Multivariate Cox Model on Main Effects (ROPRO)

The ROPRO, introduced in (Becker et al., 2020), is a prognostic score based on the Cox model. The Cox model (Cox, 1972) is a widely used model in survival analysis that estimates the hazard function based on a set of given covariates of the population. It assumes that the hazard function h(t) is composed of two terms: a baseline hazard h0(t) that does not depend on the covariates and an exponential risk term er(X)=eβX:

where X is the covariate vector and β are the model weights. The risk term integrates the interaction between the covariates and the hazard of each patient. In the case of the Cox model, the fitting focuses on the risk r(X) = βX, which is a linear function, using the following partial likelihood cost function:

where δi is the censoring indicator. It is 1 if the patient has faced the event by the end of data collection and 0 otherwise. Naturally, being a linear function, it cannot implicitly deal with nonlinearities or interaction effects between the covariates (Harrell et al., 1996). This is one of the pitfalls of the Cox model and one of the reasons that motivated the creation of other more complex models (Ridgeway 1999; Katzman et al., 2018).

The authors of ROPRO started with a Cox model with 44 covariates and applied backward selection, removing the least significant covariates, until a total of 27 covariates remained in the model. The 27 selected covariates are represented in Table 2. In this work, we used the ROPRO formula as published in (Becker et al., 2020).