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I have a data of 400 subjects from 2 groups, and each group has 200 subjects. Everyone in each group has a unique ID from 1 to 200, and subjects with the same ID are paired (e.g, each subject with ID 1 from group 1 and group 2 are paired). Each one is asked to select an option out of 5 options for a question, and the probabilities of selecting the option are calculated by a mathematical equation. Let's say, as an example, ID 1s from group 1 and group 2 select both option 3 and the probabilities are .3 and .7, respectively.

What I want to do is to compare the probabilities between two subjects in a pair. That is, I want to compare the probabilities of .3 and .7 to see if the probabilities are close enough or quite different. At the beginning, I thought of odds ratio. However, to make a conclusion, I need a distribution of the odds ratio. I know that log odds ratio follows approximately normal distribution, but in this case I don't know how the parameters to create the distribution are calculated because I don't know for sure how the contingency table looks like.

I'm wondering if the distribution of the odds ratio can be constructed or there is a better way to compare them. Any help is very appreciated.

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  • $\begingroup$ How did you conclude that you needed an odds ratio, especially since odds are typically estimated as a function of the presence or absence of a property or feature? Do you really mean a risk ratio? en.wikipedia.org/wiki/Relative_risk Why wouldn't a simple paired comparison test work? $\endgroup$ – DJohnson Nov 15 '15 at 15:00
  • $\begingroup$ @DJohnson Thanks for your comment. I thought of odds ratio because there would be two events : selecting the option vs. not selecting the option. In this case, the odds ratio would be (.3/.7)/(.7/.3). The reason I thought of odds ratio rather than risk ratio is that it is more frequently used in many studies. Frankly I don't know for sure if odds ratio is better than risk ratio. I didn't think of a paired comparison test because it also compares two values from two groups, not two individuals. $\endgroup$ – Shinzi Katoh Nov 15 '15 at 22:48

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