Proportional hazards model

Proportional hazards models are a class of survival models in statistics. Survival models relate the time that passes before some event occurs to one or more covariates that may be associated with that quantity of time. In a proportional hazards model, the unique effect of a unit increase in a covariate is multiplicative with respect to the hazard rate.

Overview

The proportional hazards model is a type of survival model, which is used in statistics to investigate the time until an event of interest occurs. This event could be anything from the failure of a machine part to the death of a patient in a clinical trial. The model is called 'proportional hazards' because the hazard function, which describes the instantaneous rate of failure at a given time, is assumed to be a function of the covariates that does not change over time.

Assumptions

The main assumption of the proportional hazards model is that the effects of the predictor variables are multiplicative with respect to the hazard rate and do not change over time. This means that the hazard for any individual is a product of a baseline hazard, which may vary over time but is the same for all individuals, and a function of the individual's specific covariates.

Mathematical Formulation

The hazard function h(t) at time t for an individual with covariates X is given by:

h(t|X) = h0(t) exp(X'β)

where h0(t) is the baseline hazard, X' is the transpose of the covariate vector X, and β is a vector of coefficients.

Applications

Proportional hazards models are widely used in biostatistics and epidemiology to analyse survival data, where the 'event' is often death. They are also used in engineering for reliability analysis, where the 'event' is often a component failure.

See Also

• Survival analysis
• Cox proportional hazards model
• Hazard ratio
• Biostatistics
• Epidemiology