Identification of an additive interaction using parameter regularization and model selection in epidemiology

Background In epidemiology, indicators such as the relative excess risk due to interaction (RERI), attributable proportion (AP), and synergy index (S) are commonly used to assess additive interactions between two variables. However, the results of these indicators are sometimes inconsistent in real world applications and it may be difficult to draw conclusions from them. Method Based on the relationship between the RERI, AP, and S, we propose a method with consistent results, which are achieved by constraining $e^{\theta_3}-e^{\theta_1}-e^{\theta_2}+1=0$, and the interpretation of the results is simple and clear. We present two pathways to achieve this end: one is to complete the constraint by adding a regular penalty term to the model likelihood function; the other is to use model selection. Result Using simulated and real data, our proposed methods effectively identified additive interactions and proved to be applicable to real-world data. Simulations were used to evaluate the performance of the methods in scenarios with and without additive interactions. The penalty term converged to 0 with increasing λ, and the final models matched the expected interaction status, demonstrating that regularized estimation could effectively identify additive interactions. Model selection was compared with classical methods (delta and bootstrap) across various scenarios with different interaction strengths, and the additive interactions were closely observed and the results aligned closely with bootstrap results. The coefficients in the model without interaction adhered to a simplifying equation, reinforcing that there was no significant interaction between smoking and alcohol use on oral cancer risk. Conclusion In summary, the model selection method based on the Hannan-Quinn criterion (HQ) appears to be a competitive alternative to the bootstrap method for identifying additive interactions. Furthermore, when using RERI, AP, and S to assess the additive interaction, the results are more consistent and the results are simple and easy to understand.