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What is the concept of hazard rate function in survival analysis?

Survival analysis, also known as event analysis or survival time analysis, examines the time to the occurrence of a specific event, such as the death of a patient, the onset of illness, or the failure of a device. The hazard rate function (also known as the risk function or the hazard function) is a central concept in survival analysis.

The hazard rate function describes the probability of the occurrence of the event per unit of time, assuming that the event has not yet occurred by a certain point in time. So it indicates how "dangerous" or risky it is to experience the event at a certain point in time. The hazard rate function can be thought of as a kind of "instantaneous rate" of event occurrence.

Mathematically, the hazard rate function is often represented using the symbol λ(t) or h(t), where t is time. It is defined as the quotient of the conditional probability of the event occurring in a very small time interval around t divided by the length of this interval. In formal terms:

λ(t) = lim(Δt→0) [P(t ≤ T < t+Δt | T ≥ t) / Δt]

Where T is the random variable representing the time to the event, and P() denotes the probability.

The hazard rate function can take different forms depending on the evolution of the risk over time. A constant hazard rate function (λ(t) = λ) would mean that the risk remains constant regardless of time. An increasing hazard rate function would suggest that risk is increasing over time, while a decreasing hazard rate function would suggest that risk is decreasing.

Analysis of the hazard rate function allows researchers to identify patterns over time of event occurrence, determine risk factors, and make predictions about the probability of event occurrence.