A Review of the Connections Among Logistic Regression, the Cox Model, Competing Risk Models, and Multi-State Models for Estimating Hazards and Survival Risks

Survival analysis has been widely used to understand the etiology of chronic diseases. However, traditional survival analysis models only a single endpoint, whereas the development of chronic diseases is a multi-state process. In this review, we examine the connections among logistic regression, the Cox model, competing risk models, and multi-state models in estimating hazards and survival risks, with findings summarized in three aspects. First, logistic regression and the Cox model are connected. The conditional likelihood of a conditional logistic regression stratified by risk set is equivalent to the partial likelihood used by the Cox model. The survival risks can be derived from the Cox model using the Breslow estimator. Alternatively, pooled logistic regression can be used to estimate risk within small time interval that approximate discrete-time hazard, with survival risks estimated using the Kaplan-Meier (K-M) estimator. Survival risk estimated from the Cox model is more accurate, whereas discrete-time hazard is more flexible for creating complex statistics (such as counterfactual survival risks in causal inference) and provides an approach for integrating machine learning into survival analysis. Second, for nonparametric estimation of survival risks from hazards, the cumulative incidence functions (CIFs) used in competing risks and the Aalen-Johansen (A-J) estimator used in multi-state process are extensions of the K-M estimator used in single disease endpoint. Third, the cause-specific Cox model and the Markov Cox model are extensions of the Cox model to competing risk and multi-state settings, respectively. Correspondingly, multinomial pooled logistic regression and a discrete-time split-state framework extend pooled logistic regression to competing risk and multi-state settings. Our paper can serve as a tutorial to illustrate the connections between survival analysis and multi-state modeling. We anticipate that multi-state models will play an increasingly important role in understanding chronic disease dynamics and advancing precision prevention and prediction.