This work illustrates how you can apply quantile process regression (QPR) methods to perform causal inferences. QPR builds the probability distribution model for a response variable conditional on explanatory covariates by fitting quantile regression models in the entire quantile-level range from 0 to 1. For treatment-control comparison studies, you can use QPR to predict counterfactual distribution of the treatment response (that counterfactually assumes the treatment-group subjects were not receiving the treatments) by scoring the fitted distribution model of the control-group response on the treatment covariates.

Because QPR estimates the entire response distribution, you can then evaluate treatment effects and subjects-selection bias by using a variety of statistical standards such as mean difference, median different, Mann-Whitney-Wilcoxon U score, and so on. In addition, when mediation is involved, QPR can also predict the distributions of the mediation variables and perform causal mediation analysis. The work uses two real-data studies to illustrate the QPR causal inference methods. The first example analyzes the impact of smoking on newborn body weights, and the second example evaluates the employment-and-salary effect of the 1975-1979 National Supported Work Demonstration project.