Interrupted Time Series Analysis: Segmented Regression, Autocorrelation, and Lag

Interrupted time series (ITS) analysis evaluates whether an intervention changed an outcome by comparing the trend before the intervention to the trend after it — using segmented regression to separate an immediate jump (level change) from a change in the slope of the trend (slope change). Because the data are ordered in time, the errors are almost never independent of each other the way plain OLS assumes, which is the single most consequential technical detail in fitting an ITS model correctly: get the autocorrelation and lag structure wrong, and a real effect can look spuriously significant, or a real null result can look like a finding.

When Interrupted Time Series Is the Right Design

ITS is a quasi-experimental design for evaluating a policy, program, or clinical intervention when random assignment to a control group is not feasible — a hospital rolling out a new hand-hygiene protocol, a formulary change restricting a medication class, a state law taking effect on a fixed date. It requires repeated observations of the same outcome, in the same population, at regular intervals, spanning a clearly-defined intervention point. What makes it credible as a causal design at all is that the pre-intervention trend acts as the counterfactual: instead of comparing intervention to no-intervention, ITS compares the observed post-intervention trajectory to the trajectory the pre-intervention data would have predicted if nothing had changed.

See experimental vs. quasi-experimental design for where ITS sits relative to a randomized trial, and difference-in-differences for the closest sibling design when a comparable untreated group’s data are also available.

The Segmented Regression Model

The standard specification, widely used since Wagner et al.’s methods paper popularized it for health-services and pharmacy research, is:

Y_t = β₀ + β₁·T + β₂·X_t + β₃·(T−T₀)X_t + e_t

• T — time elapsed since the start of the observation period (1, 2, 3…), continuous across the whole series.
• X_t — the intervention indicator: 0 for every time point before the intervention, 1 from the intervention point onward.
• (T−T₀)X_t — time since the intervention, where T₀ is the time point of the intervention: 0 before the intervention, then counting up (0, 1, 2…) after it.

Each coefficient answers a distinct substantive question:

• β₀ — the baseline level of the outcome at the start of the series.
• β₁ — the pre-existing trend: how much the outcome was already changing per time unit, absent the intervention. This matters because a naive before/after comparison of two means silently assumes β₁ is zero — if the outcome was already trending in the intervention’s favored direction, ignoring β₁ overstates the effect.
• β₂ — the level change: the immediate, one-time jump (or drop) in the outcome right at the intervention point, holding the trend fixed. This is the coefficient people usually mean by “did the intervention have an effect,” and it is estimated as the vertical gap between where the pre-intervention trend line would have landed at T₀ and where the post-intervention trend line actually starts.
• β₃ — the slope change: how much the trend’s steepness itself changed after the intervention. This captures effects that build gradually rather than arriving all at once — a training program that takes several months to change practice, for instance, may show little level change but a substantial slope change.

Level and slope change are not interchangeable evidence for “the intervention worked.” A policy can produce a real, sustained effect through slope change alone with β₂ near zero, and a level change with no slope change describes a one-time shift that then resumes the old trend — a genuinely different clinical or policy story. Report both coefficients, their confidence intervals, and the counterfactual comparison (what the model predicts the series would have done without the intervention, projected from β₀ and β₁ alone) rather than either coefficient in isolation.

Worked Example (Synthetic Data, for Illustration Only)

The numbers in this section are a constructed, illustrative dataset built specifically for this guide — not measurements from any real study, institution, or publication. Every value below was generated by a fixed, reproducible formula and the regression was actually fit to it; nothing here is an invented “looks-right” statistic.

Suppose a 24-month hospital-acquired infection rate (per 1,000 patient-days), 12 months before and 12 months after an infection-control bundle rolls out in month 13. Fitting the segmented regression model above by ordinary least squares to this synthetic series returns:

• β₀ (baseline level) = 8.24
• β₁ (pre-existing monthly slope) = −0.089
• β₂ (level change at intervention) = −1.30
• β₃ (post-intervention slope change) = −0.053
• R² = 0.999

Read this as: the infection rate was already declining slightly before the bundle (β₁), the bundle produced an immediate drop of about 1.3 infections per 1,000 patient-days at rollout (β₂), and the downward trend got modestly steeper afterward (β₃). But before trusting any of those estimates’ significance, the residuals from this same fit need to be checked for autocorrelation — which is exactly the problem the next two sections cover, using this same synthetic series: the residuals came back with a Durbin-Watson statistic of 0.69 and a lag-1 residual autocorrelation of 0.61, both clearly signaling the standard OLS standard errors on β₂ and β₃ above cannot be trusted as computed.

