Interrupted series: distinguish immediate change from a new trend
A series can keep rising after an intervention while remaining below the path it would otherwise have followed. An immediate drop and a slower subsequent rise are different parameters. A simple before-and-after average can obscure both, especially when the underlying trend was already increasing.
The deliverable is a segmented-mean worksheet with an explicit time origin: the immediate level change, the slope change and the model-implied gap at each requested post-intervention period. Keep the residual dependence model and the causal interpretation separate from that mean accounting.
Fix time zero and the outcome definition
Assume a hypothetical research workflow reports monthly average completion time in minutes using a consistent eligible-document population and timing rule. A new review procedure starts in the first fully exposed month. Let k=0 in that month, negative k before it and positive k afterward. Define P=1 when k≥0 and P=0 before.
The intervention date comes from implementation records, not a search for the strongest-looking break in the outcome. Record partial rollout, anticipation and plausible lag. If the first calendar month was only partly exposed, it needs a prespecified treatment rather than an automatic full-post label.
Attach a calendar-to-index map showing the last pre-month, first fully exposed month and six subsequent periods. Check whether the implementation starts counting at zero or one and whether missing calendar months have accidentally been compressed into adjacent observations, changing what a slope “per month” means.
Lopez Bernal, Cummins and Gasparrini’s original ITS tutorial, Steps 2 and 4–6, discusses impact-model choice, segmented regression and time-series diagnostics. The original article is the 2017 journal version, DOI 10.1093/ije/dyw098 (PMCID PMC5407170). Read it together with the authors’ corrigendum, which corrects the time-interaction parameterization needed for an immediate-level interpretation. The correction was published online in September 2020, DOI 10.1093/ije/dyaa118 (PMCID PMC7750921), and appeared in the June 2021 issue. The numbers below are independently constructed.
Write the centered mean model
For a continuous outcome, suppose the model is:
Y[k] = 100 + 2k − 5P − 1(kP) + error[k].
The no-intervention continuation is 100+2k. The coefficient −5 is the immediate jump at k=0 because kP is zero at that point. The interaction coefficient −1 changes the slope from +2 minutes per month to +1, rather than making the entire post-period slope −1.
The intercept 100 is the pre-intervention line’s continuation at k=0, not the previous month’s observed value. The errors have conditional mean zero under the model; their variance and serial dependence are additional specifications, not encoded by the four mean coefficients.
| Post period k | Original-trend continuation | Post-model mean | Gap versus original trend |
|---|---|---|---|
| 0 | 100 | 95 | −5 |
| 1 | 102 | 96 | −6 |
| 6 | 112 | 101 | −11 |
| 12 | 124 | 107 | −17 |
At any post k, the gap is −5−k: level change plus slope change times elapsed post time. The −11 at month 6 is not an immediate effect and is not the sum of all earlier monthly gaps. It is the difference at that particular month.
Check why adjacent periods and averages differ
The final pre-period k −1 has expected value 98. The first post-period has expected value 95, so the adjacent-period difference is −3. The immediate jump is nevertheless −5 relative to the original line’s value 100 at k=0. Comparing 95 to 98 skips the +2 increase expected between adjacent months.
For a separate descriptive check, the six pre-period means at k −6 through −1 average 93. The first six post means at k=0 through 5 average 97.5. A naive before/after difference is +4.5 even though every post-model mean lies below its original-trend continuation.
The corresponding original-trend average over k=0 through 5 is 105, giving an average modeled gap −7.5. Immediate change −5, month 6 gap −11, average first-six gap −7.5 and raw before/after +4.5 are four distinct quantities. Label the window and comparison line rather than calling all four “the effect.”
Reparameterization changes coefficient meaning
Suppose an analyst instead uses an absolute time index T with intervention T0=20, so k=T−20. The same fitted mean can be written:
Y[T] = 60 + 2T + 15P − 1(TP) + error[T].
The uncentered P coefficient is +15. It is not the immediate jump. At T=20, the actual discontinuity is 15−20=−5. The centered and uncentered equations can describe the same lines; the mistake is giving their P coefficients the same interpretation.
