I am writing my own function, designed to allow you to manually fix confidence intervals in metagen (rather than metagen estimating these automatically from given effect sizes).
I have everything I need to manually calculate the weights. However, I need to know how metagen creates the POOLED confidence interval to write the rest of my custom function.
That is, I already know how metagen estimates the confidence intervals of each study, I know how it calculates the weights assigned to each study, and I know how it calculates the pooled point estimate. What I don't know is exactly how it calculates the confidence interval around this pooled point estimate.
I've looked deeply into the documentation and this isn't explained. Please, could someone point me to where to find this, or tell me how it is done?
Why do you need to know this? Based on your previous question, you are attempting to pool SIRs, but you want poisson exact standard errors (Byar's approximation) instead of those calculated by metafor? Why don't you just calculate the poisson exact SE for each SIR, then pool these using generic meta-analysis? Writing a function to manually set confidence intervals seems needlessly complicated - meta simply estimates the standard error for each study then gets CIs as +/- 1.96*SE. Alternatively, you can actually input confidence intervals for each study into meta directly, using the lower = and upper = arguments to metagen(). You could calculate the Byar confidence intervals by hand, then simply input these. It sounds like your issue is with the precision of the studies involved rather than the pooled standard error.
To directly answer your question, it depends on whether a fixed or random effects model is used. The confidence interval is calculated as ES +/- 1.96*pooled SE. The standard error of the pooled estimate is calculated by meta as the square root of the pooled variance. The pooled variance is calculated as 1/sum(weights). For a random effects model, this incorporates both within-study variance and between-study variances (tau^2). Therefore, for a random effects model, it will also depend in part on which estimator is used for tau^2.
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