Instrumental Variable Estimation When Compliance is not Deterministic:   The Stochastic Monotonicity Assumption
  The instrumental variables (IV) method is a method for making causal inferences about the effect of a treatment based on an observational study in which 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 that is often made for the IV method is deterministic monotonicity, which is an assumption that for each subject, the level of the treatment that a subject would take if given a level of the IV is a monotonic increasing function of the level of the IV. Under deterministic monotonicity, the IV method identifies the average treatment effect for the compliers (those subject who would take the treatment if encouraged to do so by the IV and not take the treatment if not encouraged). However, deterministic monotonicity is sometimes not realistic. We introduce a stochastic monotonicity condition which relaxes deterministic monotonicity in that it does not require that a monotonic increasing relationship hold within subjects between the levels of the IV and the level of the treatment that the subject would take if given a level of the IV, but only that a monotonic increasing relationship hold across subjects between the IV and the treatment in a certain manner. 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 the stochastic monotonicity assumption. 
Dylan Small, Zhiqiang Tan, Scott Lorch and Alan Brookhart
stat.ME