hazardmodel

A hazard model is a statistical framework used to describe the hazard function, the instantaneous risk of an event occurring at time t given that it has not yet occurred. In survival analysis, the hazard function h(t|X) depends on covariates X, and the survival function S(t|X) is linked to the hazard by S(t|X) = exp(-∫0^t h(u|X) du). The model focuses on the timing of events rather than only whether they occur.

One widely used form is the Cox proportional hazards model, written as h(t|X) = h0(t) exp(β'X). The baseline hazard h0(t) is left unspecified, and estimation relies on partial likelihood to infer the regression coefficients β. Hazard ratios exp(β) describe the relative risk associated with covariates. Parametric hazard models specify both the time dependence and the covariate effects directly, with examples such as exponential, Weibull, Gompertz, or log-logistic forms, enabling explicit estimates of survival and hazard functions.

Extensions and variants include time-varying covariates, stratified models, and models that accommodate non-proportional hazards. More complex structures address competing risks (subhazards), frailty (unobserved heterogeneity), and multi-state processes that model transitions between distinct states (for example, health, disease, and death).

Key data considerations include censoring, particularly right censoring, which is common in survival data. Assumptions such as non-informative censoring and, in the case of Cox models, proportional hazards, are important for valid inference; diagnostics and model checks often use residuals and goodness-of-fit tests.

Hazard models are used across biostatistics, clinical research, reliability engineering, and risk assessment to understand the timing and risk factors of events.