Why do we treat study standard errors as known in a meta-analysis?

Question

I wonder if anyone knows why it is that we treat study standard errors as known when conducting a meta-analysis? When conducting analysis of trial data, the error is treating as a quantity to be estimated (at least in the Bayesian code I've seen). In a meta-analysis however, we might say:

y[i]  ~ N(delta,sigma[i])
delta ~ N(u,tau)

leaving sigma as data instead of identifying it as a parameter. Is there a reason for this? Would we just never get stable estimates? Are there any consequences to this approach (i.e., underestimating uncertainty?)

Answer 1

(This answer is the same as the comment.)

This is because meta-analysis uses the aggregated data. If the meta-analysis is performed using a random effect model, the estimate of between-study heterogeneity is produced in response to the amount of errors estimated from analysis of trial data. I think your code should be changed: y[i] ~ N(u,sigma) should read y[i] ~ N(u,sigma[i]), where sigma[i] was data which was extracted from the original article along with y[i].

Answer 2

It's much easier -- especially back in the days before modern computing -- and it's a pretty good approximation, because small differences in the uncertainties don't have much impact on the final result.

Here's an example. Start with the meta-analysis that's in the logo of the Cochrane collaboration

Now adjust the standard errors for each trial by multiplying by independent $U[0.8,1.2]$, changing them up or down by up to 20%. Re-do the meta-analysis. These are the confidence intervals from 100 replications (on the log odds ratio scale)

Even a 20% error in the SE doesn't make a big difference to the overall meta-analysis. So people didn't worry.

Now that meta-analysis can easily model the uncertainties in the standard errors it's probably worth doing, but there were good reasons not to worry too much in the past.