Controlled Interrupted Time Series (CITS) Methodology
In citsr: Controlled Interrupted Time Series Analysis and Visualization

Overview

This vignette provides an overview of the Controlled Interrupted Time Series (CITS) methodology, explaining its analytic principles, model structure, and workflow.
CITS extends the standard Interrupted Time Series (ITS) framework by introducing a comparison (control) group that did not receive the intervention, improving causal interpretation.

1. Purpose

CITS evaluates how an intervention affects an outcome variable over time by comparing trends between a treatment group and a control group.
Unlike a single‐group ITS model, the CITS design helps adjust for external shocks or secular trends through the inclusion of a parallel control series.

2. Data Structure

A typical CITS dataset includes:

y: Outcome variable
T: Time index
I: Intervention indicator (0 = before intervention, 1 = after intervention)
E: Group indicator (0 = control, 1 = treatment)

3. Basic Model

The canonical CITS specification is:

$$ y_{it} = \beta_0 + \beta_1 T_t + \beta_2 I_t + \beta_3 E_i + \beta_4 (T \times I)t + \beta_5 (E \times T){it} + \beta_6 (E \times I){it} + \beta_7 (E \times T \times I){it} + \varepsilon_{it} $$

Where:

( \beta_1 ): baseline trend
( \beta_2 ): immediate intervention effect (control group)
( \beta_3 ): baseline level difference (treatment vs control)
( \beta_4 ): post-intervention trend change (control group)
( \beta_5 ): treatment–control trend difference before intervention
( \beta_6 ): treatment–control level difference at intervention
( \beta_7 ): post-intervention trend difference (treatment vs control)

The error term ( \varepsilon_{it} ) may exhibit temporal autocorrelation.
The citsr package addresses this by fitting the model using generalized least squares (GLS) with optional ARMA(p,q) correlation structures.

ARMA Handling

The cits() function:

Searches candidate ARMA(p,q) models over a user-specified range
Selects the structure minimizing AIC
If all ARMA fits fail, automatically falls back to standard GLS

4. Analysis Workflow

Data preparation
Construct variables ( y, T, I, E ) and their interaction terms.
(The cits() function creates interactions automatically if not provided.)
Model fitting
Estimate the CITS model using GLS or ARMA‐GLS via the cits() function.
Intervention effect estimation
Coefficients ( \beta_2, \beta_4, \beta_6, \beta_7 ) describe level and trend changes attributable to the intervention.
Counterfactual prediction
Counterfactual trajectories for the treatment group are generated by setting
( I = 0 ) after the intervention time and recomputing relevant interaction terms. This enables comparison of actual vs. no‐intervention outcomes.

5. Advantages

Controls for unobserved external shocks using a comparison group
Captures both level and slope changes before and after intervention
Supports autocorrelation modeling through GLS and ARMA structures
Provides interpretable counterfactual predictions

6. Considerations

Sufficient pre/post intervention observations are essential
Parallel trends assumption between groups should be evaluated
Additional covariates may be necessary in applied contexts
Autocorrelation should be tested and accounted for when present

References

Linden, A., & Adams, J. L. (2011). Applying a propensity score–based weighting model to interrupted time series data: Improving causal inference in program evaluation. Journal of Evaluation in Clinical Practice, 17(6), 1231–1238.

Lopez Bernal, J., Cummins, S., & Gasparrini, A. (2018). Use of controls in interrupted time series studies of public health interventions. International Journal of Epidemiology, 47(6), 2082–2093.