Estimating intervention impacts when timing is unclear: an AR-LagDT model with distributed lags

Interrupted Time Series (ITS) analysis is widely used to evaluate intervention impacts, but conventional methods like the classical segmented regression (CSR) and "prediction" models (ARIMA, ETS, etc.) have limitations. CSR assumes an immediate effect, while "prediction" models rely on counterfactual estimates that become unreliable over time. To address this, we propose the Autoregressive with Lag Distributed Term (AR-LagDT) model, which explicitly accounts for intervention effect ambiguity using distributed lag functions. By incorporating varying fuzzy statuses (the onset and duration ambiguity of the intervention effect) and multiple distribution patterns, the AR-LagDT model provides more precise estimates of intervention dynamics. We validate the model through an empirical case study on Japan’s COVID-19 State of Emergency and its impact on the Human Mobility Index in Tokyo. To enhance usability, we develop a user-friendly software application for AR-LagDT model. Our findings demonstrate the model’s ability to capture the temporal progression of intervention effects, providing estimates of maximum impact, cumulative effect, and time-dependent changes. In the empirical analysis, AR-LagDT (e.g., MSE: 7.0139) demonstrate superior in-sample fit to CSR (e.g., MSE: 14.2741) while achieving a broadly similar performance level to the average of "prediction" models (e.g., MSE: 8.1845, ranging from 5.6944 to 14.1337). Moreover, the AR-LagDT model offers clearer interpretability of time-varying impacts. Despite challenges in selecting appropriate fuzzy statuses and distribution patterns, this model offers a robust and flexible framework for impact evaluations in various disciplines and enhances accessibility for researchers conducting intervention analyses in epidemiology, public health, and policy evaluation.