Hazard Rate

A hazard rate, also known as a hazard function or hazard ratio, is a concept that arises largely in the literatures on SURVIVAL ANALYSIS and EVENT HISTORY[Page 453]ANALYSIS, and this entry focuses on its usage and estimation in these literatures. However, the estimator of the hazard ratio also plays a key role in the MULTIVARIATE modeling approach to correcting for sample SELECTION BIAS proposed by Heckman (1979).

The mathematical definition of a hazard rate starts with the assumption that a continuous RANDOM VARIABLET defined over a range tmin ≤ T ≤ tmax has a cumulative distribution function (CDF), F(t), and a corresponding PROBABILITY DENSITY FUNCTION, f(t) ≡ dF(t)/dt. Frequently, the random variable T is assumed to be some measure of time. The complement of the cumulative probability function, called the survivor function, is defined as S(t) ≡ 1 − F(t). Because it gives the probability that T ≥ t (i.e., “survives” from tmin to t), it is also known as the survivor probability. The hazard rate (or hazard ratio), h(t), is then defined as

Speaking roughly and not rigorously, the hazard rate tells the chance or probability that the random variable T falls in an infinitesimally small range, t ≤ T < t + Δt, divided by Δt, given that T ≥ t, when Δt is allowed to shrink to 0. (Mathematically, one takes the limit as Δt approaches 0.)

Because both the probability density function and the survivor function cannot be negative, the hazard rate cannot be negative. The hazard rate is often regarded as analogous to a probability, and some authors even refer to a “discrete [time] hazard rate” (rather than to a probability) when T is taken to be a discrete random variable. However, for a continuous random variable T, the hazard rate can exceed 1, and, unlike a probability, it is not a unit-free number. Rather, its scale is the inverse of the scale of T. For example, in event history analysis, T might denote the age at which a person experiences some event (marriage, death) or the duration at which someone enters or leaves some status (starts or ends a job). In such instances, the hazard rate might be expressed in units “per year” or “per month.”

Because the survivor function equals 1 at tmin, the hazard rate equals the probability density function at tmin. In addition, because the survivor function is a monotonically nonincreasing function and equals 0 at tmax (unless T has a “defective” probability distribution), the hazard rate tends to become an increasingly larger multiple of the probability density function as t increases.

The LIFE TABLE or actuarial ESTIMATOR of the hazard rate in a specified discrete interval [u, v) has a very long history of usage in demography and is still utilized in many practical applications. Its definition is

where d(u, v) is the number of observations of t falling in the interval [u, v); n(u) is the number of observations that survived until u (i.e., for which T ≥ u); and c(u, v) is the number of observations censored in the interval [u, v). (See CENSORING AND TRUNCATION for a definition and discussion of censored observations.) One-half appears in the divisor on the right-hand side of the equation because observations within the interval [u, v) are assumed to occur uniformly over the interval.

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