Why Multiple Controls Require Careful Trend Comparisons
A comparative interrupted time-series (CITS) analysis compares an outcome trajectory for a treated unit with one or more untreated control trajectories before and after an intervention. With several controls, a researcher must decide how to combine their outcomes and how to represent the treated-control difference before the intervention. These choices matter because the treated and control series need not have the same baseline trend.
A constant difference between treated and control outcomes is not required for a CITS analysis. Instead, a segmented model can describe how their difference changes over time before the intervention, then estimate a departure from that projected baseline after the intervention. The resulting estimate is conditional on that trend specification: it compares the observed post-intervention contrast with the contrast predicted by extending the pre-intervention pattern.
Weights may be equal, prespecified, or chosen to improve fit to the treated unit during the pre-intervention period. Whatever the method, the weights should be determined without using post-intervention outcomes to make the treated-control contrast look favorable. A good pre-intervention match is useful evidence about comparability, but it does not establish that the post-intervention counterfactual trend would have continued in the same way.
A Segmented Model With Differential Baseline Trends
Index time so that \(k_t=0\) is the last pre-intervention period and \(k_t=1\) is the first post-intervention period. Define \(I_t=1\) for post-intervention periods and \(I_t=0\) otherwise, and let \(r_t=k_t-1\) in post-intervention periods. A common model for the contrast is:
Here, \(\alpha+\beta k_t\) is the projected pre-intervention contrast. The coefficient \(\beta\) is its baseline change per time period, so a nonzero \(\beta\) explicitly allows differential baseline trends: the treated series can be increasing or decreasing relative to the controls before the intervention. At \(k_t=1\), the treatment-associated level change relative to the projected baseline is \(\tau\). After that first post-intervention period, \(\gamma\) is the change per period in the contrast’s slope relative to its baseline slope. Thus, the post-intervention slope of the contrast is \(\beta+\gamma\).
This parameterization makes the level-change interpretation direct because \(r_t=0\) at the start of the post period. Some software instead uses an interaction with \(k_t I_t\); under that parameterization, the post-period level effect at \(k_t=1\) combines the intervention indicator and interaction coefficients. Always check the time coding before interpreting coefficients.
Worked Example: Combining Three Controls
Suppose a hypothetical region receives an intervention and three unaffected regions are candidate controls. For illustration, use fixed weights \(0.50\), \(0.30\), and \(0.20\) for controls A, B, and C. Their recorded outcomes and the treated outcome are shown below. The intervention begins at \(k=1\).
| Time \(k\) | Control A | Control B | Control C | Treated |
|---|---|---|---|---|
| -2 | 10 | 8 | 12 | 13.8 |
| -1 | 11 | 10 | 12 | 15.4 |
| 0 | 12 | 12 | 12 | 17.0 |
| 1 | 13 | 14 | 12 | 20.6 |
| 2 | 14 | 16 | 12 | 22.6 |
| 3 | 15 | 18 | 12 | 24.6 |
At \(k=-2\), the weighted control outcome is \(0.50(10)+0.30(8)+0.20(12)=9.8\), so the contrast is \(13.8-9.8=4.0\). Repeating this calculation gives weighted control outcomes \(10.9,12.0,13.1,14.2,15.3\), and contrasts \(4.0,4.5,5.0,7.5,8.4,9.3\), in time order.
The three pre-intervention contrasts increase by \(0.5\) per period, so their fitted line is \(D_t=5.0+0.5k_t\). At the first post period, this line predicts \(5.0+0.5(1)=5.5\). The observed contrast is \(20.6-13.1=7.5\), making the estimated first-period level change \(7.5-5.5=2.0\).
From \(k=1\) to \(k=2\), the observed contrast increases by \(8.4-7.5=0.9\); its projected baseline would increase by \(0.5\). The slope change is therefore \(0.9-0.5=0.4\) per period. The fitted contrast model is \(D_t=5.0+0.5k_t+2.0I_t+0.4r_tI_t\). For example, at \(k=3\) it predicts \(5.0+0.5(3)+2.0+0.4(2)=9.3\), matching the observed contrast. This calculation is internally consistent with the selected weights and the listed data.
The interpretation is conditional: the contrast is estimated to be 2.0 outcome units above its projected baseline in the first post period, with a further 0.4-unit increase in that departure per period. A causal interpretation requires the assumptions discussed below, not just this arithmetic.
How Weighting Changes the Comparison
Different weights can produce different control trajectories, and therefore different contrasts and projected baselines. Even if the first-period estimate happens to be similar across specifications, the full post-period pattern may differ. Weight choices should be justified by the design and pre-intervention information, and results should be examined for sensitivity to defensible alternatives.
Worked Example: Comparing Equal and Unequal Weights
Consider hypothetical outcomes for two controls, A and B, and a treated unit. The time points are \(k=-2,-1,0,1\); \(k=0\) is the last pre-intervention period. There are only two controls, so this example compares two possible ways of combining them.
| Time \(k\) | Control A | Control B | Treated |
|---|---|---|---|
| -2 | 10 | 10 | 15 |
| -1 | 11 | 12 | 17 |
| 0 | 12 | 14 | 19 |
| 1 | 13 | 16 | 22 |
With equal weights, \(C_t=0.50Y_{At}+0.50Y_{Bt}\), the control outcomes are \(10,11.5,13,14.5\). Subtracting them from the treated outcomes gives contrasts \(5,5.5,6,7.5\). The pre-intervention line is \(D_t=6+0.5k_t\), so its prediction at \(k=1\) is \(6.5\). The estimated first-post-period departure is \(7.5-6.5=1.0\) outcome unit.
