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I have between-subject design with 3 different groups where I capture interval data (on scale 1-4 where equal spacing amongst scores can be assumed). The data in groups is not normally distributed with distributions of different shape across all three groups.

This means I can not use one way ANOVA to compare means or Kruskal Wallis to compare medians as my data violates key assumptions of both tests.

Which test should I use? Or how should I proceed in analysing the data?

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    $\begingroup$ What do you want to test? Means? Medians? Shape? Something else? $\endgroup$ – Peter Flom Sep 15 '17 at 10:58
  • $\begingroup$ 1. I'll leave confirmation of this to a statistician on this site, but my understanding is that similar shape is not an assumption of Kruskal-Wallis. It is only an assumption of K-W if you wish to interpret the result as a difference in medians. Otherwise the interpretation is something like a stochastic difference among groups. 2. However, I have heard people with the opinion that K-W and other rank-based tests may not be appropriate if there are many ties, like in this case where there are only 4 levels in the dependent variable. Again, I'll defer to a statitician on this point. $\endgroup$ – Sal Mangiafico Sep 15 '17 at 14:42
  • $\begingroup$ A thought: It may be helpful to treat your dependent variable as an ordinal variable. Ordinal regression is a great tool for handling this. It is relatively easy to conduct in some software package (R), and maybe not in some others. But with your design, it may be a good approach to arrange your data in a contingency table and use an extended Cochran-Armitage test. The test tests for association between an ordinal variable and a nominal variable. The original test allows for only two levels in the nominal variable, but some software has implementation for large tables. $\endgroup$ – Sal Mangiafico Sep 15 '17 at 15:34
  • $\begingroup$ First thanks for replies! I would like to compare means or medians as this is the common way of presenting data in my filed (Human Computer Interaction). Would it be correct if, irrespective of shape violation, I present my data as medians and based on results of K-W test conclude if medians are significantly different? $\endgroup$ – klen Sep 16 '17 at 12:57

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