Bounding the Effects of Continuous Treatments for Hidden Confounders

Causal inference involves the disentanglement of effects due to a treatment variable from those of confounders, observed as covariates or not. Since one outcome is ever observed at a time, the problem turns into one of predicting counterfactuals on every individual in the dataset. Observational studies complicate this endeavor by permitting dependencies between the treatment and other variables in the sample. If the covariates influence the propensity of treatment, then one suffers from covariate shift. Should the outcome and the treatment be affected by another variable even after accounting for the covariates, there is also hidden confounding. That is immeasurable by definition. Rather, one must study the worst possible consequences of bounded levels of hidden confounding on downstream decision-making. We explore this problem in the case of continuous treatments. We develop a framework to compute ignorance intervals on the partially identified dose-response curves, which enable us to quantify the susceptibility of our inference to hidden confounders. Our method is supported by simulations as well as empirical tests based on two observational studies.