Estimands and Design Choices in Comparative Interrupted Time Series

Why Specify the Estimand First?

A comparative interrupted time series (CITS) compares the trajectory of an aggregate unit exposed to an intervention with the trajectory of one or more unexposed control units. Before deciding how to estimate an effect, the analyst must say what change is of interest. An immediate shift at the intervention date is not the same target as a gradual change in trend, and neither alone describes the cumulative difference over a long follow-up period.

Design choices define that target. The intervention date determines which observations count as pre- and post-intervention; the follow-up horizon determines when the effect is evaluated; and the definition of the treated and control aggregates determines which population and outcome scale the result describes. These choices should follow the substantive question rather than be selected after viewing which specification gives the most favorable result.

Control-Relative Trajectories and Effects

Let \(Y_{Tt}\) be the outcome for the treated aggregate unit at time \(t\), and let \(Y_{C t}\) be the outcome for the chosen control aggregate at that time. The control aggregate may be a single unit or a fixed weighted combination of control units. Define the treated-control gap as \(D_t=Y_{Tt}-Y_{C t}\). A CITS estimand concerns how this gap departs after intervention from the trajectory it would have followed without the intervention.

For a linear pre-intervention gap trajectory, write its continuation as \(a+bt\), where \(a\) is the intercept under the chosen time coding and \(b\) is the pre-intervention change in the gap per time unit. The post-intervention departure from that continuation is the target effect at time \(t\). This definition makes the comparison explicit: it is not simply the treated unit's post-intervention change, but its change relative to the control and to the pre-intervention gap trajectory.

This is a definition of the target, not an estimation recipe. It uses a linear description of the pre-intervention gap and a linear post-intervention departure to make the level and slope components precise. Whether those features can be estimated credibly, or interpreted causally, depends on design and identification assumptions. The estimand itself does not establish those assumptions.

The level component is defined at \(k=0\). If intervention effects can only begin after a delay, the onset convention must reflect that: \(k=0\) should then refer to the first time at which an effect is considered possible, or the initial transition period should be represented separately. The slope component is measured in outcome units per time unit. For example, if the outcome is a rate per 100,000 people and time is measured in months, \(\delta_1\) has units of rate points per 100,000 people per month.

Worked Examples: Reading the Components

Choosing Timing and Follow-Up

The intervention date \(T_0\) should correspond to the time the intervention could plausibly affect the outcome, not merely the date it was announced or formally recorded. If announcement, implementation, and full exposure occur at different times, specify which event defines the target. An announcement may produce anticipation before implementation, while a phased rollout may make a single sharp onset an imperfect description.

The time unit and the follow-up horizon are part of the estimand. If observations are monthly, a horizon of six means six months under the selected indexing convention. Be explicit about whether the intervention month is \(k=0\), the first complete post-intervention month, or a transition period. A horizon should also be substantively justified: an immediate policy response and a response after one year answer different questions.

When there is a transition period, there are several defensible choices, but they target different quantities. One may set the onset at the first exposure and define a separate transition component, exclude transition observations from a particular summary, or define onset at full implementation. The choice should be reported and justified. Silently labeling partially exposed periods as untreated or fully treated changes the meaning of the estimated level and slope components.

Defining the Aggregate Units and Outcome

CITS outcomes are often measured for regions, hospitals, schools, or other aggregate units. Define the treated unit as the population or system exposed to the intervention and the control as the comparison population or system. A control aggregate formed from multiple units should be described, including whether its weights are fixed over time. If the weights change, the composition of the comparison may change as well, so a change in the gap can reflect composition as well as outcome movement.

The outcome scale matters just as much. A count, a rate per population, a percentage, and a logarithm do not have the same interpretation. For rates, state the denominator and unit. For percentages, distinguish percentage points from percent change. If the outcome is a total count, changes in population size may affect the total even when an individual-level rate is stable. The chosen scale should match the substantive question and be used consistently for treated and control series.

From Components to a Horizon-Specific Effect

The level and slope components combine because the post-intervention departure is defined at each follow-up time. At \(k=0\), the slope term is zero, so \(\tau_0=\delta_0\). At \(k=h\), the slope departure has accumulated over \(h\) time units. This yields the following direct relationship:

The expression also shows why a report of only \(\delta_0\) can be incomplete. If \(\delta_1\ne 0\), the target changes over follow-up. Conversely, reporting only a slope change hides the immediate departure. A complete description gives both components and at least one substantively chosen horizon-specific effect. For a nonlinear target trajectory, the same principle applies—define the effect at the relevant times—but the linear expression above no longer suffices.

Design Choices Are Not Identification Assumptions

An estimand says what quantity is wanted; a design and its assumptions determine whether observed data can identify it. Defining an intervention date, choosing controls, and setting a follow-up horizon do not alone guarantee a causal interpretation. In particular, the relative trajectory must support the intended counterfactual comparison. Detailed identification conditions and estimation procedures are separate questions.

A careful specification should state the treated aggregate, control aggregate and weighting, outcome and units, intervention onset convention, pre-intervention period, transition handling, follow-up horizon, and whether the target is an immediate level change, a slope change, or a horizon-specific effect. This makes later modeling choices interpretable and makes it possible to see when two analyses answer different questions despite using similar data.

Common Mistakes

• Calling the treated unit's raw change the CITS effect. A treated outcome can rise while its control-relative gap falls, or vice versa. Define the contrast against the control trajectory and the pre-intervention gap trajectory.
• Interpreting the slope component as the effect at every time. The slope component is a change per time unit. The effect at horizon \(h\) combines it with the level component as \(\delta_0+\delta_1h\).
• Leaving the time origin implicit. If one analysis counts the intervention month as \(k=0\) and another counts the first complete post month as \(k=0\), their level and horizon labels may not refer to the same point. State the indexing convention.
• Changing the control composition without noticing. A time-varying weighted control may change because its members or weights change, not only because the underlying outcomes changed. Define and report the aggregation.
• Treating the estimand as proof of causality. A precisely defined target does not establish that the chosen controls represent the relevant counterfactual trajectory. Keep the target definition distinct from the assumptions needed for causal interpretation.