Hello,
Is there anyway to have the correlation coefficient of two categorical variables ,both ordinal and nominal (For example level of education and ethnicity)? I have checked the independency by chi square test, and now the strength of association is important for me.
I have heard about Cramer coefficient but my problem is that it seems STATA doesn't provide it under survey design set (svy command). Any suggestion?
Thank you very much for your help.
Test of association
If you have one ordinal and one nominal variable, you might want to consider a different test of association --- one that takes into account the ordinal nature of the one variable.
Since you have the data arranged in a contingency table, the Cochran-Armitage test is ideal. The original test works for only a table where the nominal variable has two levels, but there are versions of the test that allow for larger tables.
For one ordinal and one nominal variable, Kruskal-Wallis is probably a better test of association than would be a chi-square test of association.
Correlation coefficients and effect sizes
If you were to use Kruskal-Wallis, you can use an effect size for the degree of association. See the link at the bottom of this answer, but the topic of the appropriate effect size for K-W seems to be a disputed one.
Cramer's V is easy to calculate. See the link at the end of this answer. But it is really meant for two nominal variables, not a nominal variable with an ordinal variable.
I don't think there is a common correlation coefficient for one ordinal and one nominal variable. The following post is the best discussion I've seen. Cramer's V or eta-squared may do the job. The link discusses a statistic called Freeman's theta, but this statistic doesn't seem very common or particularly easy to calculate. https://stats.stackexchange.com/questions/73065/correlation-coefficient-for-non-dichotomous-nominal-variable-and-ordinal-numeric .
Paper on effect sizes with formulae for Cramer V and eta-squared
http://www.tss.awf.poznan.pl/files/3_Trends_Vol21_2014__no1_20.pdf
Thank you very much for the answer and the links. I will study them and try to apply on my data.
In my data, most of the variables are based on Likert scale . I considered them as ordinal variables. However , I was wondering if converting them to numeric ranks and consider them as continuous variables might be a right approach.
You will find that there is a lot of contradictory information about the best way to analyze Likert item data.
Really the best approach is to use either A) ordinal regression. This is easy with some software, but may not be easy with other software. And your audience may or may not be familiar with the idea. Or B) tests specific for tables with ordinal data: Linear-by-linear test, Cochan-Armitage test. Some comments as ordinal regression: Your software may not support these, and your audience may not be familiar with them.
I think it's fine to use rank-based tests with Likert item data. Not everyone agrees. This has the advantage of being familiar for the audience and being pretty universally easy in statistical packages. These would be classic nonparametric tests: Kendall correlation (with tau-b that handles ties), Mann-Whitney, Kruskal-Wallis (with Dunn post-hoc). Then there are tests for paired data: Paired Rank-sum, Friedman, Quade.
Another advantage to using the traditional nonparametric tests is that you are treating the Likert item responses as a dependent variable, and so don't need to worry about the effect sizes for correlations. You can present appropriate effect sizes (like eta-squared), but the size of the effect can simply be shown with a difference in means or in, say, box plots. In theory, you can report quantile information for ordered categories, like you could say "the median education level was 'Bachelors' and the 75th percentile was 'Masters'". This can get a little weird, tho.
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