# What is the best procedure to conduct a meta-analysis of proportions in within-subject designs?

Most procedures I found use only one proportion per study. But all the studies I am analyzing have within-subject designs. They compare the proportion of successes in the manipulation with the proportion of successes in the control condition. All sample sizes are known.

The meta-analysis method for dichotomous models asks me for sample sizes of two groups, but that does not make sense for within designs, right? The same people performed the manipulation and the control condition.

What is the appropriate procedure to conduct a meta-analysis with within designs that compare conditions and have proportion as data?

• I would consider a multivariate meta-analysis model (1st choice), or a meta-regression/weighted approach (2nd choice) Nov 9, 2022 at 7:03
• Do they also tell you the proportion who changed from positive to negative and vice versa? Nov 9, 2022 at 11:13
• Participants had to perform severaI tasks (0=failed, 1=success) in both conditions. I have the proportions of successes in the both conditions and the sample size is the same across conditions because there were not missing data. Nov 9, 2022 at 11:54

## 1 Answer

This is a somewhat unusual case, but an appropriate effect size measure for this kind of meta-analysis is the (log) odds ratio based on paired data. You can find a discussion of this under the help page of the escalc() function in the metafor package: https://wviechtb.github.io/metafor/reference/escalc.html#-b-measures-for-dichotomous-variables-2

Note that you either need the paired 2x2 table or the marginal table (which you have if you know the proportions and the sample size) and also an estimate of the correlation (phi coefficient for the paired 2x2 table). Since that correlation is unknown when you only have the marginal table, you could make an educated guess (maybe based on the raw data from a study that you have conducted yourself or have access to) and then conduct a sensitivity analysis.

Once you have the 'matched pairs marginal log odds ratio' (and its variance) for each study, you can proceed with standard meta-analytic methods.