Correlation between two ordinal categorical variables

What is the best statistical test for investigating if there is any correlation between 2 categorical variables?

Both are satisfaction scores:

1st variable is:

Overall satisfaction with the service.

1: Not at all satisfied; 10: Completely satisfied

2nd variable is:

Satisfaction with the availability of information for the service"

1: Not at all satisfied; 10: Completely satisfied.

• The question concerns ordinal variables, rather than nominal categorical ones - I think that ought to be made clear in the question. – Silverfish Jan 16 '15 at 19:26
• – Silverfish Jan 16 '15 at 19:28
• "Ordinal" added by me to the title. (Note that nobody forces you to regard these variables as ordinal and not interval.) – ttnphns Jan 16 '15 at 19:34
• @ttnphns Thanks - in that case I will tag it also. – Silverfish Jan 16 '15 at 23:50
• For categorical variables, you apply polychoric correlation. LISREL program and FACTOR software could do the polychoric correlation. – Emma Nov 15 '18 at 1:44

I would go with Spearman rho and/or Kendall Tau for categorical (ordinal) variables.

Related to the Pearson correlation coefficient, the Spearman correlation coefficient (rho) measures the relationship between two variables. Spearman's rho can be understood as a rank-based version of Pearson's correlation coefficient.

Like Spearman's rho, Kendall's tau measures the degree of a monotone relationship between variables. Roughly speaking, Kendall's tau distinguishes itself from Spearman's rho by stronger penalization of non-sequential (in context of the ranked variables) dislocations.

Both of these have enough levels that you could just treat them as continuous variables, and use Pearson or Spearman correlation. You can then calculate a significance (p) value based on your correlation and sample size.

If you really want to treat the data as categorical, you want to run a chi-squared test on the 10x10 matrix of overall satisfaction vs. availability satisfaction. You will need a decent amount of data for this (~thousands), since the majority of the cells should contain at least 5 observations for the test to be valid. This would allow for more general types of dependence between the two measures, in which even nearby levels show different relationships (e.g. rating1=9 tends to predict rating2=4, rating1=8 tends to predict rating2=10) which are probably not likely in your data.

I went and searched for it, found this from John Ubersax: http://www.john-uebersax.com/stat/tetra.htm

and some papers