Abstract
The conventional Q statistic, using estimated inverse-variance (IV) weights, underlies a variety of problems in random-effects meta-analysis. In previous work on standardized mean difference and log-odds-ratio, we found superior performance with an estimator of the overall effect whose weights use only group-level sample sizes. The Q statistic with those weights has the form proposed by DerSimonian and Kacker. The distribution of this Q and the Q with IV weights must generally be approximated. We investigate approximations for those distributions, as a basis for testing and estimating the between-study variance (τ2 ). A simulation study, with mean difference as the effect measure, provides a framework for assessing accuracy of the approximations, level and power of the tests, and bias in estimating τ2 . Two examples illustrate estimation of τ2 and the overall mean difference. Use of Q with sample-size-based weights and its exact distribution (available for mean difference and evaluated by Farebrother's algorithm) provides precise levels even for very small and unbalanced sample sizes. The corresponding estimator of τ2 is almost unbiased for 10 or more small studies. This performance compares favorably with the extremely liberal behavior of the standard tests of heterogeneity and the largely biased estimators based on inverse-variance weights.
Highlights
In meta-analysis, many shortcomings in assessing heterogeneity and estimating an overall effect arise from using weights based on estimated variances without accounting for sampling variation
SSW, outperformed estimators that use inverse-variance-based (IV) weights. Those weights use estimates of the between-study variance ( 2) derived from the popular statistic discussed by Cochran3, which uses inverse-variance weights and which we refer to as
We discuss approximations to the distribution of and estimation of 2 when the are arbitrary positive constants. Because it is most tractable, but still instructive, we focus on a single measure of effect, the mean difference (MD)
Summary
In meta-analysis, many shortcomings in assessing heterogeneity and estimating an overall effect arise from using weights based on estimated variances without accounting for sampling variation. Our studies of methods for random-effects meta-analysis of standardized mean difference 1 and log-odds-ratio 2 included an estimator of the overall effect that combines the studies’ estimates with weights based only on their groups’ sample sizes. SSW, outperformed estimators that use (estimated) inverse-variance-based (IV) weights. Parallel to SSW, we investigate an alternative, , in which the studies’ weights are their effective sample sizes. This is an instance of the generalized statistic introduced by DerSimonian and Kacker, in which the weights are fixed positive constants
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