Is a randomized trial required?

No; interrupted time series is often quasi-experimental.

Can the effect grow over time?

Yes, when the model includes a slope change.

AI Glossary / Statistics and probability / Advanced

Interrupted time series

Definition

Interrupted time series is a study design that examines repeated measurements before and after a defined intervention or event. It assesses whether the outcome's level, trend, or both change relative to a specified continuation of the earlier pattern. The design can support intervention evaluation, but a change occurring at the same time does not by itself establish causation.

What it means

A simple before-and-after average can confuse an intervention with an improvement already underway. Interrupted time series uses the sequence of observations to describe the earlier trajectory and compare later observations with a proposed counterfactual trajectory.
That counterfactual is a model-based description of what might have happened without the interruption. It is not observed. Its credibility depends on the study context, measurement consistency, competing events, and the adequacy of the modeled trend.

How it works

Define the intervention date, outcome, observation frequency, and expected response pattern. Decide whether an immediate shift, a gradual change, or a delayed response is plausible before choosing a model from the observed results.
For a simple continuous outcome, write the modeled mean as b0 + b1*t + b2*D + b3*(t-t0)*D. Here D is zero before the intervention and one from t0 onward. The coefficient b2 represents the immediate level change at t0; b3 represents the change in slope. The post-intervention slope is b1 + b3.
The time interaction must be centered consistently with that interpretation. Inspect seasonality, serially correlated errors, and changes in how outcomes are recorded. Counts or rates may require a different outcome model. Report uncertainty for the effect at the decision-relevant time.

Three practical examples

A service team evaluates an AI routing launch using monthly handling times before and after deployment. It considers whether an existing improvement trend would have continued.
An operations team studies a monitoring-policy change using repeated incident counts. It checks whether reporting rules changed at the same time.
A training program examines assessment outcomes over successive cohorts. A comparison series may help assess broader changes, provided its relationship to the intervention and competing events is credible.

Think of it this way

Imagine extending the earlier direction of travel as a reference line, then examining how the later path departs from it. The reference is an informed projection, not a second journey that was actually observed.

Worked example

Consider an invented series of average task-handling times, measured in minutes. Let t run from 1 to 24, and let a new workflow begin at t0 = 13. For illustration, suppose the fitted mean before the change is 30 minus 0.5t.
Without an intervention effect, the projected mean at month 13 would be 30 minus 6.5 = 23.5 minutes. At month 18 it would be 21 minutes, and at month 24 it would be 18 minutes.
Now stipulate a fitted immediate change b2 = minus 3 minutes and slope change b3 = minus 0.2 minutes per month. The post-intervention fitted mean becomes 30 minus 0.5t minus 3 minus 0.2 times (t minus 13).
At month 13, elapsed post-intervention time is zero. The fitted mean is 20.5 minutes, exactly three minutes below the projected 23.5. This isolates the level-change interpretation.
At month 18, five months have elapsed since month 13. The fitted mean is 21 minus 3 minus 1 = 17 minutes. The difference from the projected counterfactual is minus four minutes.
At month 24, eleven months have elapsed. The fitted mean is 18 minus 3 minus 2.2 = 12.8 minutes. The modeled difference is minus 5.2 minutes. The overall post-intervention slope is minus 0.5 minus 0.2 = minus 0.7 minutes per month, not minus 0.2.
Compare that with a crude average contrast. Across months 1 through 12, the mean time index is 6.5, so the mean fitted outcome is 26.75 minutes. Across months 13 through 24, the mean time index is 18.5. The post-intervention fitted average is 30 minus 9.25 minus 3 minus 1.1 = 16.65 minutes.
The crude post-minus-pre difference is minus 10.1 minutes. However, the average post-period counterfactual is 20.75 minutes, so the modeled average intervention-associated difference is only 16.65 minus 20.75 = minus 4.1 minutes. The remaining six minutes in the crude contrast reflect continuation of the earlier downward trend.
Check the coding another way. Expanding minus 3*D minus 0.2*(t-13)*D gives minus 0.4*D minus 0.2*t*D. In this uncentered form, the coefficient minus 0.4 on D is not the immediate change at month 13. Evaluating both terms there gives minus 0.4 minus 2.6 = minus 3. Correct interpretation depends on the parameterization.
These numbers are stipulated fitted means, not actual observations or a completed statistical analysis. No confidence interval can be calculated from them alone. If a staffing increase occurred at month 13 as well, these calculations would not separate its contribution from the workflow's contribution.

Illustrative case study

Fictional company Alder Routing launches an assistant while recording handling times monthly. Analysts discover that a staffing policy changed during the same month. They document this competing explanation and investigate a comparison service that experienced the staffing policy but not the assistant launch.
They assess whether the comparison is suitable rather than assuming any untreated service will work. Their report distinguishes observed changes, model-based contrasts, and the assumptions needed for attribution. This case is fictional.

Common misconceptions and mistakes

A visually striking break is not sufficient proof of an intervention effect.
More transactions within one month do not automatically provide more independent time points.
The slope-change coefficient is not the complete post-intervention slope.
A control series can help address shared events, but an unsuitable control can introduce bias or fail to address the relevant event.
Trying many interruption dates or response shapes and reporting only the strongest result can misrepresent the evidence.

Business use

Use the design when a rollout has a defined timing and consistent historical measurements. Preserve the underlying series, operational calendar, and record of concurrent changes. Explain what continuation was assumed and how the estimated difference develops over time before translating it into an operational benefit claim.

Check your understanding

What is the immediate example change?

Answer: Minus three minutes.

What is the post-intervention slope?

Answer: Minus 0.7 minutes per month.

Why is the crude average difference larger?

Answer: It includes the pre-existing downward trend.

Related terms

Autocorrelation
Segmented regression
counterfactual
seasonality
level change
slope change
controlled interrupted time series

Learn next

Study serial-correlation diagnostics, intervention timing, comparison-series selection, and uncertainty for time-specific model contrasts.

Sources

Lopez Bernal, Cummins and Gasparrini, tutorial and published correction: https://www.ag-myresearch.com/2017_lopezbernal_ije.html . Their study of controls: https://pubmed.ncbi.nlm.nih.gov/29982445/ . Numerical model and company case are original illustrations.

Previous: Interpretation Bias Next: Intersection Of Events