# Can Mann-Whitney test be used for post-hoc comparisons after Kruskal-Wallis?

I have a simulation where an animal is placed in a hostile environment and timed to see how long it can survive using some approach to survival. There are three approaches it can use to survive. I ran 300 simulations of the animal using each survival approach. All simulations take place in the same environment but there's some randomness so it's different each time. I time how many seconds the animal survives in each simulation. Living longer is better. My data looks like this:

Approach 1, Approach 2, Approach 2
45,79,38
48,32,24
85,108,44
... 300 rows of these


I'm unsure of everything I do after this point so let me know if I'm doing something stupid and wrong. I'm trying to find out if there's a statistical difference on lifespan using a particular approach.

I ran a Shapiro test on each of the samples and they came back with tiny p values, so I believe the data isn't normalized.

Data on rows have no relationship to each other. The random seed used for each simulation was different. As a result, I believe the data isn't paired.

Because the data is not normalized, not paired and there were more than two samples, I ran a Kruskal Wallis test which came back with a p-value of 0.048. I then moved on to a post hoc, selecting Mann Whitney. In really not sure if Mann Whitney should be used here.

I compared each survival approach with each other approach by performing the Mann Whitney test i.e. {(approach 1, approach 2), (approach 1, approach 3), (approach 2, approach 3)}. There was no finding of statistical significance between the pair (approach 2, approach 3) using a two tailed test but there was significance difference found using a one tailed test.

Problems:

1. I don't know if using Mann Whitney like this makes sense.
2. I don't know if I should be using a one or two tailed Mann Whitney.
• Do you have any a priori hypothesis about the relative strength of different approaches (e.g. approach1>approach2>approach3)? This is crucial to answer your questions. – amoeba says Reinstate Monica Aug 14 '14 at 16:48
• I have the mean, median and standard deviation and it appears that approach 3 is better because it has a higher median and mean but it also has a much higher standard deviation so I'm not sure. But I had no way of knowing this before hand. – Phlox Midas Aug 14 '14 at 17:07
• Or is it also known as the Bonferroni correction? – Phlox Midas Aug 15 '14 at 13:23
• Phlox: if there was "no way of knowing this before hand", you should absolutely not use a one-tailed test, only two-tailed (as @Alexis mentioned in his reply as well). – amoeba says Reinstate Monica Aug 15 '14 at 15:32
• @amoeba "her" ;) – Alexis Sep 23 '14 at 17:38

No, you should not use the Mann-Whitney $U$ test in this circumstance.

Here's why: Dunn's test is an appropriate post hoc test* following rejection of a Kruskal-Wallis test. If one proceeds by moving from a rejection of Kruskal-Wallis to performing ordinary pair-wise rank sum (i.e. Wilcoxon or Mann-Whitney) tests, then two problems obtain: (1) the ranks used for the pair-wise rank sum tests are not the ranks used by the Kruskal-Wallis test; and (2) the rank sum tests do not use the pooled variance implied by the Kruskal-Wallis null hypothesis. Dunn's test does not have these problems

Post hoc tests following rejection of a Kruskal-Wallis test which have been adjusted for multiple comparisons may fail to reject all pairwise tests for a given family-wise error rate or given false discovery rate corresponding to a given $\alpha$ for the omnibus test, just as with any other multiple comparison omnibus/post hoc testing scenario.

Unless you have reason to believe that one group's survival time is longer or shorter than another's a priori, you should be using the two-sided tests.

Dunn's test can be performed in Stata using dunntest (type net describe dunntest, from(https://www.alexisdinno.com/stata)), and in R using the dunn.test package.

Also, I wonder if you might take a survival analysis approach to assessing whether and when an animal dies based on different conditions?

* A few less well-known post hoc pair-wise tests to follow a rejected Kruskal-Wallis, include Conover-Iman (like Dunn, but based on the t distribution, rather than the z distribution, implemented for Stata in the conovertest package, and for R in the conover.test package), and the Dwass-Steel-Citchlow-Fligner tests.

• Thanks for your answer. Is the Dunn test also known as the Nemenyi-Damico-Wolfe-Dunn test or is that a separate test? – Phlox Midas Aug 15 '14 at 12:26
• I ask because I can't find any implementation of the Dunn test. – Phlox Midas Aug 15 '14 at 13:02
• @PhloxMidas I don't know about the "Nemenyi-Damico-Wolfe-Dunn test," but Wikipedia implies it is an appropriate post hoc test following rejection of an omnibus test in a repeated measures design—e.g. following a Friedman test. Also, see my comment about Stata. – Alexis Aug 15 '14 at 18:04

A unifying generalization of Kruskal-Wallis/Wilcoxon is the proportional odds model, which admits general contrasts with either pointwise or simultaneous confidence intervals for odds ratios. This is implemented in my R rms package's orm and contrast.rms functions.

You can also use the critical difference after Conover or the critical difference after Schaich and Hamerle. The former is more liberal whereas the latter is exact but lacks a bit of power. Both methods are illustrated on my website brightstat.com and brightstat's webapp also lets you calculate these critical differences and perform the post-hoc tests right away. Kruskal-Wallis on brightstat.com

If you are using SPSS, do the post-hoc Mann-Whitney with Bonferroni correction (p value divided by the number of groups).

• The Mann-Whitney suffers from the two problems I identify in my answer, and is an inappropriate post hoc test for Kruskal-Wallis. – Alexis Sep 28 '15 at 17:58