P-hacking in meta-analyses: A formalization and new meta-analytic methods

As traditionally conceived, publication bias arises from selection operating on a collection of individually unbiased estimates. A canonical form of such selection across studies (SAS) is the preferential publication of affirmative studies (i.e., those with significant, positive estimates) versus nonaffirmative studies (i.e., those with nonsignificant or negative estimates). However, meta-analyses can also be compromised by selection within studies (SWS), in which investigators “p-hack’’ results within their study to obtain an affirmative estimate. Published estimates can then be biased even conditional on affirmative status, compromising existing methods that only consider SAS. We propose two new analysis methods that accommodate joint SAS and SWS; both analyze only the published nonaffirmative estimates. First, we propose estimating the underlying meta-analytic mean by fitting “right-truncated meta-analysis’’ (RTMA) to the published nonaffirmative estimates. This method essentially imputes the entire underlying distribution of population effects. Second, we propose conducting a standard meta-analysis of only the nonaffirmative studies (MAN); this estimate is conservative (negatively biased) under weakened assumptions. We provide an R package, phacking. Our proposed methods supplement existing methods by assessing the robustness of meta-analyses to joint SAS and SWS.