## The Annals of Statistics

- Ann. Statist.
- Volume 36, Number 3 (2008), 1464-1507.

### Statistical modeling of causal effects in continuous time

#### Abstract

This article studies the estimation of the causal effect of a time-varying treatment on time-to-an-event or on some other continuously distributed outcome. The paper applies to the situation where treatment is repeatedly adapted to time-dependent patient characteristics. The treatment effect cannot be estimated by simply conditioning on these time-dependent patient characteristics, as they may themselves be indications of the treatment effect. This time-dependent confounding is common in observational studies. Robins [(1992) *Biometrika* **79** 321–334, (1998b) *Encyclopedia of Biostatistics* **6** 4372–4389] has proposed the so-called structural nested models to estimate treatment effects in the presence of time-dependent confounding. In this article we provide a conceptual framework and formalization for structural nested models in continuous time. We show that the resulting estimators are consistent and asymptotically normal. Moreover, as conjectured in Robins [(1998b) *Encyclopedia of Biostatistics* **6** 4372–4389], a test for whether treatment affects the outcome of interest can be performed without specifying a model for treatment effect. We illustrate the ideas in this article with an example.

#### Article information

**Source**

Ann. Statist., Volume 36, Number 3 (2008), 1464-1507.

**Dates**

First available in Project Euclid: 26 May 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1211819571

**Digital Object Identifier**

doi:10.1214/009053607000000820

**Mathematical Reviews number (MathSciNet)**

MR2418664

**Zentralblatt MATH identifier**

1360.62511

**Subjects**

Primary: 62P10: Applications to biology and medical sciences

Secondary: 62M99: None of the above, but in this section

**Keywords**

Causality in continuous time counterfactuals longitudinal data observational studies

#### Citation

Lok, Judith J. Statistical modeling of causal effects in continuous time. Ann. Statist. 36 (2008), no. 3, 1464--1507. doi:10.1214/009053607000000820. https://projecteuclid.org/euclid.aos/1211819571