The instrumental variables (IV) method provides a way to estimate the causal effect of a treatment when there are unmeasured confounding variables. The method requires a valid IV, a variable that is independent of the unmeasured confounding variables and is associated with the treatment but which has no effect on the outcome beyond its effect on the treatment. An additional assumption often made is deterministic monotonicity, which says that for each subject, the level of the treatment that a subject would take is a monotonic increasing function of the level of the IV. However, deterministic monotonicity is sometimes not realistic. We introduce a stochastic monotonicity assumption, a relaxation that only requires a monotonic increasing relationship to hold across subjects between the IV and the treatments conditionally on a set of (possibly unmeasured) covariates. We show that under stochastic monotonicity, the IV method identifies a weighted average of treatment effects with greater weight on subgroups of subjects on whom the IV has a stronger effect. We provide bounds on the global average treatment effect under stochastic monotonicity and a sensitivity analysis for violations of stochastic monotonicity. We apply the methods to a study of the effect of premature babies being delivered in a high technology neonatal intensive care unit (NICU) vs. a low technology unit.

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