Hazardfunktio

Hazardfunktio, in English often called the hazard function, is a concept in survival analysis and reliability theory. It describes the instantaneous rate at which an event occurs at time t, conditional on survival up to that time. In other words, it is the risk of the event happening in a very small time interval given that it has not occurred before.

Formally, if T denotes the time to an event, with density f(t) and survival function S(t) = P(T > t), the hazard function is defined as h(t) = f(t) / S(t). The cumulative hazard function is H(t) = ∫0^t h(u) du, and the survival function can be expressed as S(t) = exp(-H(t)). The density and hazard are related by f(t) = h(t) S(t) = h(t) exp(-H(t)).

The hazard function reflects the instantaneous risk and can vary with time. A constant hazard corresponds to the exponential distribution, while increasing or decreasing hazards model aging or improvement effects. Hazard ratios compare hazards between groups and are central to regression models in survival analysis.

In practice, the hazard is estimated and modeled in several ways. Nonparametric methods include the Nelson-Aalen estimator for the cumulative hazard and the Kaplan-Meier estimator for the survival function. Parametric models specify a functional form for h(t), such as exponential, Weibull, Gompertz, or log-logistic. The Cox proportional hazards model expresses the hazard as h(t|X) = h0(t) exp(β′X), separating baseline hazard h0(t) from covariate effects X.

Hazardfunktio is fundamental for interpreting time-to-event data in medicine, engineering, and reliability research, where understanding instantaneous risk informs prognosis, decision-making, and design.