I am working on a paper right now where meta-analysis is a key element. Here is an excerpt from the methods section. The equations didnt copy so I used some short hand notation instead.
The goal is to estimate the reporting rate (p^) of electronic tags using success/failure as reported in 53 journal articles as data.
We analyzed the PSAT reporting rates using re-sampling methods (Efron 1979, 1988; Efron and Tibshirani, 1986; Manly, 2001; Gurevitch and Hedges, 1999). We assumed heterogeneity, so that each study had its own reporting rate and variance. Then for each observed reporting rate, pi, and sample size, ni, we generated 3000 binomial random variables, r, from:
r~ bin(ni,pi) . (14)
Then for each species, manufacturer and depth class we computed the sample size weighted reporting rate:
p =sum(ni*pi/n) (15)
where there are k studies, each with its own ni deployments and reporting rate pi, and a total of n PSAT deployments. The percentiles of the bootstrap sampling distributions were then used to summarize the reporting rates for each species, depth class and manufacturer.