# Shouldn't the adjusted p value be less than the normal p value?

I am following an SPSS tutorial for Kruskall-wallis test. In the example give, we are comparing the driving reaction time for 3 groups (each group contain 10 subjects) following a drink. 1st grp was given water, 2nd coffee, 3rd alcohol. Kruskall-wallis was significant. A Mann-whitney test was carried on each pair of grps. As multiple tests are being carried out, SPSS makes an adjustment to the p-value. The adjustment here is to multiply each Mann-Whitney p-value by the total number of Mann-Whitney tests being carried out (Bonferroni correction). But as I know, bonferroni correction should decrease the p value and its formula is p value/nb of tests. So the adj sig should be smaller than the sig. So the sig and adj sig columns should be reversed. Where am I mistaken?

• It is not appropriate to run separate WIlcoxon tests after a K-W test because the Wilcoxon tests re-rank the variables resulting in non-transitivity e.g. A > B > C > A. More at hbiostat.org/doc/bbr.pdf which points out how its better to use the regression model generalization of the K-W and Wilcoxon tests, the proportional odds model. Aug 16, 2020 at 11:12
• @Ahmad Leaving aside the issue Frank raised, in general when accounnting for multiple testing, either the significance level should go down (while p-values remain the same) or p-values should be adjusted upward (while significance levels remain the same) Aug 16, 2020 at 12:15

The Bonferroni correction adjusts the $$\alpha$$-level, the threshold required to declare a result significant. This makes it harder to reject the null hypothesis in order to prevent a true null from slipping throttle cracks just because you kept on trying to find a false null among many true nulls e.g. the XKCD on jelly beans.

Alternatively, the p-value could be adjusted up while keeping $$\alpha$$ where it was.

Either way, the goal is to make it harder to reject the null hypothesis.

• So in the picture attached, we adjusted up the p-value but kept the level of significance at 0.05. If we were to go with the Bonferroni correction, we would adjust α -level for the water-coffee pair, for example, at 0.012. Right? Aug 16, 2020 at 15:13
• Where are you getting $0.012$?
– Dave
Aug 17, 2020 at 2:18
• For the first pair: 0.036/3= 0.012. Shouldn't the adjusted sig be lower than the sig? Aug 17, 2020 at 9:27
• I think that “sig” column is the p-value, so the appropriate arithmetic is to multiply by three, not divide. You’re trying to make rejection harder.
– Dave
Aug 17, 2020 at 9:53