The Autocorrelation Problem Specific to Time-Series Data

OLS standard errors are derived assuming the regression errors are independent of each other. Time-ordered data routinely violates that assumption: whatever made this month’s outcome higher or lower than the model predicts (a staffing shortage, a seasonal surge, measurement drift) tends to still be present next month, so consecutive residuals correlate with each other — almost always positively in real surveillance and utilization data. As with any regression, autocorrelation does not bias the coefficient estimates themselves (β₂ and β₃ stay, on average, centered on their true values), but it invalidates the standard-error formula OLS uses to build confidence intervals and p-values. Positive autocorrelation, the far more common case, makes those standard errors too small — which inflates significance and can turn a level or slope change that is really noise into something that looks statistically significant. See the Durbin-Watson test guide for the full mechanics of reading that statistic and its inconclusive zone; the same 0–4 scale and d ≈ 2(1−ρ̂) relationship applies directly to ITS residuals.

Detecting it in an ITS model specifically means examining the residuals in time order after fitting the segmented regression, not the raw outcome series — the outcome series is expected to trend and jump by design; it’s the leftover, unexplained variation that should look like noise if the model is well specified. A Durbin-Watson statistic well below 2 (as in the worked example above), a plotted autocorrelation function (ACF) with a slowly decaying pattern, or a significant Breusch-Godfrey/Cumby-Huizinga test (which, unlike Durbin-Watson, remains valid when a lagged outcome is among the regressors) are the standard checks.

Correcting for Autocorrelation

Three approaches are in common use, and they trade off differently between simplicity and efficiency:

• Newey-West (HAC) standard errors. Keeps the OLS point estimates for β₀–β₃ exactly as fit, but replaces the standard-error formula with one that stays valid under autocorrelation (and heteroskedasticity) up to a chosen maximum lag. This is usually the first thing to try: it requires no assumption about the specific form the autocorrelation takes, and it’s a direct drop-in replacement for the standard-error step, not a re-fit of the model.
• Cochrane-Orcutt / Prais-Winsten estimation. Explicitly models the error term as a first-order autoregressive process, e_t = ρe_t−1 + u_t, estimates ρ, then transforms the data to remove that structure before re-fitting. Prais-Winsten retains the first observation (Cochrane-Orcutt drops it), which matters more in a short ITS series where every pre-intervention point is scarce.
• ARIMA/ARMA error models (Box-Jenkins). When the autocorrelation structure is more complex than a single AR(1) term — genuine seasonal cycles, moving-average components, or higher-order dependence — the segmented-regression covariates are fit jointly with an ARIMA error model identified via the ACF/PACF plots. See ARIMA models for research time series for the full AR/I/MA identification process this approach depends on.

Whichever correction is used, report the corrected standard errors and confidence intervals for β₂ and β₃ specifically — those are the coefficients the intervention claim rests on, and they are exactly the ones the uncorrected OLS standard errors are most likely to have gotten wrong.

Choosing an Appropriate Lag Structure

“Lag structure” in an ITS model covers two related but distinct decisions:

• The order of the autoregressive error term. A first-order AR(1) error (this month’s error correlates with last month’s) is the default starting assumption and is often sufficient for monthly administrative or clinical data. Higher-order structure shows up as a slowly-decaying ACF and a PACF that only cuts off after more than one lag — when that pattern appears, fit an AR(p) or full ARIMA error model rather than forcing an AR(1) correction onto data that needs more lags. Information criteria (AIC/BIC) across a small set of candidate lag orders are the standard way to choose p without overfitting; adding lags indefinitely will always improve in-sample fit while degrading out-of-sample validity and burning through the already-limited degrees of freedom a short ITS series has.
• A lag between the intervention and its effect. Some interventions take effect immediately at T₀; others have a genuine implementation delay (a policy announced in month 12 but not enforced until month 14, or a clinical protocol that takes several weeks to be adopted in practice). Where a real delay is documented, some analysts either exclude an explicit “washout” window around the intervention point from the model, or shift T₀ to the date enforcement actually began rather than the announcement date — whichever choice is used, it should be justified by the intervention’s actual implementation timeline, not chosen after seeing which specification produces a larger effect.

