What are the assumptions of the Cochran-Armitage test for trend? 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?"):




answer
Men
Women




not important at all
124
127


slightly important
4
6


important
6
6


very important
3
4


extremely important
5
16




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):




answer
Men
Women




not important at all
118.41196013
132.58803987


rather important
4.71760797
5.28239203


important
5.66112957
6.33887043


very important
3.30232558
3.69767442


extremely important
9.90697674
11.09302326




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.
Thanks!
 A: I don't think C-A is too useful/valid unless your percentages are monotonically increasing or decreasing or at least close to that.  For example, percentages in 3 groups that are 50, 33, and 67 are clearly not showing a linear trend, more like a check mark ✔️, yet C-A says that they are linear for all n > 210.
In your case, the percentages of women in each of your 5 categories are:
50.6
60.0
50.0
57.1
76.2
If you didn't have slightly important and/or important, then maybe C-A could be useful.  If you combine those two groups, then they'd be 54.5% and C-A makes more sense, and maybe that's somehwat what it's picking up on.  There's a "sense" that the % of women is increasing, even if it's not perfect.
A: The Cochran Armitage Test-for-Trend is just the score test for a logistic regression with a single continuous adjustment for the ordinal category. As a test, it is most powerful when the trend is in fact logistic linear (i.e. follows an S shape), and when the distribution of counts is inversely proportional to the variance for a particular stratum. These aren't "assumptions" per se. One can show that, even if the trend were monotonic, even if not linear, then the "trend" slope-of-best-fit is some non-zero value and so the test still has power. You could robustify the test by fitting said logistic model and estimating the robust sandwich-based standard error and associated p-value.
For your data, if one were to plot the influence and leverage functions for these observations, we could see it's the last "extremely important" category that has the highest impact on the analysis, where the proportion deviates from the homogeneous 50% that's typical of the first 4 categories.
Where I think you've gone wrong is that it seems to me that the "X" variable should be gender and the "Y" variable the ordinal Likert response. Fitting a simple T-test, or cumulative logit model should tell you if there's a difference in the average response from men and women which is more sound.
