Causal Mediation Analysis for an Interrupted Time Series: Stabilized Mediator Weighting with an Application to a Vehicle Emissions Policy

Abstract: Population-level policies are introduced at a fixed time and evaluated from a single series of aggregate outcomes, and the interrupted time series design estimates the total shift in an outcome after the intervention. When the policy is expected to act through a measurable pathway, the total effect is less informative than its decomposition into direct and indirect effects. We formulate causal mediation for a single interrupted time series and study stabilized mediator weighting as the estimator of the natural direct and indirect effects. Because the intervention is a deterministic function, the exposure weight equals one and the exposure contrast is identified through the segmented-regression level shift, so only the mediator pathway is weighted. We add a cumulative mediator weight that carries the lagged confounder history, incorporate a concurrent event as a second interruption, and replace variance formulas that treat the estimated weights as fixed with a block-residual bootstrap that keeps the deterministic exposure timing intact and resamples the mediator and outcome residuals in moving blocks. In a simulation calibrated to daily data, the unweighted product-of-coefficients estimator is biased under mediator-outcome confounding, with an indirect-effect bias near $0.19$ and coverage of $0.003$, whereas stabilized weighting reduces the bias to about $0.03$ and improves indirect-effect coverage from near zero to about $0.83$. Applied to the 2019 termination of Ontario's Drive Clean vehicle emissions testing program, the method estimates a direct reduction in ground-level ozone of $2.113$ parts per billion (95\% interval $-3.384$ to $-0.841$) that is robust across four Toronto regions, a positive but heterogeneous indirect effect through nitrogen dioxide, and a total effect near the boundary of significance; a pre-pandemic sensitivity analysis agrees with the primary results.