hazard rate: one construct, many systems

In demography and actuarial science the force of mortality, μ(x), is the instantaneous rate of death at exact age x among those who have survived to that age, the limiting death rate over a vanishingly small age interval. Gompertz's 1825 On the nature of the function expressive of the law of human mortality treated this instantaneous mortality intensity and its roughly exponential rise with age. The referent is an age-specific instantaneous event rate conditional on survival so far, exactly the same mathematical object that reliability engineers call the failure rate and statisticians the hazard function.[1]

Unit

The instantaneous death rate at exact age x, conditional on survival to x

Recovery means

Not applicable

Measured by

Derived from the life table as minus the derivative of the log survival function; estimated from age-specific death rates

In reliability engineering the failure rate, or hazard rate, is the instantaneous rate at which a component still working at time t fails in the next instant, which traces out the familiar bathtub curve of early, random, and wear-out failures. Barlow and Proschan's Mathematical Theory of Reliability is the classic treatment. It is the identical construct as the demographic force of mortality, an instantaneous event rate conditional on survival, applied to manufactured parts instead of people.[2]

Unit

The instantaneous failure rate of a component, conditional on having survived to time t

Recovery means

Not applicable

Measured by

Estimated from lifetime-test data as failures per unit time among survivors, and characterised across the bathtub curve

In survival statistics the hazard function h(t) generalises the idea to any time-to-event process: it is the instantaneous rate of the event at time t given survival up to t, the quantity that links the survival and density functions. Cox and Oakes's Analysis of Survival Data is a standard account. It is the same instantaneous-conditional-rate construct as the force of mortality and the engineering failure rate, stated for arbitrary events (death, relapse, default, or churn).[3]

Unit

The instantaneous event rate h(t) at time t, conditional on survival to t

Recovery means

Not applicable

Measured by

Estimated nonparametrically or by models from censored time-to-event data; related to S(t) by h(t) = −d/dt ln S(t)