Concept Architecture
Concept
Theoretically, Interrupted Time Series is a quasi-experimental design used to estimate the effect of an intervention introduced at a clearly defined point in an ordered sequence of observations. It is based on counterfactual reasoning, whereby the pre-intervention trend is projected forward to represent the outcome expected in the absence of the intervention. The design distinguishes immediate changes in outcome level from changes in the underlying trend following implementation.
Mathematically, Interrupted Time Series is commonly represented using segmented regression. The model estimates the baseline outcome level, the pre-intervention trend, the immediate level change at the intervention point and the change in trend after the intervention. Extensions may account for autocorrelation, seasonality, non-linear trends, count outcomes and comparison series.
In practice, Interrupted Time Series is estimated using repeated observations collected at regular intervals before and after an intervention. Health economists apply the method to evaluate policy reforms, reimbursement changes, prescribing restrictions, public health programmes and healthcare delivery interventions when randomised evaluation is infeasible. Model specification, intervention timing, underlying trends, concurrent events and serial correlation must be assessed carefully.
Purpose
Used to estimate the immediate and longer-term effects of healthcare policies or interventions by comparing observed post-intervention outcomes with the counterfactual trajectory projected from the pre-intervention period.
Mathematical Formulae
Primary Formula
Y? = ?? + ??T? + ??X? + ??(T? ? X?) + �?
where:
Y? = outcome observed at time t
T? = continuous time from the beginning of the series
X? = intervention indicator, equal to 0 before the intervention and 1 after the intervention
?? = baseline outcome level at T? = 0
?? = pre-intervention trend
?? = immediate change in outcome level following the intervention
?? = change in trend following the intervention
Supporting Formulae
Post-intervention trend:
?? + ??
Counterfactual outcome after the intervention:
??? = ??? + ???T?
Estimated intervention effect at post-intervention time t:
??? = ??? + ???(T? ? T?)
First-order autoregressive error structure:
�? = ? �??? + u?
Related Mathematical Methods
Segmented Regression
Generalised Least Squares
Autoregressive Integrated Moving Average Models
Newey-West Standard Errors
Controlled Interrupted Time Series
Difference-in-Differences
Poisson Regression
Negative Binomial Regression
Example
A national prescribing restriction is introduced to reduce expenditure on a high-cost medicine. Monthly expenditure is observed for 36 months before the restriction and 24 months afterwards. Before implementation, expenditure increased by �120,000 per month. Segmented regression estimates an immediate reduction of �2,400,000 and a post-intervention trend change of ?�80,000 per month.
The estimated effect 12 months after implementation is:
???? = ?�2,400,000 + (?�80,000 ? 12)
???? = ?�3,360,000
Monthly expenditure after 12 months is therefore estimated to be �3,360,000 lower than the projected counterfactual expenditure without the restriction.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| IF | =IF(A2>=$H$2,1,0) | Create an indicator identifying observations after the intervention date |
| MAX | =MAX(0,B2-$H$3) | Calculate time elapsed since the intervention |
| LINEST | =LINEST(C2:C61,D2:G61,TRUE,TRUE) | Estimate segmented regression coefficients for level and trend changes |
| TREND | =TREND($C$2:$C$37,$B$2:$B$37,B38) | Project the pre-intervention trend as the counterfactual outcome |
| SUMXMY2 | =SUMXMY2(C38:C61,H38:H61) | Calculate squared deviations between observed and predicted outcomes |
| CORREL | =CORREL(I3:I61,I2:I60) | Assess first-order serial correlation in regression residuals |
VBA (Optional)
VBA can automate the creation of intervention variables, segmented regression estimation and counterfactual projections across multiple healthcare outcomes or intervention dates.
Sources
- Bernal JL, Cummins S, Gasparrini A. Interrupted Time Series Regression for the Evaluation of Public Health Interventions: A Tutorial. International Journal of Epidemiology. 2017;46(1):348?355.
- Wagner AK, Soumerai SB, Zhang F, Ross-Degnan D. Segmented Regression Analysis of Interrupted Time Series Studies in Medication Use Research. Journal of Clinical Pharmacy and Therapeutics. 2002;27(4):299?309.
- Linden A. Conducting Interrupted Time-Series Analysis for Single- and Multiple-Group Comparisons. The Stata Journal. 2015;15(2):480?500.
- NICE. Health Technology Evaluations: The Manual.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
Related Concepts (2)
Frequently Asked Questions (6)
What is interrupted time series?
A quasi-experimental design analysing observations before and after an intervention, testing for a significant change in level or trend at that point.
Source: Wagner et al. 2002
Why does an interrupted time series use the pre-intervention trend as a comparison?
An interrupted time series takes a run of measurements before an intervention and projects the trend they establish forward, then compares what actually happened after the intervention against that projected continuation. The pre-intervention trend serves as the counterfactual, showing where the outcome would have gone had nothing changed, so a departure from it, a jump in level or a change in slope, marks the intervention's effect. Using the series' own prior trend as the comparison is what lets it work without a separate control group. The past predicts the counterfactual. Bernal and colleagues (2017) describe this.
Source: Bernal et al. 2017
How is an interrupted time series analysed?
An interrupted time series is analysed by modelling the outcome over time, typically with segmented regression, which fits the trend before the intervention and tests for changes in the level, an immediate shift, and in the slope, a change in trend, after the intervention. The pre-intervention trend provides the counterfactual for what would have happened without the intervention. The analysis accounts for features such as seasonality and autocorrelation. So interrupted time series uses statistical modelling of the series to detect and quantify changes attributable to the intervention at its point of introduction.
Source: Wagner et al. 2002
Why is interrupted time series used?
Interrupted time series is used to evaluate the impact of interventions or policies applied at a known time to a population, when a randomised trial is impractical, by using the population's own prior trend as a comparison. It is common in public health and policy evaluation, such as assessing the effect of a regulation or programme. By controlling for the underlying trend, it distinguishes the intervention's effect from pre-existing patterns, providing stronger evidence than a simple before-and-after comparison. So it suits population-level interventions where repeated outcome data over time are available.
Source: Shadish, Cook & Campbell 2002
What are the strengths of interrupted time series?
The strengths of interrupted time series include its use of the pre-intervention trend as a counterfactual, controlling for underlying patterns and distinguishing the intervention's effect from the existing trajectory; its ability to detect both immediate level changes and changes in trend; its suitability for evaluating population-level interventions where randomisation is impossible; and its use of routinely collected repeated data. These strengths make it a robust quasi-experimental approach for policy and public health evaluation, providing stronger inference than a simple before-and-after comparison by accounting for the trend and testing for specific types of change.
Source: Wagner et al. 2002
What are the limitations of interrupted time series?
The limitations of interrupted time series include vulnerability to other events occurring at the same time as the intervention, which could account for observed changes, so attributing effects to the intervention alone requires ruling these out; the need for enough data points before and after to model the trend reliably; and the influence of seasonality and autocorrelation, which must be handled. Without a control series, confounding by concurrent events remains a concern. These limitations mean interrupted time series is strengthened by adding a control series and careful modelling, with causal conclusions interpreted cautiously.
Source: Wagner et al. 2002
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 20 Nov 2025
Content version: 1.0.0
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