# Is it possible to meta-analyze means on different scales?

I have study-level means/SDs reflecting depression symptom severity from multiple single-group studies. I do not have access to participant-level data. The studies all use different measures, and I have the sample size and the minimum and maximum possible scale scores for each. I would like to be able to aggregate the means using meta-analysis to indicate the average severity of depression across these studies.

I was thinking about rescaling the means to a 0-100 scale so they roughly represent % of maximum severity but I'm not sure what I could use as the variance, because I can't compute a rescaled SD without participant-level data. Is there any other way I could go about this?

I'm using metafor to calculate other effect sizes for this project but could also calculate by hand. I'm assuming it would be inappropriate to treat the rescaled means as if they were percentages or event counts with ni or ti = 100, respectively, because that wouldn't take into account the actual variation in estimates, but if I'm wrong I'd love to know.

## 2 Answers

Ideally, one would want to use proper test equating methods to properly link the scores (and hence means and SDs) of different measures to each other. But since you only have the means, SDs, and possible ranges of the different studies, you are limited in what you can do. As you suggested, you could rescale each measure, so that it has the same range (0 to 1) with: $$y_i = \frac{\bar{x}_i - \mbox{min}_i}{\mbox{max}_i - \mbox{min}_i},$$ where $$\bar{x}_i$$ is the observed mean in study $$i$$ and $$\mbox{max}_i$$ and $$\mbox{min}_i$$ are the maximum and minimum possible scores on the measure used in the study (note that this must be the range of the possible scores, not the observed scores!). Then the sampling variance of $$y_i$$ is given by: $$\mbox{Var}[y_i] = v_i = \frac{\mbox{SD}^2_i}{n_i (\mbox{max}_i - \mbox{min}_i)^2},$$ where $$\mbox{SD}^2_i$$ is the observed SD in the study and $$n_i$$ is the sample size.

If you prefer a 0 to 100 range, multiply $$y_i$$ by 100 and $$v_i$$ by $$100^2$$.

You can then feed these estimates and corresponding sampling variances (or their square-root if the standard errors are the required input) into the meta-analysis software of your choice.

If you have the mean and the standard deviation of each score, you just need to divide the mean by the standard deviation. By proceeding this way, you get a standardized score, that can be comparable (and combinable) with other standardized scores of other studies despite those variables were initially measured on a different scale.

• Could you explain a little more how to interpret that measure and why it is valid for different scales? Or maybe you can mention a source or the name of that measure? It reminds me on en.wikipedia.org/wiki/Coefficient_of_variation but here it is sd/ mean not mean/ sd – machine Jun 4 '19 at 14:09
• Don't do this; the resulting values are not comparable. – Wolfgang Jun 4 '19 at 14:35