you compute it based on $\chi$ statistic versus based on $\chi^2$ statistic. | |
| Dec 2, 2023 at 11:15 | comment | added | Richard Hardy | Are these the same densities we are talking about? Can your example be interpreted as an instance of a test statistic falling in some place of its null distribution? Because I think I am talking about the latter. | |
| Dec 1, 2023 at 14:42 | history | answered | Sextus Empiricus | CC BY-SA 4.0 |