I have data summarizing proportion of events from 5 different papers. The proportions are in %'s. For example, in paper number 1, there was a report of 2000 people and a proportion of 10%. I want a general proportion and CI's. I tried using a generalized linear mixed model with PROC GLIMMIX in SAS, but for some reason, the probability estimate I got was equal to the average of all proportions, and not the weighted average as I would expect. I used a binomial link function, and declared: model proportion/n = / solution cl; and random intercept / subject=paper; I took the model's estimate, and calculated the probability using (1)/(1+e^-estimate). As I mentioned, I got the exact mean of the percentages, and not the weighted mean like my logic say I should have got. I can easily calculate the weighted mean by hand, but I need a CI. Can you help me do it, preferably with SAS, if not, then with R ?
The method of Freeman and Tukey would be appropriate here or indeed using another transformation like the logit. These are widely available in R packages of which there is a list on the CRAN Task View.
The reason why the fixed effect model differed from the random effects is that the random effects using weights which incorporate a component depending on between study variability leading to weights which are more similar than the fixed effects ones.
Disclaimer: I maintain the Task View mentioned above.