Vector incremental treatment effects

Abstract

Causal questions involving multiple binary treatments are poorly served by standard causal approaches. Potential outcomes grow exponentially with the number of treatments, many treatment combinations are rarely or never observed, and fixing everyone to one combination can require implausible positivity assumptions. We extend incremental propensity score interventions to vector-valued binary treatments, defining causal effects through counterfactual distributions that scale the odds of each treatment by a user-specified factor. Because marginal odds constraints do not uniquely determine the joint distribution, we define it as the minimum-divergence projection of the observed law onto the constrained set, for a user-specified separable divergence, preserving the observed conditional support. The resulting counterfactual mean outcome is identified under consistency and conditional mean exchangeability, without requiring every treatment combination to have positive conditional probability. For a class of smooth divergences, we derive the efficient influence function and construct cross-fitted one-step estimators that are consistent at the parametric rate and asymptotically normal when marginal treatment probabilities are exactly 0, exactly 1, or bounded away from those values. We relax this requirement by specifying a different estimand and deriving a corresponding estimator, which is asymptotically normal under a convergence-rate condition on the estimated active set. We also extend sensitivity analysis to address unmeasured confounding. Applying the framework to 159,466 New York Medicaid beneficiaries with serious mental illness, we estimate how healthcare costs would change under uniform improvements in the odds of six recommended quality-of-care processes. Increasing recommended care is associated with lower total, inpatient, and mental-health cost and higher outpatient cost.