Regression Not-to-the-Mean: An Oddity of Regression, Illustrated With the Risk of Overdose Deaths

Recent works in econometrics have shown that there can be issues with applying a constant treatment effect model in longitudinal settings with staggered treatment and heterogeneous treatment effects. We focus on the issue that the estimated constant treatment effect may be a weighted average, with some negative weights, of treatment effects that are heterogeneous across treatment durations. When this issue arises, the estimated constant treatment effect and estimated heterogeneous treatment effects may result in conflicting results. Through the example of estimating the effect of drug-induced homicide (DIH) prosecutions reported by media on unintentional drug-overdose deaths in the United States, we illustrate how the negative weighting issue can lead to conflicting results in practice. Moreover, although research has shown that the negative weight issue may arise in linear regression models, we show this issue may also arise in logistic regression models. Using a linear link, we estimated a constant treatment effect risk ratio of 0.977 (95% CI: [0.866, 1.101]) and an average risk ratio of 0.728 (range: 0.507-0.979) over different treatment durations. Using a logistic link, we estimated a constant treatment risk ratio effect of 1.064 (95% CI: [0.972, 1.165]) and an average risk ratio of 0.739 (range: 0.538-1.008) over different treatment durations. Under both models, the estimated constant treatment effect is either smaller in magnitude or has a different sign than almost all estimated heterogeneous treatment effects, suggesting a negative weighting issue is present. Our results suggest additional care is needed when applying constant treatment effect models in longitudinal settings.