Taking Into Account the Susceptible Window for Exposures in Cox Proportional Hazards Model: A Way Using Splines to Model the Lag-Risk Relationship.

External exposures take time to accumulate and affect human health, leading to a time lag between initial exposure and the manifestation of clinical signs. Susceptible time window analysis aims to describe the relationship between the time since exposure and its effects. To obtain unbiased estimates in susceptible time window analysis, researchers may need to adjust for exposures measured at different times simultaneously. Conventional Cox proportional hazards models often fail in this regard due to the strong multicollinearity between lagged exposure terms. In addition to estimation challenges, missing data in historical exposures present another obstacle for long-term analysis. To address these challenges, we first impose flexible parametric models on the effects across historical exposures by assuming that they follow restricted cubic splines. Based on our proposed restricted cubic spline model, we develop two model selection algorithms to help researchers determine the optimal knot locations in the splines. Additionally, we propose using multiple imputations to handle the missing exposure data. We conducted simulation studies to evaluate the performance of the restricted cubic spline models and the imputation algorithms in various realistic settings. The results show that our model demonstrates strong performance, offering great flexibility and robustness to misspecified maximum length of effect. Compared to the complete-person-time analysis, our multiple imputation algorithm exhibits significantly lower variance, with only a slight increase in bias. Finally, as an illustrative example, we apply our proposed models to analyze the effect of folate consumption on colorectal cancer incidence in the Nurses' Health Study from 1976 to 2016.