Association between an indicator’s inequality d̄ and β from simulated data. Across simulation runs, we vary one parameter at a time, considering five different increments that satisfy the criteria θ << 0 , φ << 0 , ω > 0, τ ≥ 0 and δ > 1 (0.5, 0.75, 1.0, 1.25 and 1.5), while holding the remaining parameters of equations ( 1 ) and ( 2 ) constant at 1. For each parameter configuration, we generate 1,000 posterior draws of the estimated regression coefficient of β * on d̄ * (see text for the standardization) to display estimation uncertainty (in total, 21 × 1,000 regression slopes). The colour represents the density over the regression lines extracted from simulated β and d . The density ranges from 0 (dark blue) to 1 (bright yellow); for relative interpretation, we standardized density values by dividing each by the maximum density. The mean regression coefficient is 8.573 ± 0.209 (95% confidence interval around the mean regression coefficient), R 2 = 0.973. The inset shows the same result for the slope of d (0.695 ± 0.023 and R 2 = 0.849).
