# 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.

• Is your interest just in median survival for a single defined condition, or is it in estimating relative median survival between two (or more) treatments/conditions? Also, how many studies are you combining in your meta-analysis? How much do their median-survival estimates differ? Please provide that information by editing the question, as comments are easy to overlook and can be deleted.
– EdM
Jul 25 at 16:48
• @EdM, thank you for your comment. The primary interest was in providing pooled estimates for single defined conditions (if I understood you correctly). That being several pooled medians for pre-specified treatment combinations (e.g. cancer treatment type, cancer type, outcome metric etc.). The collected data DOES include sample size, so in order to compute these medians I employed the methods of McGrath et al 2019 onlinelibrary.wiley.com/doi/abs/10.1002/sim.8013 Jul 29 at 16:34
• However, there is also interest in select comparisons, which I do not believe can be computed in the way McGrath et al (2019) describe. Therefore I thought that my proposed approach could be used to fit a meta-regression model with moderators to model the specific effects of interest. Jul 29 at 16:45

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.

• thankyou for your advice. I had assumed that this approach you kindly described could equally be adopted using the lower confidence bound also? However, when I do this the results do not match when calculated using the upper bound. Hence, I would have thought the most appropriate method would be to use all the information available to get a better estimate of the standard error i.e. [log(upper)-log(lower)]/(2*1.96). Aug 8 at 19:34
• This is similar to my original idea in the question but using the logs of the confidence intervals. I'm not sure if this would be incorrect however, since we are no longer using the point estimate itself...? Aug 8 at 19:34
• That seems like a reasonable way to approximate the standard error using both confidence limits. Aug 8 at 21:06
• Thank you! I'm sorry but it has actually come to my attention that SOME confidence intervals in the dataset ARE symmetric. So I am a little puzzled now whether to take the log of these particular ones (and point estimates) to estimate the SE. My hunch is that I should just use the log approach regardless (i.e. for all medians I have in the dataset) to maintain consistency. Also the meta-analysis will be done on the log scale anyway since the distribution of the set of medians from published studies is itself right skewed. Do you think that's reasonable? Aug 15 at 10:35
• Also there is one study with a very small sample size (n=11). Do you think that I should exclude this study on the basis that one cannot reliably assume the central limit theorem for such a small sample size? Aug 15 at 10:42

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.