Mastering a simulation-based parametric method to obtain bias-corrected point estimates and sampling variance for effect sizes

Abstract

Meta-analyses require an effect-size estimate and its corresponding sampling variance from primary studies. For some effect size statistics, however, estimators of sampling variance are unavailable, requiring new derivations. Traditionally, such formulas are obtained using hand-derived Taylor expansions (the delta method), but this approach can be technically demanding and inaccessible to many applied researchers. Building on the idea of single-fit parametric resampling, we introduce the SAFE bootstrap: a Single-fit, Accurate, Fast, and Easy simulation recipe that replaces complex algebra with four intuitive steps: fit, draw, transform, and summarise. SAFE is a model-based parametric bootstrap applied to summary statistics, so its performance depends on how well the assumed sampling model approximates the sampling distribution of the reported statistics. Here, we focus on two-group effect sizes that can be parameterised from standard reported summaries, using Gaussian working models for continuous outcomes and binomial or multinomial models for discrete outcomes. Within this framework, SAFE yields model-based, bias-corrected point estimates and standard errors for familiar effect sizes, and readily extends to less common statistics. We first demonstrate SAFE with a simple example, then apply it to common effect sizes, such as the standardised mean difference and the log odds ratio, as well as several less common measures. With additional coding, SAFE can also accommodate zero values and small sample sizes, albeit with important caveats. Our tutorial, accompanied by

code supplements, aims both to clarify the logic of simulation-based estimation from summary statistics and to provide a practical workflow when only summary statistics are available.