Interrupted Time Series

Interrupted time series (abbreviated ITS) is a quasi-experimental longitudinal evaluation architecture in epidemiology, biostatistics, and public health policy research used to quantify the population-level causal impact of an intervention introduced at a clearly defined point in calendar time (T₀). Formalized by Donald T. Campbell and Julian C. Stanley (1963), Glass, Willson, and Gottman (1975), and AK Wagner et al. (2002), an interrupted time series collects a sequence of equally spaced aggregate population observations (t = 1, 2, ..., T₀, ..., T—such as weekly hospital admissions, monthly mortality rates, or quarterly antimicrobial prescribing volumes) across multiple time points both before and after the intervention. Rather than comparing a single pre-intervention mean against a single post-intervention mean (a naive before-and-after comparison that is invalidated by any pre-existing secular decline or rise), segmented regression projects the pre-intervention baseline level and secular slope forward into the post-intervention era as an explicit counterfactual trajectory. Deviations from this extrapolated counterfactual are decomposed into an immediate step-change in level (β₂) at T₀ and a sustained change in slope (β₃) after T₀.

When an intervention is enacted simultaneously across an entire municipality, nation, or hospital network, individual-level randomization is impossible. Interrupted time series designs are widely deployed across environmental health, infectious disease control, and pharmacoepidemiology to evaluate:

• Environmental and Tobacco Legislation: Quantifying immediate reductions in acute myocardial infarction hospitalizations following municipal indoor smoke-free legislation, or drops in pediatric asthma emergency visits after low-emission zone implementation and coal power plant closures (reducing ambient PM_2.5, SO₂, and NO₂).
• Fiscal and Nutritional Health Policies: Evaluating declines in per-capita sugar-sweetened beverage purchases following excise taxation, or reductions in neural tube defect birth prevalence after mandatory folic acid grain fortification.
• National Immunization and Antimicrobial Stewardship Rollouts: Measuring reductions in pediatric pneumonia and invasive pneumococcal disease following conjugate vaccine introduction, or declines in Clostridioides difficile infection rates after hospital fluoroquinolone restriction policies.
• Harm-Reduction and Suicide Prevention: Assessing drops in fatal analgesic overdoses after statutory paracetamol (acetaminophen) blister-pack size limits.

Standard Linear and Generalized Linear Segmented Models

Let Y_t denote the population health outcome (such as age-standardized incidence rate or log-transformed count with population offset) at equally spaced time index t ∈ {1, 2, ..., T}, where the intervention occurs at time T₀. The canonical segmented regression model specifies three core time variables:

1. Time_t = t (a continuous variable indexing elapsed time from the start of the observation series, t = 1, 2, ..., T).
2. Intervention_t = I(t ≥ T₀) (a binary step indicator equal to 0 for all pre-intervention periods t < T₀ and 1 for all post-intervention periods t ≥ T₀).
3. PostTime_t = max(0, t − T₀) (a continuous spline variable equal to 0 prior to the intervention and counting elapsed time intervals since the intervention took effect: 0, 1, 2, ...).

The segmented regression equation is:

E[Y_t] = β₀ + β₁ Time_t + β₂ Intervention_t + β₃ PostTime_t + γ^TZ_t

where Z_t represents seasonal Fourier harmonic terms or time-varying covariates. When Y_t consists of discrete disease counts C_t in a population of size N_t, the model is fitted as a Poisson or negative binomial generalized linear model (GLM) with log link ln(E[C_t]) and offset ln(N_t), so that exp(β₂) estimates the immediate incidence rate ratio (IRR) and exp(β₃) estimates the per-period multiplicative slope ratio.

Absolute and Relative Counterfactual Effect at Horizon k

At k time units after the intervention (t = T₀ + k), the expected outcome under the intervention is:

Ŷ_T0+k^(treated) = β₀ + β₁(T₀ + k) + β₂ + β₃(k)

Whereas the projected counterfactual outcome had the intervention never occurred (Intervention = 0, PostTime = 0) is:

Ŷ_T0+k^{(counterfactual)} = β₀ + β₁(T₀ + k)

Subtracting the two yields the total absolute intervention effect at post-policy horizon k:

ΔY(k) = β₂ + β₃ × k

Serial Autocorrelation (ARMA / Prais–Winsten / Newey–West)

Unlike independent cross-sectional observations, consecutive points in a longitudinal time series exhibit serial autocorrelation: the residual error ε_t at time t is correlated with residuals at preceding lags ε_t−1, ε_t−2, ..., ε_t−p due to pathogen transmission momentum, environmental persistence, and healthcare capacity inertia:

ε_t = ρ₁ ε_t−1 + ··· + ρ_p ε_t−p + u_t

Positive autocorrelation (ρ₁ > 0) does not bias point estimates in ordinary least squares, but it systematically underestimates standard errors, inflating Type I false-positive rates. Epidemiologists test residuals using the Durbin–Watson statistic, the Breusch–Godfrey test, and partial autocorrelation function (PACF) plots, and correct estimation via Prais–Winsten generalized least squares (GLS) for AR(1) errors, Newey–West heteroskedasticity and autocorrelation consistent (HAC) standard errors, or autoregressive integrated moving average (ARIMA) models.

