The Annals of Statistics

Mimicking counterfactual outcomes to estimate causal effects

Judith J. Lok

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In observational studies, treatment may be adapted to covariates at several times without a fixed protocol, in continuous time. Treatment influences covariates, which influence treatment, which influences covariates and so on. Then even time-dependent Cox-models cannot be used to estimate the net treatment effect. Structural nested models have been applied in this setting. Structural nested models are based on counterfactuals: the outcome a person would have had had treatment been withheld after a certain time. Previous work on continuous-time structural nested models assumes that counterfactuals depend deterministically on observed data, while conjecturing that this assumption can be relaxed. This article proves that one can mimic counterfactuals by constructing random variables, solutions to a differential equation, that have the same distribution as the counterfactuals, even given past observed data. These “mimicking” variables can be used to estimate the parameters of structural nested models without assuming the treatment effect to be deterministic.

Article information

Ann. Statist., Volume 45, Number 2 (2017), 461-499.

Received: November 2013
Revised: December 2015
First available in Project Euclid: 16 May 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62P10: Applications to biology and medical sciences
Secondary: 62M99: None of the above, but in this section 62N02: Estimation

Causality in continuous time dynamic treatments longitudinal data observational studies panel data rank preservation stochastic differential equations structural nested models


Lok, Judith J. Mimicking counterfactual outcomes to estimate causal effects. Ann. Statist. 45 (2017), no. 2, 461--499. doi:10.1214/15-AOS1433.

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Supplemental materials

  • Web-Appendix with “Mimicking counterfactual outcomes to estimate causal effects”. This Web-Appendix provides mathematical details about mimicking counterfactual survival outcomes, additional information on the simulation study, and theorems used in the main text.