In the comparative interrupted time series design (also called the method of difference-in-differences), the change in outcome in a group exposed to treatment in the periods before and after the exposure is compared with the change in outcome in a control group not exposed to treatment in either period. The standard difference-in-difference estimator for a comparative interrupted time series design will be biased for estimating the causal effect of the treatment if there is an interaction between history in the after period and the groups; for example, there is a historical event besides the start of the treatment in the after period that benefits the treated group more than the control group. We present a bracketing method for bounding the effect of an interaction between history and the groups that arises from a time-invariant unmeasured confounder having a different effect in the after period than the before period. The method is applied to a study of the effect of the repeal of Missouri’s permit-to-purchase handgun law on its firearm homicide rate. We estimate that the effect of the permit-to-purchase repeal on Missouri’s firearm homicide rate is bracketed between 0.9 and 1.3 homicides per 100,000 people, corresponding to a percentage increase of 17% to 27% (95% confidence interval: 0.6, 1.7 or 11%, 35%). A placebo study provides additional support for the hypothesis that the repeal has a causal effect of increasing the rate of state-wide firearm homicides.
From the aDepartment of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA
bDepartment of Health Policy and Management, Johns Hopkins Bloomberg School of Public Health, Johns Hopkins University, Baltimore, MD.
Submitted May 14, 2018; accepted January 29, 2019.
The authors report no conflicts of interest.
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Data and code availability: The data is provided in Table 1. The code that produced the results can be found in the electronic supplementary materials.
Correspondence: Dylan S. Small, Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104. E-mail: email@example.com.