# Which transformation should be used when conducting meta-analysis of single proportions?

The goal of my meta-analysis project is to combine proportions from studies without a control group.

In Practical Meta-Analysis, Lipsey and Wilson mentioned that the logit transformation should be used when proportions are less than 0.2 or more than 0.8 instead of using raw proportions. But they didn't mention the double arcsine transformation. Barendregt et al. have claimed the double arcsine transformation to be a preferred option. I wonder which method you guys would prefer to use and how you would do to decide which one to use.

I tried to find out how to determine which method to choose by trying to replicate the results of a handful of studies using both methods (I also used the arcsine square root method). Here is my approach: For each study, I first transformed the original data using the logit and the double arcsine transformation. I then generated Q-Q plots and histograms of these transformed data. After that, I compared their skewness and kurtosis in order to see if they were in the acceptable range. Then I conducted the Shapiro-Wilk test. By now, I would have a basic idea of which method would be better for me to use, but I wonder how you would make the decision.

The thing is, sometimes, both methods could lead to very similar results; sometimes, neither of the methods was able to pass the Shapiro-Wilk test; sometimes, both methods failed to normalize the data; sometimes, the arcsine square root transformation seemed to have done a better job. In these situations, what would you do?

• What is your goal?! Some of those transformations are more for treatment comparisons. – Björn May 16 '17 at 17:38
• I just want to combine the single proportions. – Naike Wang May 16 '17 at 18:49
• Why not using instead a Poisson model without any transformation? – Joe_74 May 17 '17 at 8:10
• To Joe_74, logit and double arcsine are the most commonly used methods in these situations. But could you elaborate on your point a little bit more please? – Naike Wang May 19 '17 at 12:48

If you assume the proportion to be the exact same in all trials, you can just consider this as $y:=y_1+y_2+\ldots$ successes out of $n:=n_1+n_2+\dots$ attempts with an estimate of $\hat{\pi}=y/n$. A CI for the proportion can then just be obtained e.g. using logistic regression or Clopper-Pearson or any number of simple asymptotic approximations for a single proportion.