interrupted time series

Many public health measures cannot be tested by randomly assigning some people to receive them and others not to. A city cannot be half vaccinated at random, and a law applies to everyone at once. Interrupted time series is one of the designs used instead. A run of measurements of the outcome, month by month or year by year, is used to establish an underlying trend, and that trend is interrupted by the intervention at a known point in time.

The comparison is not before against after in the crude sense. It is the data actually observed after the intervention against the counterfactual: what the earlier trend would have produced had nothing changed. Deviation from that expected path is the estimated effect. A good analysis adjusts for things that move the numbers on their own, seasonality above all, since diseases and injuries have their own annual rhythm.

The design's strength is that each place acts as its own comparison, which removes differences between populations. Its weakness is the assumption it rests on: that the pre-existing trend would have continued unchanged. Anything else that shifted at the same moment, including changes in how cases are reported or how people seek care, can be mistaken for the effect of the intervention, so honest papers state that possibility explicitly.