Sensitivity analysis for publication bias in meta-analyses

We propose sensitivity analyses for publication bias in meta-analyses. We consider a publication process such that 'statistically significant' results are more likely to be published than negative or "non-significant" results by an unknown ratio,eta. Our proposed methods also accommodate some plausible forms of selection based on a study's standard error. Using inverse probability weighting and robust estimation that accommodates non-normal population effects, small meta-analyses, and clustering, we develop sensitivity analyses that enable statements such as 'For publication bias to shift the observed point estimate to the null, "significant" results would need to be at least 30 fold more likely to be published than negative or "non-significant" results'. Comparable statements can be made regarding shifting to a chosen non-null value or shifting the confidence interval. To aid interpretation, we describe empirical benchmarks for plausible values of eta across disciplines. We show that a worst-case meta-analytic point estimate for maximal publication bias under the selection model can be obtained simply by conducting a standard meta-analysis of only the negative and 'non-significant' studies; this method sometimes indicates that no amount of such publication bias could 'explain away' the results. We illustrate the proposed methods by using real meta-analyses and provide an R package:PublicationBias.