Background: I am working with a data set that requires a transformation. It's prevalence data so I have proportions to deal with, and as the proportions are quite low, I'm using the Freeman-Tukey transformation. My aim is to perform a meta analysis on the prevalence data.
I have transformed the proportions, and found confidence intervals using the transformed data.
I have a forest plot with CIs calculated exactly, and another with CIs calculated after a backtransformation. The largest difference between the two sets is 0.07, so they are very similar.
My issue is deciding whether I should be reporting the exact confidence intervals, or those that have been back transformed. There are ten studies in my data, so an approximation is not appropriate.
Question: In order to gain the correct confidence intervals, do I have to perform a back transformation?
I currently have two sets of answers and I'm not sure of the correct method.
Example: Let's say I have a proportion:
(1) In calculating the exact CIs without transformation, I get:
p=0.01245443; LB=0.01036126; UB=0.01484199
(2) After transforming the original data, and using
(p-z*SE(p), p+z*SE(p)), where
SE(p)=sqrt(1/(n+0.5)), I get:
p=0.224109; LB=0.2043868; UB=0.2438312
(3) Back transforming gives:
p=0.01245443; LB=0.01035768; UB=0.01474083
But which of these three results is correct?