The corrigendum uses the centered interaction (T−T0)P so the P coefficient corresponds to the level change at intervention. Preserve the exact variables used by the software. A column named “post trend” may start at zero or one, and that choice changes what the intercept difference represents.
The time unit matters too. A slope is per month in this example. Changing to weeks requires consistent time conversion and outcome aggregation. A permanent level step, a temporary pulse and a delayed slope change are different mean models; choose the expected impact shape before inspecting which version gives the smallest p value.
Diagnose residuals after specifying the mean
Plot the observed series, fitted segments and residuals in time order. An obvious seasonal pattern can be a missing mean component rather than unexplained serial noise. Assess residual dependence after reasonable trend and seasonal terms are considered, not only correlation in the raw rising series.
The original tutorial’s Step 5 discusses seasonality and residual autocorrelation, including autoregressive approaches when dependence remains. A mean model can have the correct level/slope interpretation while ordinary independent-error standard errors remain inappropriate. Conversely, an elaborate residual model does not fix a wrongly specified mean break or a changed outcome definition.
Retain the error method, diagnostic plots and uncertainty procedure. There is no universal minimum number of months that makes all interrupted series reliable. A long history spanning a different operating regime can distort the relevant pre-trend, while a short post-period may provide little information about a new slope.
Missing months are not zero outcomes. Changing monthly document mix, denominator, inclusion criteria or clock software can alter the mean independently of the review procedure. Save case counts and measurement versions alongside the series so these changes can be investigated.
Keep statistical interruption and causal attribution separate
The original-trend continuation is a modeled counterfactual. A causal interpretation additionally needs a defensible claim about what would have happened without the procedure. Another intervention at the same date, an industry-wide shock or anticipatory behavior can challenge that claim even if the fitted break is precise.
For an independent boundary, suppose a reporting-system update reduces every recorded time by five minutes while the actual process is unchanged. That creates a level shift with the same statistical form as −5P. One series alone cannot distinguish measurement redefinition from real operational improvement. Source audits or suitable unaffected comparisons supply different evidence.
A controlled series, staggered introduction or other design extension may help address concurrent changes, but its own comparability assumptions need checking. Do not treat extra controls as automatic identification or convert a workflow-time result into a security-price forecast.
| Review step | Record to preserve | Response to a problem |
|---|---|---|
| Define time | implementation date, exposure and k=0 | separate rollout and lag assumptions |
| Fix measurement | outcome rule, units and monthly case counts | audit definition and composition shifts |
| Specify mean | baseline trend, level, slope and centering | reproduce values at several k |
| Check errors | residual seasonality and dependence | use appropriate uncertainty and model revision |
| Assess attribution | concurrent changes and comparison evidence | retain a descriptive interpretation if needed |
| Report | immediate, period-specific and window-average gaps | attach each target to its reference path |
Update the worksheet when implementation timing, missing data, measurement versions or the plausible continuation trend changes. Preserve the original series and specification. This lets reviewers see whether a revised result reflects new evidence about the intervention, a corrected time origin or merely a different mean/error model.
Preserve the model scale and the reporting unit
The worked example uses an identity-scale mean in minutes. A log-link model would attach different meanings to its coefficients: a post indicator coefficient is a log ratio of model means at the centered interruption, rather than a fixed subtraction of minutes. Copying the −5 coefficient into a logarithmic model would therefore define a completely different change.
For a separate scale illustration, suppose a positive mean starts at 100 and a centered log-mean model gives the immediate multiplier 0.9. Its first post mean is 90 and its relative change is −10%. At a baseline of 200, the same multiplier produces a twenty-unit difference. A constant proportional break is not a constant absolute break.
Before reviewing an exported coefficient table, record the response transformation, link, time unit and any exposure denominator. Regenerate fitted values on the reporting scale at the interruption and several later periods. The four-column worksheet remains useful, but its values must be computed from the actual model rather than assembled by adding coefficients that live on different scales.
FAQ
Is the post-intervention slope −1 in the example?
No. The slope change is −1, so the slope becomes +2−1=+1 minute per month.
Does a precise break prove the procedure caused it?
No. Concurrent changes and measurement shifts can produce the same statistical shape. Causal attribution requires additional design evidence.