# Measure dependence of categorical and ordinal variable

My purpose: When trying to find out which variables of a generic questionnaire data set will show striking differences across columns in crosstabs, I first do correlation tests (or dependence tests) across all pairs of variables (similar to this question).

Let's say that the dataset only consists of categorical (i.e. nominal) data and ordinal data (from Likert scales).

• For 1 nominal vs 1 nominal variable, I use the Fisher Exact Test or the Chi-Square Test of Independence.
• For 1 nominal vs 1 ordinal variable or vice-versa, I would like to do a non-parametric test, but which one should I use?

I thought about Kruskal-Wallis, point-biserial correlation coefficient or Eta squared test, but I cannot evaluate which makes most sense for finding differences across categories (in case the crosstab has categorical var as columns) or ordinal categories (in case the crosstab has a Likert scale as columns).

"Down-grading" the ordinal vars to categorical would be one obvious work-around, but I would prefer not to lose the ordinal information. Thank you for recommendations.

Further links:

• Do you find some usefull answers in this list: stats.stackexchange.com/search?q=+categorical+ordinal+ Sep 8, 2015 at 13:15
• Another solution that comes to my mind is standardizing the ordinal measure to make it appear like a continuos one (from 1 up to k ordinal categories). That would give the ordinal measure an explicit "direction". Then if both variables are ordinal, a Pearson correlation would make sense. If one of them is nominal, a generic t-test for equal means could do the trick. Sep 8, 2015 at 13:36

## 1 Answer

I can think of two measures of nonlinear dependencies for two discrete variables:

• The Wijayatunga coefficient (https://arxiv.org/abs/1804.07937): Departing from the fact that Pearson correlation is a normalized, Euclidean type distance between joint probability distribution of the two random variables and that when their independence is assumed while keeping their marginal distributions, this measure uses all possible maximal dependences to measure any non-linear dependence. As it can measure non-linear dependencies, it can be used with nominal variables;
• The Ordinal Classification Index (http://dx.doi.org/10.1142/S0218001411009093): This measure is more adequate to measure if the predictions about a dependent ordinal variable capture how much the result diverges from an ideal prediction. But because measures of accuracy are similar to measures of monotonic dependency, it can also probably be used in your context.

The problem with these measures is that I am not aware of any computational implementations other than the ones that I developed (e.g., https://github.com/vthorrf/wijayatunga).