Now use weights \(0.75\) for A and \(0.25\) for B. The weighted control outcomes are \(10,11.25,12.5,13.75\), so the contrasts are \(5,5.75,6.5,8.25\). The pre-intervention line is \(D_t=6.5+0.75k_t\), which predicts \(7.25\) at \(k=1\). The estimated departure is again \(8.25-7.25=1.0\) unit.
The matching first-period departures do not mean the specifications are equivalent. The pre-intervention differential trend is \(0.5\) per period with equal weights and \(0.75\) with the unequal weights. With additional post periods, different weights could also imply different projections and different slope-change estimates. Here, one post period is not enough to estimate \(\gamma\), because there is no post-intervention trajectory from which to identify a post-period slope.
A weighted average is only one approach. A regression may instead include separate control-series terms or allow control-specific trend patterns before constructing a comparison. Such models can be useful when a single weighted outcome hides important differences among controls, but their interpretation depends on how the control series enter the model. In either approach, the central question is whether the comparison represents a plausible untreated trajectory for the treated unit.
Why the Baseline Trend Must Be Calculated Correctly
The first-post-period level change is the observed post-period contrast minus the value predicted by the pre-intervention line at that same time. A useful way to check it is to calculate both quantities explicitly. The slope change is likewise the observed post-period change in the contrast minus the baseline slope; it is not simply the post-period slope.
Worked Example: A Differential Trend and a First-Period Change
Suppose a hypothetical analysis gives pre-intervention contrasts of \(14.2\), \(15.4\), and \(16.6\) at \(k=-2,-1,0\). Each adjacent difference is \(1.2\), so the baseline line is \(D_t=16.6+1.2k_t\). At the intervention start, \(k=1\), its prediction is \(16.6+1.2(1)=17.8\).
Suppose the observed contrast at \(k=1\) is \(22.4\). The first-period level change is \(22.4-17.8=4.6\), not the difference between the observed contrast and the pre-period value at \(k=0\). If the contrast at \(k=2\) is \(24.2\), the post-period change is \(24.2-22.4=1.8\). Subtracting the baseline slope gives a slope change of \(1.8-1.2=0.6\) per period.
The resulting model is \(D_t=16.6+1.2k_t+4.6I_t+0.6r_tI_t\). At \(k=1\), it predicts \(16.6+1.2+4.6=22.4\). At \(k=2\), it predicts \(16.6+2.4+4.6+0.6=24.2\). These checks show that both the level and slope calculations agree with the stated observations.
The pre-intervention trend is already increasing by 1.2 units per period. The estimated 0.6-unit slope change therefore means that the post-intervention contrast increases by 1.8 units per period, not 0.6. Keeping the baseline slope and its change distinct is essential when describing the trajectory.
Assumptions and Limits of Interpretation
The model describes contrasts; causal interpretation requires a credible counterfactual argument. In particular, the weighted controls must remain untreated by the intervention and must not be affected by spillovers that alter their outcomes. The treated and control measurements must be comparable over time, and the control weights and model specification must be chosen without tailoring them to post-intervention outcomes.
Most importantly, absent the intervention, the treated-control contrast must have continued in a way adequately represented by the extrapolated pre-intervention trend. This assumption permits differential baseline trends, but it does not permit an unmodeled change in that difference exactly when the intervention begins. A simultaneous policy, a measurement change, or a shock affecting the treated unit differently could produce the same observed departure. Several pre-intervention observations help assess the plausibility of a trend specification, but cannot rule out such competing explanations.
A linear trend is a modeling choice, not a default truth. If the pre-intervention contrast bends over time, a straight line may give a poor counterfactual projection. Alternative trend specifications may be informative, but selecting among them after inspecting post-intervention effects can compromise the comparison. Report the specification and weights, explain why they are defensible, and show whether the substantive conclusion depends on reasonable alternatives.
Common Mistakes
- Assuming the treated and controls must have parallel pre-trends. A CITS model can allow a linear differential trend through \(\beta\). The requirement is instead that the chosen pre-intervention trend provide a credible projection in the absence of intervention.
- Calling the post-intervention slope change the post-intervention slope. The slope change is \(\gamma\); the post-period slope of the contrast is \(\beta+\gamma\).
- Calculating the level change from the last pre-period contrast. Calculate the projected baseline at \(k=1\), then subtract it from the observed contrast at \(k=1\).
- Treating a good pre-period fit as proof of causality. Pre-period fit supports the comparison but cannot exclude a concurrent shock or a change in the untreated differential trend.
- Choosing weights to maximize the desired post-period result. Weights should be justified using design and pre-intervention information; post-outcome-driven choices make the comparison difficult to interpret.
Check Your Understanding
Answer each question using the definitions and model above.
- Why can a CITS model allow the treated-control contrast to have a nonzero pre-intervention slope?
- For \(D_t=\alpha+\beta k_t+\tau I_t+\gamma r_tI_t\), what does \(\tau\) represent when the intervention begins at \(k=1\)?
- If the baseline slope is \(0.8\) units per period and the slope change is \(-0.3\), what is the post-intervention slope of the contrast?
- Why might two defensible sets of control weights produce different CITS estimates?
- What assumption is needed to interpret a post-intervention departure from the projected contrast as an intervention effect?