I want to caclulate the association/dependency between 1 ordinal variable (values 0, 1 or 2) and 1 nominal variable (5 categories). Dataset will be quite small ( between 50 and 100 sample points). Is a Chi-square an appropriate test to use?
If you set this up in a 3 x 5 contingency table and no cell has fewer than five observations, using a Chi-square good-of-fit test should work fine. If you wanted to have an index of the fit, you could also use the results to compute a contingency coefficient. It would probably be more meaningful if you did a corrected contingency coefficient. That is, divide the contingency coefficient by the max contingency coefficient. If you have cells with fewer than five observations, you can collapse rows or columns to achieve the five minimum observations per cell.
James E. McLean Is the criterion no fewer than 5 observations in any cell or no fewer than 5 expected observations in any cell? I honestly can't remember.
The traditional test, if the nominal variable has two categories, is the Cochran-Armitage test. This test can can be expanded to more than two categories. (This is supported in R. I think in SAS. Not sure about other packages). However, my understanding of this test is that the spacing of the nominal categories needs to be specified. That is, you'd have to assume they are e.g. linear (-1, 0, 1), or some other spacing, like quadratic (1, -2, 1). But if you have to specify the spacing, that somewhat defeats acknowledging the ordinal nature of the variable.
Practically speaking, using the Kruskal-Wallis test, even though it specifies that one variable is the independent variable and one is the dependent variable, works well.
You _can_ use a chi-square test of association, but this will treat the ordinal variable as if it were categorical. This is useful in some cases, but may not be what you want.
Chi-square could be less appropriate if either variable has three or more ordered levels, as in your case, since it does not consider the order sequence, thus giving up statistical power if applied with an ordinal scale. Instead, a special case of Pearson’s r (called rMR) can be regarded for examining the association between an ordinal variable and a multichotomous variable. You could refer to chapter 3, “Special Cases of Pearson’s r,” of Chen and Popovich’s (2002) book, referenced below.
Chen, P. Y., & Popovich, P. M. (2002). Correlation: Parametric and nonparametric measures. Sage Publications. https://dx.doi.org/10.4135/9781412983808
Good luck,
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