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It appears common among meta-analyses to omit studies that do not provide an estimate of sample standard deviation. For example, a previous question reviews some ways to estimate $\sigma$ from different summary statistics.

But is lack of an error estimate (or standard deviation) alone sufficient justification for omitting a study from a meta-analysis? If so, is there a reference for this, and how can this potential source of bias be addressed while implementing a meta-analysis? What are statistical procedures to cope with the bias in estimation of meta-analytic value of effect-sizes and/or variability.

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It is a source of bias, and there are many ways around this. Calculate the SD from another source of variance (e.g. SE, CI), use a common SD across all the included trials, use the largest SD in the meta-analysis (e.g. largest SE of all the included trials), and most importantly... contact the original study authors.

Ahmed

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No. One needs only the effect-sizes and respective sample-sizes for a meta-analysis. If a study does not report the effect size and indicates sample size plus standard deviation, one may attempt to assess the effect-size with some assumptions.

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