Coxmodell

CoxModell, commonly known as the Cox proportional hazards model, is a semi-parametric method used in survival analysis to relate the time to an event to covariates. It does not require a specific distribution for survival times and focuses on the hazard function, the instantaneous risk of the event at time t given survival up to t.

In the CoxModell, the hazard for an individual with covariates X is h(t|X) = h0(t) exp(beta^T X), where h0(t) is an unspecified baseline hazard and beta is a vector of coefficients. The baseline hazard captures how risk changes over time without covariate effects. The model is typically estimated using partial likelihood, which allows beta to be estimated without specifying h0(t). Coefficients are interpreted as log hazard ratios; exponentiating beta_j gives the hazard ratio associated with a one-unit increase in the j-th covariate.

Key assumptions include proportional hazards, meaning the effect of covariates on the hazard is multiplicative and constant over time, and non-informative censoring, where the reason for censoring is independent of the survival process. Diagnostics often use Schoenfeld residuals and time-varying covariates to assess the proportionality assumption.

Extensions of the CoxModell include incorporating time-varying covariates, stratified Cox models to handle differing baseline hazards across groups, frailty models to account for unobserved heterogeneity, and methods for competing risks or regularized estimation in high-dimensional settings.

Applications of the CoxModell span medicine, epidemiology, and reliability analysis, where researchers compare survival or time-to-event outcomes across groups while adjusting for multiple covariates.