I have actually three questions about the Cochran-Armitage test for trend, which tests for an association between a binary variable and an ordinal one:
- Is it valid if the expected cell count is low (< 5) in many cells (> 20%) of a contingency table?
- Is it valid if there's not really a linear trend in the contingency table, but a sort of "polarization" between the two extreme values of the ordinal scale? (see example below)
- When this test is not valid, what alternative tests may be useful to show an association between a binary variable and an ordinal one (in particular when answers are "polarized" at extreme values)?
Case in point, I have the following contingency table, coming from a survey conducted on a small group of immigrants (the question was: "How important was the professional situation of your partner in your decision of coming to this country?"):
|not important at all||124||127|
Obviously there are relatively very few respondents in the categories other than "not important at all", and I suspect this may be a problem to be able to say anything about this table.
Plus, the relationship doesn't seem to be really linear, but rather a bit "polarized" between the two extreme values, i.e. things look different between men and women at the "not important at all" and "extremely important" level, but rather similar in the in-between categories. I wonder if it affects the validity of the test for trend.
If I conduct a chi-square test, the result is non-significant at an alpha level of 0.05 (p=0.249, χ²=5.397, df=4), and anyway if violates the assumption of having an expected count of 5 in at least 80% of cells (references for this assumption: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3900058/ and https://www.statology.org/chi-square-test-assumptions/).
Here's the expected counts table, where 3 cells out of 10 are under 5 (plus, most cells are very close to 5):
|not important at all||118.41196013||132.58803987|
However, if I conduct a Cochran-Armitage test for trend, the result is significant (p=0.038). But the fact that the "80%" assumption is violated for the chi-square test makes me wonder if I should take this violation into account for the Cochran-Armitage test.
I know that an alternative could be to merge some categories together (e.g. merging "very important" with "extremely important"), but I'm also asking this question for general knowledge about the Cochran-Armitage test, not only about this specific contingency table.