Cox Proportional Hazards › Summary

The Cox proportional hazards model is a semiparametric regression model for time-to-event data. It writes each individual's hazard as an unspecified baseline hazard multiplied by an exponential function of the covariates, $h(t \mid x) = h_0(t)\exp(\beta^\top x)$. The baseline cancels in every hazard ratio, producing the model's central interpretation: $\exp(\beta)$ is the multiplicative effect of a unit change in a covariate on the hazard at every moment in time.

Estimation proceeds through the partial likelihood, which is constructed by asking, at each event time, how likely the observed failing individual was given who else was at risk. The baseline hazard factors out of every term and disappears. The log partial likelihood is concave. Its score subtracts the weighted risk-set covariate mean from each failing covariate; its Hessian is a negative sum of weighted covariance matrices. Concavity is useful, but finite estimability, risk-set variation and the actual optimizer diagnostics still need checking; full column rank alone does not establish a finite unique estimate or global Newton convergence. This chapter uses approximations for tied event times; the Efron approximation is generally preferred for coarsened continuous times with moderate tie counts, and the Breslow approximation is a useful starting point.

After fitting we recover the baseline cumulative hazard with Breslow's estimator and predict absolute survival via $\hat S(t \mid x) = \exp(-\hat H_0(t)\exp(\hat\beta^\top x))$. The dimensionless hazard ratio remains a relative rate multiplier; the reconstructed survival function is a separate probability-valued quantity. A hazard ratio of 0.5 does not imply a survival ratio of 2. The survival gap depends on time and on the baseline, which is why the fitted survival curve is an essential companion to any coefficient table.

Finally, the model rests on the proportional-hazards assumption, and any serious application should verify it. Schoenfeld residuals, the Grambsch-Therneau test, and simple residual-versus-time plots detect the common failure modes. When proportionality is implausible, stratification, time-varying coefficients, or a switch to accelerated failure time models (the next chapter) offer principled alternatives. The Cox model is useful, but only when it is used as a model of relative hazards and not as a shortcut to absolute survival probabilities.