Performing hypothesis test with two categorical variables, where one variable has four ordered categories

Question

I'm working with two categorical variables and need to test whether they're related. This isn't the exact problem I'm solving but the example should help illustrate the nature of the problem I'm dealing with.

The first variable only has two categories (owns dolls or does not own dolls). The second variable is ordered (2nd grade, 3rd grade, 4th grade, 5th grade). I can perform a chi square test. The issue is that I want to understand not only whether the grade is dependent or independent from owning dolls, but whether the higher the grade, the lower the likelihood is of owning dolls.

Since the grades are ordered, would it make sense to convert them to numerical values (2, 3, 4, and 5), and then perform a t-test on the sample that owns dolls vs the sample that doesn't?

The other alternative I thought of would be to combine the grades (2nd and 3rd in one group, 4th and 5th in the other), check that the "2nd and 3rd graders" sample had a higher percentage of doll ownership, and then perform a chi square test to see if the difference is statistically significant.

Answer 1

If your predictor is ordinal (i.e. ordered) and your response is categorical, you can do a logistic regression with polynomial contrasts.

Your first idea of treating the grades as continuous numerical values is valid if you can fairly assume that grade 3 is as different from grade 2 as grade 4 is from grade 3 and so on i.e. the 'gaps' between your ordinal categories are similar. If instead you expect to see a much bigger change from (for example) grade 4 to grade 5, then this assumption does not hold and you should treat the data as ordinal and analyse it accordingly (with polynomial contrasts).

Your second idea is an inefficient use of the data and is best avoided, since there are well-established methods to deal with the problem.