# Binomial confidence intervals - which is correct?

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: 123/9876.

(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?

• This doesn't appear to be a question about programming. If you have a question about statistical modeling, you should ask over at Cross Validated, not here. – MrFlick Mar 27 '17 at 16:05
• See this great answer: stats.stackexchange.com/questions/82720/… – Tim Mar 28 '17 at 9:49
• 123/4321 is not 0.01245443. – Wolfgang Mar 28 '17 at 11:06
• Apologies, I initially used an example with 4321 but decided I wanted to make the proportion smaller to make it clear that I was using the Freeman-Tukey transformation. I have edited it now. – Tom Mar 28 '17 at 11:52
• In my real example, 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. – Tom Mar 28 '17 at 11:55