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I have a set of data which has been grouped into 5 clusters. What I would like to do is, for each variable, determine whether it's distributed differently across the clusters with respect to the responses.

I thought that levenes test would be suitable here, but the problem there is it is significant even when the average positive responses for the groups are similar to (.9,.9,.9,.9,.9). I'd like for this to be considered to be fairly uninteresting, what I'm more interested in is something like (.9,.1,.3,.2,.3), as this seems to imply that this variable was particularly related to the first group (whereas when they're all .9 that doesn't really say so much).

What test should I use for this?

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It appears that you have 5 cluster labels and a binary outcome. Just fit a logistic regression with the cluster label as the independent variable, and check for significance of the 4 coefficient estimates that pop out. You may need to do a multiple testing control, depending on what exactly you are checking for (significant differences between ALL groups vs. any pair of groups, for instance).

If the outcome were continuous, a classic ANOVA will work and the likelihood ratio test it entails will tell you if the clustering is significantly informative of the outcome or not. If categorical, do a Pearson chi-square test for independence.

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  • $\begingroup$ levenes - because i was interested in the distribution of variables. In the case of chi-squared tho, what would the expected value be? Why should I expect a particular variable to have a particular level of response? $\endgroup$
    – baxx
    Commented Oct 8, 2019 at 5:44
  • $\begingroup$ oops, see edit--it looks like your "other variable" is continuous, so ANOVA is the way to go $\endgroup$
    – Sheridan Grant
    Commented Oct 8, 2019 at 5:48
  • $\begingroup$ why does it look like that? Are you going to address my previous comment? $\endgroup$
    – baxx
    Commented Oct 8, 2019 at 9:26
  • $\begingroup$ Oh I misread your question. See edit again. $\endgroup$
    – Sheridan Grant
    Commented Oct 17, 2019 at 5:39
  • $\begingroup$ And @baxx I have no idea how to interpret your first comment--of course you're interested in the distribution of the variables. The expected value for each cluster under the null of independence is the average number of positive outcomes across all clusters. If you want to show that a variable is distributed differently across different clusters, you attempt to reject the null hypothesis that the distribution is the same across clusters. You STILL haven't clarified what type of variables you have--your second sentence says "each variable." $\endgroup$
    – Sheridan Grant
    Commented Oct 17, 2019 at 5:45

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