I have a nominal and an ordinal variable and I would like to look measure the association that is between the two. For now, I have done Chi-square test and non-parametric Kruskall-Wallis Test and I have come to the conclusion that there is an association/relationship between the two variables. How can I further measure the effect of this association? I have read about ETA, which SPSS suggests using when you have a nominal and a quantitative variable, and I have also read that you can use it with ANOVA (which is basically the parametric version of Kruskall-Wallis). Could I use this statistic (ETA) as a measure of size effect and trust the value that Crosstabs option in SPSS provides me? As a secondary question, how can I interpret the values of ETA? I came upon different guidelines on the net. Would you describe a ETA=0.321 as a medium effect? ~Thanks.
Eta is about the proportion of variance explained. If you have an ordinal outcome, you don't have a variance, so I'd say no.
Here's some more explanation. Variance is about how different the scores are. So:
1.1, 1.2, 1.3 has a small difference , hence a large variance.
1, 101, 201 has larger differences, hence larger variance.
1, 2, 10001 has even larger differences, and so the variance is even larger.
But in an ordinal measure, we don't know about differences - all we know about is the order, so for each of those variables, they go in the order 1, 2, 3. The classic example is position in a race - people came first, second, third, all we know about the winner is that they were ahead of whoever came second. Where they 0.1 seconds ahead, or 3 hours ahead. We don't know. So the times could be: 10, 11, 12
Or 10, 100, 101
Or 10, 1000, 1001.
We don't have knowledge of differences, so we don't have variance, so we can't have eta-squared.
You should (possibly) use some form of ordinal logistic regression, then you have options for effect sizes based on likelihood ratios, and/or classification probabilities.