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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 ?

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    $\begingroup$ You must look for some indication of spread in each of your papers, be it Confidence Intervals, Standard Deviations or Mean Square Errors. What do you find? $\endgroup$ – Dirk Horsten Aug 2 '15 at 21:43
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    $\begingroup$ I do not have this information, all I have is the proportion of each paper. I used an older method of Tukey and Freeman (arcsin transformation), which gave two results: fixed effect and random effect. The fixed effect was equal to the weighted mean while the random effect was not. I don't understand why. $\endgroup$ – user3275222 Aug 3 '15 at 9:48
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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.

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