## Statistical Science

- Statist. Sci.
- Volume 32, Number 4 (2017), 561-579.

### Instrumental Variable Estimation with a Stochastic Monotonicity Assumption

Dylan S. Small, Zhiqiang Tan, Roland R. Ramsahai, Scott A. Lorch, and M. Alan Brookhart

#### Abstract

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.

#### Article information

**Source**

Statist. Sci., Volume 32, Number 4 (2017), 561-579.

**Dates**

First available in Project Euclid: 28 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ss/1511838028

**Digital Object Identifier**

doi:10.1214/17-STS623

**Mathematical Reviews number (MathSciNet)**

MR3730522

**Zentralblatt MATH identifier**

1383.62290

**Keywords**

Causal inference observational study instrumental variable two stage least squares

#### Citation

Small, Dylan S.; Tan, Zhiqiang; Ramsahai, Roland R.; Lorch, Scott A.; Brookhart, M. Alan. Instrumental Variable Estimation with a Stochastic Monotonicity Assumption. Statist. Sci. 32 (2017), no. 4, 561--579. doi:10.1214/17-STS623. https://projecteuclid.org/euclid.ss/1511838028

#### Supplemental materials

- Supplement to “Instrumental Variable Estimation with a Stochastic Monotonicity Assumption”. Section A of the supplementary materials presents analogues of Propositions 1–4 when conditioning on observed covariates. Section B presents proofs for the propositions given in Section A of the supplementary materials. Section C presents proofs for the results in Section 2 of the main text. Section D shows that deterministic compliance class framework identification results are a special case of stochastic compliance class framework results. Section E discusses bounds on the global average treatment effect for a binary outcome under stochastic monotonicity. Section F gives an example in which bounds under stochastic monotonicity are tighter than bounds without stochastic monotonicity. Section G extends results in the main text to the setting of a non-binary instrumental variable. Section H discusses identification results when the measured IV is an intensity preserving proxy for a continuous causal IV satisfying deterministic monotonicity. Sections I and J give additional examples besides those given in the main text in which deterministic monotonicity is likely violated but stochastic monotonicity is plausible.Digital Object Identifier: doi:10.1214/17-STS623SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.