hazardratio

Hazard ratio is a measure used in survival analysis to compare the hazard rates between two groups. The hazard at time t is the instantaneous rate at which events occur given survival up to time t. The hazard ratio is the ratio of hazards: HR = h1(t) / h0(t). In the Cox proportional hazards model, the hazard ratio is assumed to be constant over time, an assumption known as proportional hazards.

Interpreting a hazard ratio: HR > 1 indicates a higher hazard in the numerator group, while HR < 1 indicates a lower hazard. For example, an HR of 0.75 implies a 25% reduction in the hazard of the event in the treatment group compared with control at any time under proportional hazards. The hazard ratio refers to instantaneous risk rather than cumulative incidence over a fixed period.

Estimation and inference: Hazard ratios are typically estimated using Cox regression via partial likelihood. Confidence intervals are reported on the log(HR) scale and transformed back; a 95% confidence interval that excludes 1 indicates statistical significance under a standard alpha level. Assumptions include proportional hazards, independent censoring, and correct model specification.

Non-proportional hazards can produce time-varying hazard ratios, in which case time-dependent Cox models or stratified approaches may be used. Graphical checks (such as Schoenfeld residuals) help assess the proportionality assumption.

Relation to other measures: the hazard ratio concerns instantaneous risk, not cumulative incidence, and is related but not identical to relative risk or risk ratio, which compare probabilities over a defined time horizon.

Applications: hazard ratios are widely used in clinical trials and epidemiology to compare treatments or exposures for time-to-event outcomes (e.g., time to relapse or death). Limitations include reliance on model assumptions and potential misinterpretation when hazards are not proportional.