# Correlation/Association between a polychotomous nominal variable and a continuous variable

Referring to Measures of Association How to Choose

The paper states that for the association between a polychotomous nominal variable and a continuous variable;

If the nominal variable has more than two levels, then one can calculate the point-biserial correlation between the continuous variable and all possible pairs of levels of the nominal variable; this would result in such coefficients, where k represents the number of levels of the nominal variable. For further reading, see Tate.

However, the paper doesn't state what to do with the point-biserial correlations of all the pairs of levels. I want an aggregated metric between the two variables. Would taking the arithmetic mean of the coefficients suffice?

• This sounds a bit of a waste of time. With nothing else said, one measure of the association between a continuous variable and a nominal variable such as you mention is the success of a regression model in predicting the continuous variable from the nominal variable, handled as a factor variable, i.e. a set of (0, 1) indicators. Here success can be any conventional figure of merit for a regression such as $R^2$ or RMSE. Commented May 1, 2023 at 8:22
• FWIW, the term polychotomous has been in use for some centuries, but it's based on a false parsing of dichotomous. The alternative polytomous is better Greek, and multistate is arguably an even better term. Commented May 1, 2023 at 8:23
• The average of various correlations is not often useful or even interesting Here it seems especially problematic, e.g. if the different levels of the nominal variable occur with very different frequency. Commented May 1, 2023 at 8:30
• If you mean this: having salary vs job (banker, baker, boxer), correlate the two in subsamples {banker, baker}; {banker, boxer}; {baker, boxer} and then somehow combine the three coefficients, - then this is implicitly the summary of pairwise comparisons of salary between the three jobs. Point-biserial r is an effect size measure of a difference between two means. Commented May 1, 2023 at 10:55
• @NickCox: Can you make this a formal answer? Commented May 25 at 20:55

Sorry, but this sounds a bit of a waste of time.

For one, the average of various correlations is not often useful or even interesting. Here it seems especially problematic, e.g. if the different levels of the nominal variable occur with very different frequency.

That's negative, but I have a constructive alternative.

With nothing else said, one measure of the association between a continuous variable and a nominal variable such as you mention is the success of a regression model in predicting the continuous variable from the nominal variable, handled as a factor variable, i.e. a set of (0, 1) indicators.

That could be recast or phrased as analysis of variance if anyone prefers.

Here success can be judged by any conventional figure of merit for a regression such as $$R^2$$ or RMSE.

FWIW, the term polychotomous has been in use for some centuries, but it's based on a false parsing of dichotomous, which is in essence dicho-tomous, not di-chotomous. The alternative polytomous is better use of Greek but not needed: multistate is arguably an even better term.