Posterior predictive distributions for the fraction of redMaPPer  members that are not associated with the main halo ⟨Δ_prj⟩/λ as a function of intrinsic richness λ for different redshift (panels). We plot the result of the simulation-based calibration (cyan). The 84th (97.5th) percentiles for the model with a free amplitude (prj+) are shown as orange, dashed (dotted) lines. The median (full line) and 16th and 84th percentiles (dashed line) for the model with free amplitude, and the redshift and richness trends (prj++, red) are also shown.