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The hospital I work for sends out a patient satisfaction survey after discharge (assuming a random independent draw of patients getting the survey), all multiple choice/categorical questions. Some of these are yes/no questions, others are four category (never/sometimes/often/always), and there's one 0-10 ranking.

My bosses and the regulators who watch my bosses mostly care about "top box" scores (positive answers, "always", or rating 9-10). (And yes, because there are regulators invovled, I don't get to word the actual questions. There are reasons this annoys me, but that's beyond the scope of this question.)

On top of these required questions, we do get to ask a few of our own, one of which is if the patient remembers staff members checking on them at regular intervals. (This is something our employees are supposed to do, but it does occasionally not happen, or it can happen in a way that the patient doesn't quite remember it. I'm going to assume that the patient doesn't remember, though.) I want to see just how this perception of staff involvement affects these measures of satisfaction.

So I end up with contingency tables kinda like this:

Question M
        REMEMBER    DON'T REMEMBER
Never       xxx     xxx
Sometimes   xxx     xxx
Often       xxx     xxx
Always      xxx     xxx

Question P
        REMEMBER    DON'T REMEMBER
Yes         xxx     xxx
No          xxx     xxx

I've done a series of chi-squared tests for homogeneity, one for each question, to find that yes indeed, I'm pretty sure the proportions are different in these two groups. I want to find out which questions are most "changed" by patients remembering staff involvement.

Besides comparing the p-values of all my chi-squared tests, are there other things I can do to compare these variables? Would a series of z-tests comparing the proportions in the top box for each question help? How would these tests compare against each other, and what else could I do? Should I be viewing these as "ranked" instead of just categorical, and how does that change things?

As a bonus question, what would the Bayesian instead of frequentist approach be?

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