CoxProportionalHazardsModell

The Cox proportional hazards model, named after Sir David Cox, is a widely used semi-parametric model in survival analysis. It relates individuals’ risk of an event over time to observed covariates, without requiring specification of the baseline hazard function.

The model specifies the hazard function h(t|X)=h0(t) exp(beta^T X), where h(t|X) is the instantaneous event rate at time t for a subject with covariates X, h0(t) is an unknown baseline hazard, and beta is a vector of coefficients.

Estimation is typically performed via partial likelihood, which uses the order of events rather than the exact form of h0. The coefficients beta quantify proportional effects on the hazard: exp(beta_j) is the hazard ratio associated with a one-unit increase in covariate j, holding other covariates constant. The model handles right-censored data.

Key assumptions include proportional hazards over time (the hazard ratio is constant with respect to t) and independent censoring. Diagnostics include Schoenfeld residuals and time-dependent covariates checks; violations may prompt model extensions or alternative approaches.

Extensions include stratified Cox models, time-varying covariates, and frailty models for unobserved heterogeneity. The Cox model is widely used in clinical research to compare survival across groups or to adjust for confounders, while acknowledging its limitations such as potential non-proportional hazards or unmeasured confounding.