Seasonality is a separate but related concern in monthly or weekly ITS data: an outcome that cycles annually (respiratory illness rates, academic-calendar-driven utilization) will show autocorrelation at the seasonal lag (12 for monthly data) even after the intervention-related trend is correctly modeled. The standard fix is adding harmonic terms (paired sine/cosine terms at the seasonal frequency) or monthly indicator variables to the regression itself, rather than trying to make the AR error term absorb seasonal structure it wasn’t designed to capture.

How Many Data Points the Design Needs

Segmented regression needs enough points on each side of the intervention to estimate a trend, not just a mean — a handful of points before and after can technically produce coefficients, but with almost no power to distinguish a real slope change from noise, and with residual-autocorrelation diagnostics that are themselves unreliable in a very short series. Methodological guidance in this space generally treats somewhere around eight or more observations on each side of the intervention as the practical minimum for a credible level-and-slope estimate, with published EPOC-style minimum thresholds for including a study as a valid ITS design in an evidence synthesis sometimes set lower than that recommended-for-power figure — treat “the model ran” as a much weaker bar than “the model has enough pre-intervention points to plausibly detect the effect size in question.”

Controlled ITS: Ruling Out Concurrent Events

A single-series ITS model cannot distinguish the intervention’s effect from any other event that happened to coincide with T₀ — a concurrent policy change, a seasonal event, a broader secular trend affecting the whole population. A controlled interrupted time series (CITS) design adds a comparison series that experienced the same time period but not the intervention, and tests whether the level and slope changes in the intervention series differ from the (ideally null) changes in the comparison series at the same time point — the same logic difference-in-differences uses, adapted to a segmented-regression framework. Where a credible comparison series is available, CITS is a substantially stronger design than single-series ITS for supporting a causal claim; see causal analysis for how ITS/CITS fit into the broader causal-inference toolkit, and control group for what makes a comparison series credible in the first place.

Common Pitfalls

• Treating the level-change coefficient as the whole answer and ignoring the slope change, or vice versa — report and interpret both.
• Skipping the autocorrelation check entirely because the model “ran fine” — OLS will produce coefficients and p-values regardless of whether its assumptions hold; a clean-looking regression output is not evidence the standard errors are trustworthy.
• Choosing the intervention date to maximize the apparent effect rather than fixing it to the intervention’s actual real-world implementation date before looking at the data.
• Ignoring seasonality in monthly or weekly data and attributing seasonal-lag autocorrelation entirely to the intervention’s timing.
• Running ITS with too few pre-intervention points to estimate a trend at all, then reporting a slope-change estimate with false confidence.
• Overfitting the AR error order by adding lags until the Durbin-Watson statistic looks acceptable, rather than identifying the order from the ACF/PACF pattern and an information criterion.

Frequently Asked Questions

What is interrupted time series analysis used for?

Evaluating whether a policy, program, or clinical intervention changed an outcome, in settings where a randomized controlled comparison isn’t feasible — common in health-policy, public-health, and quality-improvement research where an intervention rolls out to an entire population or facility at once.

What is the difference between level change and slope change in ITS?

Level change (β₂) is an immediate, one-time jump in the outcome right at the intervention point. Slope change (β₃) is a change in the trend’s steepness afterward, capturing an effect that builds gradually rather than arriving all at once. A real intervention effect can show up as either, both, or neither in isolation.

Why does autocorrelation matter for interrupted time series specifically?

Because the data are ordered in time, residuals from the fitted model routinely correlate with their own recent past, which violates the independence assumption behind the standard OLS standard-error formula. It does not bias the level- or slope-change point estimates, but it typically makes the standard errors too small, inflating the apparent statistical significance of the intervention effect.

How do you correct for autocorrelation in a segmented regression model?

Common approaches are Newey-West (HAC) standard errors, which correct the standard-error formula without re-fitting the model; Cochrane-Orcutt or Prais-Winsten estimation, which explicitly models a first-order autoregressive error and transforms the data; and full ARIMA error models when the autocorrelation structure is more complex than a single AR(1) term, such as genuine seasonality.

How do you choose the lag order for the autoregressive error term?

Examine the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the residuals to identify the likely order, then compare candidate lag orders using an information criterion such as AIC or BIC rather than adding lags until a diagnostic statistic looks acceptable — over-fitting the lag order burns through the degrees of freedom a short time series already has very little of.

What is a controlled interrupted time series design?

An extension of single-series ITS that adds a comparison series which experienced the same time period without the intervention, letting the analysis test whether the intervention series’ level and slope changes differ from the comparison series’ changes at the same point in time — which helps rule out a concurrent event or a broader secular trend as the real explanation.