My data is many thousands of responses to ~2000 multiple-choice (2 to 4 options) questions. Most of the questions have not been answered by most people. ie. a typical question will have <5% response rate.

I'm interested in finding out which questions have a high predictive power for which other questions. eg. People who answer option A to Q243 are five times as likely than average to answer option C to Q64.

I have constructed a (4x4) answer matrix for every question-pair combination that records (of those who answered both questions under consideration) how many people answered each possible combination of answers. This looks something like this:

                Qb
| 0  1  2  3
--|-------------
0| 3  12 5  2
Qa 1| 11 2  25 10
2| 16 5  10 7
3| 10 2  10 17


So, the Probability that a random person (who answered both questions) selected Qa option 1, given that they selected Qb option 2 is: 25 / (5 + 25 + 10 + 10) = 0.5

In order to explore the predictive power of Qb option 2 for Qa option 1. I would want to compare this number, 0.5 to the probability that a random person would choose option Qa option 1 given no further information. ie. if 50% of the population in general, chose option 2 for Qa anyway then there would be no enhanced predictive power from knowing that the random person also answered option 2 to Qb.

My question is, what would be the best population to compare to?

1. I could compare to the proportion of people who answered Qa option 1 of all those who answered Qa (so this would include people not recorded in the above comparison matrix - those who answered Qa but not Qb)
2. Or I could compare just to those who answered both Qa and Qb (ie. the sum of the above matrix)

If the questions people answered were a random subset of all questions then there should be no difference, but there is some selection bias as people could choose to skip questions they didn't want to answer.

• I wonder if you would need to take into account the test-writing process with your probabilities. Four question multiple-choice tests are typically written with an "obvious" wrong, a less obvious wrong, then two good choices where one is more correct than the other in the context of the question. As a result, the expected probabilities for each response is not .25, .25, .25, .25 but something more like .05, .15, .25,.45 – Tavrock Mar 16 '17 at 5:21