# How to report Kruskal-Wallis test?

As part of the analysis of the data collected from a survey, I was carrying out Kruskal-Wallis Tests between some Likert scale questions and demographics such as firm size and job position. I am using this test instead of One Way Anova since the data is non-normal.

For example, testing the responses of the following statement vs Job position (manager or junior): I arrive late at work. Rated on a 5-point scale from never to always.

When reporting the results of the test (if result is significant) can I state something like:

Managers are more likely to arrive late than juniors (H=14.338, p<.01)

or is this the only way one can report the result:

A statistically significant difference (H=14.338, p<.01) exists between late arrivals at work by managers and juniors.

• The Kruskal-Wallis test is used when the number of groups is three or more, otherwise it's reduced to Mann-Whitney test. However, you can interpret it like that. There's no need for post-hoc test if you have only two groups. – Germaniawerks Feb 14 '14 at 16:11
• if there are more than 2 groups, would an error graph be sufficient to see which means differ significantly? – dissertationhelp Feb 14 '14 at 16:18
• You need to apply Wilcoxon test with Bonferroni correction to the level of significance in order to compare which groups differ from which. – Germaniawerks Feb 14 '14 at 16:53
• @dissertationhelp: I have just updated my answer with an important remark, please take a look. – amoeba Feb 14 '14 at 21:35

As @Germaniawerks remarked above, if you only have two groups (managers vs juniors) you should use ranksum (aka Mann-Whitney-Wilcoxon) test and there is no need for Kruskal-Wallis. If you have more than two groups, then Kruskal-Wallis will tell you if they are significantly different, but if you want to know which pairs are significantly different between each other, you need to do a post hoc comparison, e.g. ranksum test with Bonferroni correction.

But personally, I don't think it makes a lot of sense to report U statistic (in case of comparison between two groups, it should be U of Mann-Whitney, as explained above): few people have intuitive understanding of it, and this particular number (U=14.338) does not convey anything meaningful for the reader, only taking space. Instead, I would provide the means and standard deviations of your distributions for both groups. I would also explicitly mention the test you are doing. So taking your example I would write something along these lines:

Managers are more likely to arrive late than juniors (managers: $10 \pm 5$ minutes late, juniors: $2\pm4$ minutes late, mean$\pm$SD, $N=10$ for both groups, p<.01, Mann-Whitney-Wilcoxon ranksum test)

That's a lot of information to put inside one pair of brackets, so you can split as you like. For example you can report N in the methods section, and make a boxplot figure to illustrate the distributions. Then it would suffice to write:

Managers are more likely to arrive late than juniors, see Figure 1 (p<.01, Mann-Whitney-Wilcoxon ranksum test)

Update

Note that if your data have gross outliers, than means and SDs do not have a lot of meaning and you should rather not report them. Above I assumed that there are no gross outliers in either of the groups. Otherwise situation is more complex and maybe the best way is to provide a boxplot, without giving any numbers in the text at all.

• But the test here is not comparing means. – Nick Cox Feb 14 '14 at 21:01
• @NickCox: no, of course not. Do you mean that it would be better to report median$\pm$SD when doing a ranksum test? I think the best would be to provide a boxplot, but let's imagine there is no space for it (or one wants to provide explicit numbers in addition to the figure). – amoeba Feb 14 '14 at 21:05
• @NickCox: I added an update to my answer, but would be curious to hear your opinion on the best practice in such case. – amoeba Feb 14 '14 at 21:37
• @amoeba I am using Kruskal-Wallis or One Way Anova tests (according to normality) because a statistics lecturer at my University suggested it would be the best choice for such variables. When there are more groups for example: managers, seniors and juniors and the difference in means is significant, are error bar graphs appropriate to see which means differ significantly? I did use these and was able to identify which mean was different from the means of the others in the group. – dissertationhelp Feb 15 '14 at 12:06
• @dissertationhelp: the answer to your question is no. You can of course (and should) look at boxplots to get a feeling of how much difference there is, but if you need a proper statistical test, you should conduct separate two-sample tests (Mann-Whitney) and adjust p-values for multiple comparisons. E.g. if you have three groups, then you first run Kruskal-Wallis, and then you can compare three pairs (groups 1-2, 2-3, 1-3) between each other with Mann-Whitney, but you should multiply your p-values by 3. It's called Bonferroni correction for post-hoc test. – amoeba Feb 15 '14 at 14:42

Following above very useful comments I like to add that the median should be reported instead of the mean.

The statement "Managers are more likely to arrive late than juniors (H=14.338, p<.01)" is incomplete. The only thing it says is that there is a difference between the groups. It does not specify where the difference lies or what the exact difference is. For that purpose medians are reported. Following the scenario in the question I would recommend rephrasing; On a 5-point likert scale managers reported to be more often late than juniors (H = xx, p < .01, MdnManagers = x, MdnJuniors x = )

The statement "A statistically significant difference (H=14.338, p<.01) exists between late arrivals at work by managers and juniors." is fine, but could be more informative.