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    $\begingroup$ The problem with this approach is that it is conceptually arbitrary. Why should values of a statistic in the tails of its null distribution be of interest? Why not rank the values in some other order than absolute value? That turns out to be the correct approach in the example I gave in the link, for instance. Another example of this is Fisher's examination of Mendel's pea results, which followed the null hypothesis too closely. $\endgroup$ Commented Jun 21, 2023 at 22:55
  • $\begingroup$ @Whuber I don't agree that it is arbitrary. You can choose a different statistic if you want the p-value to reflect some different property of the data, but if you want the p-value to behave as a p-value you will need to effectively rank the statistic values from the model with the null set to true. $\endgroup$ Commented Jun 22, 2023 at 1:09
  • $\begingroup$ @whuber When you say 'absolute value' are you thinking of a two-tailed p-value? For one-tailed p-values (which I prefer) the test statistics are not converted to the absolute values (i.e. negatives are made positive), but are ranked on their actual value. $\endgroup$ Commented Jun 22, 2023 at 1:11
  • $\begingroup$ @whuber When you say "The problem with this approach" are you referring to my explanation of p-values or to the use of p-values? $\endgroup$ Commented Jun 22, 2023 at 1:12
  • $\begingroup$ (1) "Effectively rank" is what makes the approach arbitrary. (2) I used absolute value to reflect your example that refers to "small" and "large" values. (3) "Approach" refers to your post. $\endgroup$ Commented Jun 22, 2023 at 13:02