Improving causal inference in interrupted time series analysis: the triple difference design

TL;DR

This paper formalizes the triple-difference interrupted time series design, enhancing causal inference in policy evaluation by adding a second control group and providing practical guidance and implementation updates.

Contribution

It introduces a formal framework for DDD-ITSA, clarifies the estimand, and demonstrates its application with an example, improving causal inference in complex observational studies.

Findings

Triple-difference estimand showed a significant reduction in cigarette sales.

Control groups were balanced on pre-intervention levels and trends.

Implementation updates to the itsa Stata package facilitate use.

Abstract

Background: Interrupted time series analysis (ITSA) is widely used to evaluate health policy and intervention effects. While multiple-group ITSA (MG-ITSA) improves causal inference by incorporating a control group, residual confounding from unmeasured time-varying factors may remain. The triple-difference interrupted time series (DDD-ITSA) design extends this approach by adding a second control group to further isolate treatment effects, but it remains underutilized and lacks formal guidance. Methods: We formalize the DDD-ITSA framework, specify the regression model, define key parameters for estimating level and trend effects, and clarify interpretation of the triple-difference estimand. We illustrate the approach using a worked example evaluating California's Proposition 99 cigarette tax and its impact on per-capita cigarette sales. Results: In the example, all groups were balanced on pre-intervention level and trend. The triple-difference estimand indicated a statistically significant annual reduction of -1.76 per-capita cigarette packs in California relative to the secondary control (P = 0.020; 95 percent CI: -3.24, -0.28), consistent with results from the primary comparison. Differences between control groups were not significant. Conclusions: DDD-ITSA strengthens causal inference when two-group comparisons may be confounded by leveraging an additional control group to remove remaining biases and assess heterogeneity. Implementation is facilitated by updates to the itsa Stata package. Careful attention to control selection, baseline balance, and autocorrelation remains essential.