Background and Problem
This is a continuation of a question posted earlier (here) in which I describe a meta-analysis combining between- and within-subject designs using log-odds ratios (OR) as the metric of interest. In short, I indicated that I would have access to the raw data from a series of experiments on the same topic using either design. I was asked to aggregate them, and also to test for design effects. The formula necessary to calculate a within-subject OR comparable to typical between-subject OR calculations was helpfully provided in one of the answers.
I have since received the data and discovered that the structure is not quite as anticipated. In my earlier post, I said that the within-subject condition would involve two responses (i.e., a binary response for each of two conditions). This remains true. However, I had anticipated that the between-subject condition would involve only a single, binary response (in one of the two conditions), but have discovered it involves two binary responses (each in the same condition). This means that whereas the within-subject OR must be based on only a single response for each condition, the between-subject OR based on two responses in a single condition for each subject.
With this in mind, I would like to ask:
- Is it possible to calculate a log-odds ratio from binary data involving multiple responses from a given subject comparable to the within-subject log-odds ratio described here? Or is this comparison no longer possible using an OR?
- While I have reason to prefer a standard meta-analytic approach, if this is no longer possible, could my question instead be resolved using a logistic multi-level modelling approach with participant as a random effect nested within experiment?