CoxproportionaleHazardsModellierung

The Cox proportional hazards model, introduced by Sir David Cox in 1972, is a semiparametric model used in survival analysis to relate time-to-event data to covariates. It models the hazard function, the instantaneous risk of the event at time t, given covariates.

The model specifies h(t|X) = h0(t) exp(beta^T X), where h0(t) is the baseline hazard and X is a vector of covariates. The baseline hazard is left unspecified, allowing flexible hazard shapes. Coefficients beta are estimated by partial likelihood, which uses the order of event times and accommodates right-censored data.

Key assumption is proportional hazards: the hazard ratio between any two individuals with covariate vectors X and X' is exp{beta^T (X - X')}, independent of time. Violations can be addressed by including time-dependent covariates or by stratification. Diagnostics include Schoenfeld residuals and time-varying coefficient tests.

Interpretation: exp(beta_j) is the hazard ratio associated with a one-unit increase in covariate j, holding other covariates fixed. The model provides relative risk over the follow-up period, not a direct probability of the event.

Extensions and variants include stratified Cox models, time-dependent effects, frailty models, competing risks, and joint models. It is widely implemented in statistical software such as R (survival), SAS, Stata, and Python (lifelines).

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