Estimation of standard error of the median from median and confidence intervals for meta-analysis? I have a dataset of median survival time and their confidence intervals. One way of pooling these medians using meta-analysis requires an estimate of the standard error of the median / variance of the median of the raw data. I am assuming the unseen raw data are skewed since they are survival data. However, if I assume the sampling distribution of medians is normal (for a large sample size), is it reasonable to estimate the standard error of the median from the confidence intervals e.g.
(Upperbound CI - lowerbound CI) / (2*1.96) ?
One reason I am hesitant is that the confidence intervals for the medians are asymmetric around the median value and so I'm not sure whether this would make sense.
Any guidance would be very much appreciated.
 A: If the confidence intervals are not symmetric about the point estimate, then it is likely that a log link function was used before inverting a Wald test to construct the confidence interval, i.e.
exp$[$log$\{\hat{\theta}\}\pm1.96\cdot\hat{\text{se}}/\hat{\theta}]$,
where $\hat{\theta}$ is the median point estimate, $\hat{\text{se}}$ is the estimated standard error of $\hat{\theta}$, and  $\hat{\text{se}}/\hat{\theta}$ is the estimated standard error of log$\{\hat{\theta\}}$ based on the delta method.  If you have the upper confidence limit you can solve for the estimated standard error, i.e.
$\hat{\theta}_U$= exp$[$log$\{\hat{\theta\}}+1.96\cdot\hat{\text{se}}/\hat{\theta}]\implies \hat{\text{se}}=[\text{log}\{\hat{\theta}_U\}-\text{log}\{\hat{\theta}\}]\frac{\hat{\theta}}{1.96}$
You can then pool the point estimates and standard errors and ultimately construct a meta-analytic confidence interval by inverting a Wald test, perhaps with a log link function.
A: Yes, it's reasonable but not ideal.
It's likely that the interval was constructed from a confidence interval around a Kaplan-Meier estimator for the proportion surviving (because that's how it's commonly done in software).  If so, you might be better off meta-analysing the survival curves and extracting a confidence interval from the summary survival curve. But you might not have the survival curve information needed to do that, and what you describe makes sense.  My survey package for R estimates standard errors for quantiles from confidence intervals this way.
As another possibility, Charles Gray has shown that (at least in uncensored data) you get surprisingly good results by just picking a reasonable parametric model and using the standard error of the median based on the density in that model.
