# Identify relations between categorical and ordinal/continuous variables

I'm evaluating a survey regarding opinions. Now I check for relations/similarities between the variables.

My German workbook names the following condition for a Spearman rank correlation without further explanation:

"At least one variable is ordinal-scaled and/or not normally distributed."

But when I look at how Spearman rank correlation works, it only makes sense to use the test if both variables are at least ordinal-scaled.

Why does the German workbook tell otherwise? Is there something I am missing?

Now I'm looking for another appropriate test to test relations between the variables with the following properties:

• not normally distributed
• variable a: dichotomous or categorical (>2 categories)
• variable b: ordinal scaled or continuous.

I considered Mann Whitney U test and Kruskall-Wallis test. Are there more appropriate tests to identify relations between the variables?

And can I use the same tests for testing relations between the independent and dependent variables?

Thank you.

• Maybe the book says "at least one variable must be ordinal scaled" for cases where one axis only has 2 categories (then order doesn't matter). Then this would be similar to a T-Test in case of Pearson and similar to a U-test in case of Spearman. Feb 25, 2020 at 12:58

The German workbook is trying to give you simple guidance, but in the process of simplifying, it's actually being a little misleading.

Spearman correlation requires the variables be at least ordinal in nature. That is, they can be ordinal (ordered category), or continuous (interval or ratio). (Assuming the method can handle ties well for ordinal data). In addition, if one of the variables is dichotomous, that will work the same as an ordinal variable with two levels.

The workbook is trying to say, "If at least one of your variables is ordinal, and not continuous, then you want use Spearman correlation rather than Pearson."

The normality criterion isn't quite correct, but Pearson is may be most useful when the data are approximately bivariate normal, and when this isn't the case, Spearman may be desirable.

I would also mention that Spearman is useful when you are looking for a nonlinear, but monotonic relationship between two variables.

Mann-Whitney and Kruskal-Wallis work well with an ordinal dependent variable and a nominal independent variable. (Again, assuming the method handles ties well).

If you are looking for a test of association between two variables, one ordinal and categorical, then the Cochran-Armitage test (which can be extended to more than two categories) is useful. For the size of the association, there are a few different effect size statistics, like Cliff's delta (rank biserial correlation) or Vargha and Delaney's A for two categories; or maximum CDA or VD, or epsilon squared or Freeman's theta for more categories.

• Thank you a lot. It is good to know that Spearman rank correlation works fine with a dichotomous independent variable. The Cochran-Armitage test seems nice for the other case, but I think it requires normal distribution of the data. Is this correct? One other small question besides the posted one just to be sure: Kruskall-Wallis test makes no sense if the independent variable is ordinal I guess because I think it treats the independent variable as categorical? Feb 26, 2020 at 10:54
• No, I don't think the Cochran-Armitage "test of trend" requires normal data. It's data are arranged in a contingency table. But I think the spacing between the ordered categories is assumed equal unless otherwise specified. Be careful as there are different tests pwith the name Cochran.... Right, KW needs a nominal independent variable. There is a similar test for when there is an ordinal independent variable: Cuzick test, and I think Jonckheere-Terpstra. Feb 26, 2020 at 12:08