# Meta-analysis of means and medians in R?

I've been asked to review the literature for all studies that estimate the mean and median Blood Lead Level (BLL) in a certain country, and then perform a meta-analysis to come up with one overall value for the mean and median level.

Is such a thing typically done? I'm not interested in any "effect size," just a univariate measure.

I'd like to create a forest plot and do some of the standard tests of heterogeneity across studies. Is there a package and function in R that will allow me to do a meta-analysis of mean and median BLLs?

Just to clarify, this would be a separate meta-analysis for means, and a separate meta-analysis for medians.

• I get the impression that the gmeta package (tutorial PDF here) would help you estimate a population mean, but I can't tell whether it would work for estimating a population median... – Nick Stauner Mar 5 '14 at 2:09
• Ahhh… it looks like the example on the bottom of page 4 of that tutorial PDF covers it, right? – Slyron Mar 5 '14 at 2:25
• Sure looks like it! I found it interesting that they discuss using sample medians to estimate a population mean, but I didn't follow the idea completely (I just skimmed, so that's to be expected). You may want to read further in case that would be a better method in your case too (for estimating the population mean...I don't see any indication that it would work for the median). – Nick Stauner Mar 5 '14 at 2:34
• Well, much obliged sir – Slyron Mar 5 '14 at 2:52

I thought I would summarize people's suggestions and what I found on my own. It looks like there are meta-analysis methods for analyzing means but not for analyzing medians. These are some sources that are useful for meta-analyzing means:

I was mostly interested in analyzing medians because BLL measurements are almost always highly skewed. However, provided the sample sizes of individual studies are not small, and you are meta-analyzing many studies, the central limit theorem allows you to collapse the individual study means into an overall mean. See the following paper for more explanation: Julian P. T. Higgins et al., Meta-analysis of skewed data: Combining results reported on log-transformed or raw scales, Statist. Med. 2008; 27:6072–6092.

• The first link is not working. – Amer Oct 14 '15 at 0:57

If you have medians and range, then you can get the formula for converting to mean and SD from this sentinel paper by Hozo et al., 2005 (http://www.biomedcentral.com/1471-2288/5/13).

• Would it be possible to do a meta-analysis of medians though? I ask because blood lead values are highly skewed in all populations. Usually studies report geometric means or medians, but hardly ever standard means. I'd like to create a forest plot of medians – Slyron Mar 5 '14 at 14:51
• To the best of my knowledge 'no'. All the methods I have seen are for means rather than medians. – abousetta Mar 5 '14 at 17:37

I came through a useful paper by Wan et al, 2014 and I thought it will be very useful for people interested in doing meta-analysis and are reading this post. The paper has formulas for converting median, range and/or IQR to mean and standard deviations (Click here for the paper). The method presented has improvements over the Hozo et al, 2005.

Since this question was posted and answered there has been more work in this area. In two papers McGrath and colleagues discuss working directly with the medians and compare this with the transformation methods referenced in other answers. Their papers are "One-sample aggregate data meta-analysis of medians" available here and "Two-sample aggregate data meta-analysis of medians" available here. These are pre-prints and I believe the first at least may be in press.

The two papers are too long to summarise here but in the first paper on summarising in the one-sample case in a nutshell they show that in their simulations working directly with the median is superior to transforming to the mean. They also use an example data-set where the methods give very different results: delay times in care seeking in China. In the second paper things are more complex but it seems that if the majority of the primary studies have presented medians then a median based method is better and if the majority have presented means then the converse applies. They also present methods based on linear quantile mixed models and methods using the quantiles from each primary study to estimate the underlying unknown distribution in each study.

The methods have been provided in an R package metamedian available from CRAN here

Perhaps also worth noting here that they also reference a later paper with a better transformation method by Luo and colleagues entitled "Optimally estimating the sample mean from the sample size, median, mid-range, and/or mid-quartile range" available here