Regression models for survival data (Cox proportional hazards and parametric alternatives)
CS2 tests regression on survival data using Cox's semiparametric model and parametric alternatives, with emphasis on hazard ratios, partial likelihood and model checking.
Cox model
The Cox model assumes h(t|x)=h₀(t)exp(xβ). The baseline hazard h₀(t) is left unspecified and cancels from the partial likelihood, which compares, at each event time, the covariate of the case failing with those still at risk. exp(β_k) is a hazard ratio for a one-unit increase in x_k, adjusted for other covariates. Ties are handled by Breslow or Efron approximations. The proportional hazards assumption can be checked with Schoenfeld residuals, plots of log(−log Ŝ(t)) by group, or time-dependent covariates.
Parametric alternatives
Exponential and Weibull models are parametric PH models; Weibull is also an accelerated failure time (AFT) model. In AFT form log T=μ+xβ+σW, so covariates multiply survival time. Log-logistic and lognormal AFT models allow non-monotone hazards. Compare models by AIC, BIC, likelihood and residual checks, not by R².
Worked example
A Cox model with a binary treatment covariate gives β=0.588. The hazard ratio is e^{0.588}=1.80, so treated subjects have 80% higher hazard at every time. If the baseline survival at 5 years is S₀(5)=0.6, then S(5|treated)=0.6^{1.80}=0.399. The same calculation is invalid for a model without proportional hazards.
Traps
A hazard ratio is not generally a ratio of medians or mean survival times. The Cox partial likelihood does not estimate h₀(t). Do not ignore informative censoring or immortal time. Check that a time-dependent covariate is not defined using future information.