Bounds improve estimates of treatment effects with hidden variables
Sharp Bounds on Continuous Treatment Effects under Unobserved Confounding
Machine Learning
Summary
When studying how changing a treatment level affects outcomes, hidden factors can make it hard to know the true effect. The authors develop mathematical limits that tightly capture what can be known about treatment effects even when some influencing factors are unmeasured. These limits do not need assumptions like everyone having a chance to receive all treatments or effects moving only in one direction. This work helps give clearer, more reliable conclusions about treatment effects in tricky real-world settings.
What this means in practice
- •For health data analysts: Estimate treatment effects from observational health data even with hidden patient factors influencing treatment choice.
- •For marketing teams analyzing promotions: Better bounds on how varying discount levels affect sales when customer preferences are imperfectly observed.
Authors
Junzhe Zhang
Abstract
We study partial identification of conditional treatment effects with a continuous treatment under unobserved confounding. We target the conditional average potential outcome (CAPO) E[Y_x|z] and the conditional average treatment effect (CATE) between two treatment levels. We assume a bounded outcome and Lipschitz continuity of the response functions and of the propensity scores given the latent confounder, and show that these conditions imply the smooth treatment response and smooth treatment selection conditions of Kim et al. (2018). The combined bound of Kim et al. uses only two reference points on the identified diagonal and is generally loose. Bounding through every diagonal reference point instead gives explicit bounds whose average is the sharp identified set of the CAPO at every treatment, inside or outside the observational support. Capping the difference between the upper and lower bounds gives sharp causal bounds on the CATE. These bounds are informative without monotonicity or overlap assumptions.