I administered a customer service satisfaction survey, and one of the questions is a check-all-that-apply asking respondents to check the boxes for the services that they liked most (medication assistance, linkage to medical doctor, housing assistance, therapy, and additional funds). I wanted to determine not only if there were significant differences in the proportions of "yes" to "no" response counts as a function of age group (55-60, 61-65, 66-70, 71+) for each of the above services, but also to see if there were any significant decreasing or increasing trends in proportion of "yes" to "no" responses as a function of increasing age for each category. I received insight into this at this post. Here is the link to my post to get clarity on tests to run for these research questions.
I conducted chi Square tests to investigate if any significant differences in popularity of the above services, and separately Cochran-Armitage tests (linear-by-linear association test for trends) to answer the question about whether there are any trends for each of the categories.
Now I wonder if anyone can offer insight into the following finding for the "medication assistance" choice. I'm using an alpha of .05 as the threshold for rejecting the null hypothesis, and the linear-by-linear association finding shows that the result is significant (p=.035) while the associated Pearson Chi-Square is not (p=.213). How can there be both a significant trend of decreasing proportion of popularity of medication assistance, but at the same time a non-significant difference in proportions of "yes" to "no" responses as a function of age group? Isn't the finding of a significant trend dependent on the significant differences between the proportions of "Yes" to "no" responses of each level of age group?