Supplementary file 1_Beyond the pooled mean: distributional structure in the attentional-focus meta-analysis.docx

This study evaluates whether distributional tools applied to study-level effect-size posterior mixtures can capture evidence structures—such as directionally contrary evidence and shape consistency—that conventional mean-focused meta-analytic models obscure.</p>Methods<p>We reanalyzed 200 effect sizes from the published attentional-focus literature using three distributional frameworks applied to study-level effect-size posterior mixtures rather than participant-level raw distributions: Wasserstein-1 (W₁) distances, a distributional heterogeneity measure H_w, and distributionally robust optimization (DRO).

Against a zero-effect benchmark, W₁ was decomposed into the pooled mean and a non-negative residual equal to twice the expected negative part of the mixture, quantifying the proportion of evidence opposing the recommended instructional focus. H_w was compared against the location-only prediction 1−I²/100, and DRO stability margins (expressed as signal-to-noise ratios) were benchmarked against conventional trim-and-fill publication-bias corrections.</p>Results<p>The directional residual derived from accounted for 36.4% of the transport distance for transfer outcomes, 20.5% for retention, and 13.5% for immediate performance; its bootstrap confidence intervals excluded zero across all evaluated outcomes, demonstrating that no sub-literature in this sample is fully summarized by its mean alone.