Meta analysis of proportions that has control group I have data from a couple studies. The outcome I have is ordinal/likert. For each value I have the proportion of people who had each of the values at a measured end point. For example, in a study I have a record saying 12 people had value 1, 52 had value 2, 28 had value 3. I have these records for every study and each study had 2 conditions that people were in. What sort of meta analysis models would I be using here? I saw some example of meta analysis involving proportions here, but this seems like a different thing than what I'm looking for.
 A: In this situation, you can do an individual-patient/subject-data meta-analysis, because you can simply recreate the data per person (just create one record per each of the people). Then, you use whatever analysis model you consider appropriate for an individual study (e.g. ordinal proportional odds model), stratify by study (i.e. you assume for the control group outcomes that the probability of each category could be completely different in each study), and assume that your effect measure of interest is either the same in every study (fixed effects meta-analysis) or varies across studies (random effects meta-analysis). Exactly which model you would use would depend on what you want to assume and what effect measure you are interested in.
Alternatively, you could analyze the individual patient data from each study (or take an estimate that the study authors have produced) to obtain the effect measure of interest (e.g. log-odds-ratio for the ordinal proportional odds model; note log transformation is important here, but you can back-transform the final results to the odds-ratio scale) and its standard error. Then, you do a standard inverse variance (for fixed effects) or random effects meta-analysis for the estimates and their standard errors.
The first approach should be superior, if there are small numbers in some categories in some studies, but if all categories are filled with enough people, the two approach may end up giving very similar results. If numbers are very small and/or you have perfect separation (e.g. in some study group 1 has 10% in category 1, while groups 2 has 100% in category 3) or issues like everyone being in a single category in all groups in one study, then perhaps a Bayesian version of the first approach with vague priors might help.
