Observational studies with unknown time of treatment

TL;DR

This paper develops a methodology to estimate causal effects in observational studies where the exact timing of treatment is unknown, using plausible treatment times to bound effects and ensure valid inference.

Contribution

It introduces formal assumptions and inference methods for causal analysis with uncertain treatment times, addressing a gap in observational study methodology.

Findings

Bounded the last plausible treatment time in case studies

Provided valid causal estimates despite treatment time uncertainty

Applied methods to online marketing data involving snowfall and search behavior

Abstract

Time plays a fundamental role in causal analyses, where the goal is to quantify the effect of a specific treatment on future outcomes. In a randomized experiment, times of treatment, and when outcomes are observed, are typically well defined. In an observational study, treatment time marks the point from which pre-treatment variables must be regarded as outcomes, and it is often straightforward to establish. Motivated by a natural experiment in online marketing, we consider a situation where useful conceptualizations of the experiment behind an observational study of interest lead to uncertainty in the determination of times at which individual treatments take place. Of interest is the causal effect of heavy snowfall in several parts of the country on daily measures of online searches for batteries, and then purchases. The data available give information on actual snowfall, whereas the natural treatment is the anticipation of heavy snowfall, which is not observed. In this article, we introduce formal assumptions and inference methodology centered around a novel notion of plausible time of treatment. These methods allow us to explicitly bound the last plausible time of treatment in observational studies with unknown times of treatment, and ultimately yield valid causal estimates in such situations.