I am working on a random effects meta-analysis covering a number of studies which do not report standard deviations; all studies do report sample size. I do not believe it is possible to approximate or impute the SD missing data. How should a meta-analysis which uses raw (unstandardized) mean differences as an effect size be weighted when standard deviations are not available for all studies? I can, of course still estimate tau-squared and would like to incorporate that measure of between-study variance in whatever weighting scheme I use to stay within the random-effects framework.
A little more information is included below:
Why raw mean differences might still be useful: The data is reported in a an intrinsically meaningful scale: US dollars per unit. So, a meta-analysis of mean differences would be immediately interpretable.
Why I cannot approximate or impute the SD data: The studies for which standard deviation data is missing do not include enough data to approximate a standard deviation (i.e. median and range are never reported in the literature). Imputing the missing data seems inadvisable as a large portion of the studies are missing the sd, and because the studies differ greatly in terms of geographical region covered and survey protocol.
What is typically done with raw mean differences in meta-analysis: Study weights are based on the standard error of the mean difference (typically computed with sample-size term and the pooled variance). I don't have this. In a random-effects meta-analysis, study weights also include a term for between-study variance. I have this.
Can simple inverse-sample-size weighting be used in this context? How would I incorporate my estimate of tau-squared (or some other measure of between-studies dispersion) into the weighting?