Z-Curve Plot: A Visual Diagnostic for Publication Bias in Meta-Analysis

Abstract: Publication bias undermines meta-analytic inference, yet visual diagnostics for detecting model misfit due to publication bias are lacking. We propose the z-curve plot, a publication-bias-focused absolute model fit diagnostic. The z-curve plot overlays the model-implied posterior predictive distribution of $z$-statistics on the observed distribution of z-statistics and enables direct comparison across candidate meta-analytic models (e.g., random-effects meta-analysis, selection models, PET-PEESE, and RoBMA). Models that approximate the data well show minimal discrepancy between the observed and predicted distribution of z-statistics, whereas poor-fitting models show pronounced discrepancies. Discontinuities at significance thresholds or at zero provide visual evidence of publication bias; models that account for this bias track these discontinuities in the observed distribution. We further extrapolate the estimated publication-bias-adjusted models to the absence of publication bias to obtain a posterior predictive distribution in the absence of publication bias. From this extrapolation, we derive three summaries: the expected discovery rate (EDR), the false discovery risk (FDR), and the expected number of missing studies (N missing). We demonstrate the visualization and its interpretation using simulated datasets and four meta-analyses spanning different degrees of publication bias. The method is implemented in the RoBMA R package.