Estimating an overall summary statistic (mean) for a value reported by ten published studies Background
I am writing a systematic review and meta-analysis of the association of exposure to X with the outcome Y. I have identified ten studies that report on this subject. This will be a meta-analysis of the published literature, not of individual participant data. I want to report the overall mean age and standard deviation of all patients in all of the studies together. The studies all have different sample sizes. How should I do this?
Possibilities
(1) I imagine that I cannot just add up the reported mean values and divide by ten. This would give smaller studies an undue weight in contributing to the overall mean.
(2) I thought perhaps that I could just do a simple weighted average, using the sample size of each study as the weight. Is this appropriate? Can I also just do a weighted average of the reported standard deviations and ranges? If not, how would I compute those?
(3) Another thought I had was actually taking each of the ten reported mean weights along with their standard deviations and actually plugging them into a meta-analysis program to get a "pooled weight" with a fixed effects model. But perhaps this would be a strange approach?
Any advice on which technique to use? Is there another possibility that I'm not thinking of? Thanks!
 A: Your goal of obtaining a combined effect estimate is the central problem of meta-analysis, as you probably know. Before going too far, it may help you to have a textbook such as this one at hand to use as a reference and to give you some illustrative examples. 
Regarding your question:


*

*As you correctly state later, simply averaging the studies will not give appropriate weight to more accurate studies. So scratch this approach.

*This is closer to the typically taken approach in meta-analysis (good intuition). Often a weighted average of the observed effect sizes is calculated where the weights correspond to the inverse standard deviations of the effect size estimates from your studies. This is also the approach used in a fixed-effects regression that meta-analyses often include.

*It is not entirely clear what you mean by "reported mean weights" but I suppose it is the observed effect sizes from your studies. As mentioned in the previous bullet, it is indeed common to perform a weighted fixed effects regression, using the inverse standard deviation or variance as weight. Below, I will point you to some options.


A very useful R package for performing meta-analysis is metafor. This will provide you with many tools. Of particular interest might be the following.


*

*If you have the observed effect sizes and their variances, you can use the function rma.uni with method = "FE" to perform a fixed effects meta-analysis.

*A fixed effect analysis is not appropriate if there is heterogeneity in the studies. Use the Q-test to assess this. If there is heterogeneity, you should consider using a random effects model if you have enough studies. Set method = "REML" in rma.uni to perform one. 

*You may also want to examine your data for publication bias. Once you have a model from rma.uni you can create a funnel plot to see if you might have this problem (funnel plots do not prove publication bias is present, but indicate visually that it might be).

*You can also use your output from rma.uni to create a forest plot, which may be helpful in visualizing your studies.

