Approximately unbiased estimators of the inverse variance from sample summary statistics

Background In meta-analysis, researchers often pool the results from a set of similar studies. A number of studies, however, often tend to report only the minimum and maximum values, median, and/or the first and third quartiles. Recently, many methods have been discussed for estimating the mean and standard deviation from those sample summaries. However, these methods may provide a substantially biased estimate of the inverse variance that is needed for the meta-analysis. Research Design We use Basu’s theorem to derive unbiased estimators for σ −2 from the most commonly used sample summaries from the normal distribution. While there are no closed formulas for these estimators, we use simulations to obtain simple approximations for the estimators. Results The proposed approximate estimators still show a little to no bias for normally distributed data and generally show smaller bias than the usual methods even for some non-normal distributions. The proposed estimators have lower mean squared error. Conclusions The proposed estimators are recommended for the purpose of obtaining inverse-variance weights, particularly in the context of meta-analyses.