Seasonality Adjustment and Implementation Lag Windows

Many infectious, respiratory, and cardiovascular outcomes follow pronounced annual cycles (peaking in winter and troughing in summer). If an intervention is enacted in spring and evaluated over the subsequent 6 months without seasonal adjustment, natural summer declines will be misattributed to the policy. Seasonal variation is removed by including Fourier harmonic terms—pairs of sine and cosine functions sin(2π*kt* / S) and cos(2π*kt* / S) where S = 12 for monthly data or S = 52 for weekly data—or calendar-month indicator variables, provided at least 2 to 3 full annual cycles (≥ 24–36 monthly observations) are observed before and after T₀.

Furthermore, when an intervention exhibits a biological latent period or gradual logistical rollout (such as a vaccine requiring two doses separated by 8 weeks), the transition interval [T₀, T₀ + L] is either modeled as a phase-in spline or censored as an implementation washout period so delayed onset does not attenuate β₂ and β₃.

The primary threat to internal validity in a single-group ITS is history bias (co-interventions or concurrent shocks occurring at T₀, such as an economic recession, a meteorological anomaly, or an unrelated diagnostic coding revision). To guard against concurrent shocks, investigators employ two extensions:

• Controlled Interrupted Time Series (CITS): Incorporates an untreated comparison series (an adjacent state or province that did not enact the law) modeled with three-way interaction terms (Group × Intervention and Group × PostTime), bridging ITS with difference-in-differences while relaxing the static parallel-trends assumption to allow differing pre-intervention linear slopes. When multiple donor regions exist, CITS can be generalized via the synthetic control method.
• Negative Control Outcome Series: Evaluates the same segmented regression at T₀ on an outcome recorded in the same hospital database that cannot biologically respond to the intervention (such as appendicitis or hip fracture admissions during a smoke-free indoor air ban).

How many pre- and post-intervention time points are required for an interrupted time series?

Methodological guidelines (Wagner et al., 2002; Bernal et al., 2017) recommend a minimum of 8 to 12 time points before the intervention and 8 to 12 time points after the intervention to reliably estimate baseline linear slopes and test AR(1) autocorrelation. When outcomes exhibit seasonal periodicity (such as monthly respiratory infections), at least 24 pre-intervention and 12 to 24 post-intervention monthly observations (with at least 50 to 100 events per time point) are recommended to decouple seasonal harmonics from the policy step and slope parameters.

How does Controlled ITS differ from standard Difference-in-Differences?

A standard two-group, two-period difference-in-differences estimator requires that the treated and control groups follow parallel pre-intervention slopes (β_1,treated = β_1,control). In contrast, a Controlled Interrupted Time Series (CITS) explicitly estimates separate pre-intervention slopes for both the treated and control groups and tests whether the change in level (Δβ₂) and change in slope (Δβ₃) at T₀ differ between groups, controlling for group-specific linear secular trends.

1. Wagner, A. K., Soumerai, S. B., Zhang, F., & Ross-Degnan, D. (2002). Segmented regression analysis of interrupted slowly changing time series studies in medication use research. Journal of Clinical Pharmacy and Therapeutics, 27(4), 299–309. https://doi.org/10.1046/j.1365-2710.2002.00430.x
2. Bernal, J. L., Cummins, S., & Gasparrini, A. (2017). Interrupted time series regression for the evaluation of public health interventions: a tutorial. International Journal of Epidemiology, 46(1), 348–355. https://doi.org/10.1093/ije/dyw098
3. Kontopantelis, E., Doran, T., Springate, D. A., Buchan, I., & Reeves, D. (2015). Regression based quasi-experimental approach when randomisation is not an option: interrupted time series analysis. BMJ, 350, h2750. https://doi.org/10.1136/bmj.h2750
4. Lopez Bernal, J., Cummins, S., & Gasparrini, A. (2018). The use of controls in interrupted time series studies of public health interventions. International Journal of Epidemiology, 47(6), 2082–2093. https://doi.org/10.1093/ije/dyy135