Fig. 1

Distributions used for covariate X. (a) symmetric distributions (normal and uniform), (b) lognormal distributions, (c) gamma distributions, (d) bimodal distributions (mixture of two normal distributions). The panels display the kernel densities based on 1 million observations randomly sampled from each distribution. \( \left[\mathrm{N}\left({\mu}_1,{\sigma}_1^2\right),\mathrm{N}\left({\mu}_1,{\sigma}_1^2\right)\right] \) represents a homogeneous mixture of 50% \( \mathrm{Normal}\left({\mu}_1,{\sigma}_1^2\right) \) and 50% \( \mathrm{Normal}\left({\mu}_1,{\sigma}_1^2\right) \). For figures with boxplots, the top and bottom 0.025 percentiles were truncated to avoid extreme values in order to facilitate the visual comparison of the boxplots