Cox Proportional Hazards Regression

When the outcome is the time until an event, such as death, relapse, or recovery, linear and logistic regression do not fit, because they ignore timing and censoring. Cox proportional hazards regression is the standard tool: it models the hazard, the instantaneous event rate, as a function of predictors and reports each predictor's effect as a hazard ratio. A hazard ratio above 1 means the predictor speeds up the event, below 1 slows it, for example 'each disease stage carries a hazard ratio of 1.5 for death, adjusted for age'. Its strengths are exactly what survival data need. It uses censored observations, meaning patients lost to follow-up or still event-free at the study's end, efficiently. It does not require assuming a specific shape for the baseline survival curve, which is why it is called semi-parametric. And, like the other regressions, its multivariable form adjusts for confounders, yielding adjusted hazard ratios. This makes it the backbone of prognostic research and of clinical trials with time-to-event endpoints. The central assumption is in its name: proportional hazards, meaning the hazard ratio between groups is assumed roughly constant over time. If the survival curves cross or diverge dramatically, this assumption fails and a single hazard ratio is misleading; analysts check it (for example with Schoenfeld residuals or log-log plots) and turn to alternatives, such as time-varying covariates or the restricted mean survival time, when it does not hold. Report each hazard ratio with its 95% confidence interval, and pair the model with Kaplan-Meier curves so readers see the absolute survival picture rather than only a ratio. As always, the hazard ratio shows association adjusted for the measured confounders, not proven causation. In trials with time-to-event endpoints the adjusted hazard ratio from a Cox model is often the primary effect estimate, and modern reports increasingly add an absolute companion, such as the difference in survival at a landmark time or the restricted mean survival time, so clinicians see both the relative and the absolute picture.