Introducing activation functions into segmented regression model to address lag effects of interventions

The interrupted time series (ITS) design is widely used to examine the effects of large-scale public health interventions and has the highest level of evidence validity. However, there is a notable gap regarding methods that account for lag effects of interventions. To address this, we introduced activation functions (ReLU and Sigmoid) to into the classic segmented regression (CSR) of the ITS design during the lag period. This led to the proposal of proposed an optimized segmented regression (OSR), namely, OSR-ReLU and OSR-Sig. To compare the performance of the models, we simulated data under multiple scenarios, including positive or negative impacts of interventions, linear or nonlinear lag patterns, different lag lengths, and different fluctuation degrees of the outcome time series. Based on the simulated data, we examined the bias, mean relative error (MRE), mean square error (MSE), mean width of the 95% confidence interval (CI), and coverage rate of the 95% CI for the long-term impact estimates of interventions among different models. OSR-ReLU and OSR-Sig yielded approximately unbiased estimates of the long-term impacts across all scenarios, whereas CSR did not. In terms of accuracy, OSR-ReLU and OSR-Sig outperformed CSR, exhibiting lower values in MRE and MSE. With increasing lag length, the optimized models provided robust estimates of long-term impacts. Regarding precision, OSR-ReLU and OSR-Sig surpassed CSR, demonstrating narrower mean widths of 95% CI and higher coverage rates. Our optimized models are powerful tools, as they can model the lag effects of interventions and provide more accurate and precise estimates of the long-term impact of interventions. The introduction of an activation function provides new ideas for improving of the CSR model.