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

Abstract

Publication bias undermines meta-analytic inference, yet visual diagnostics for detecting and understanding model misfit due to publication bias are lacking. We propose the z-plot, a visual publication bias-focused absolute model fit diagnostic. The z-plot overlays the model-implied distribution of z-statistics on the observed distribution of z-statistics. Models that approximate the data well show minimal discrepancy between the observed and predicted distributions of z-statistics, whereas models that approximate the data poorly show systematic discrepancies. Discontinuities in the observed distribution of z-statistics at significance thresholds or at zero provide visual evidence of publication bias; models that account for this bias track these discontinuities. In addition, the z-plot facilitates visual model fit comparison of competing meta-analytic models within a single figure. We demonstrate the visualization and its interpretation with a Bayesian random-effects meta-analysis, a Bayesian PET model, a Bayesian three-parameter selection model, and RoBMA on simulated datasets and a real meta-analysis. The method is implemented in the RoBMA R package.