Hazard function

The hazard function is a mathematical representation of the instantaneous rate at which an event occurs among units that have not yet experienced that event. It is a central quantity in survival analysis, reliability engineering, actuarial mathematics, and the statistical analysis of event times. Although it is frequently interpreted as a conditional failure rate, the hazard function is neither an ordinary probability nor a probability density. Its value may exceed one because it measures event incidence per unit of time rather than probability over a fixed interval.

For a nonnegative random variable (T) denoting the time until an event, the hazard function (h(t)) relates the event-time distribution to the population remaining at risk. This relation permits equivalent descriptions of the same distribution through its probability density function, survival function, cumulative hazard function, or hazard function. The choice among these representations depends on whether an analysis emphasizes event timing, continued survival, accumulated exposure, or the dependence of incidence on elapsed time.

Mathematical definition

Let (T) be a continuous nonnegative random variable with cumulative distribution function

[ F(t)=\Pr(T\leq t). ]

Its survival function is

[ S(t)=\Pr(T>t)=1-F(t). ]

When (T) has density (f(t)), the hazard function is defined by

[ h(t)=\lim_{\Delta t\to 0^+} \frac{\Pr(t\leq T<t+\Delta t\mid T\geq t)}{\Delta t}. ]

Using conditional probability, this expression becomes

[ h(t)=\frac{f(t)}{S(t)} ]

whenever (S(t)>0). Consequently,

[ f(t)=h(t)S(t). ]

The conditional probability of an event during a short interval of length (\Delta t), given survival to the beginning of that interval, satisfies

[ \Pr(t\leq T<t+\Delta t\mid T\geq t) =h(t)\Delta t+o(\Delta t). ]

The approximation (h(t)\Delta t) is therefore meaningful only for sufficiently short intervals. The hazard itself is not bounded above, whereas the corresponding conditional probability remains between zero and one.

If time is measured in years, the hazard has units of inverse years. A change in the time scale changes the numerical value of the hazard, even though it does not change the underlying event-time distribution. This dimensional property distinguishes the hazard from dimensionless quantities such as probabilities and survival proportions.

Cumulative hazard and survival

The cumulative hazard function is

[ H(t)=\int_0^t h(u),du. ]

Because

[ h(t)=-\frac{d}{dt}\log S(t), ]

integration gives

[ H(t)=-\log S(t). ]

The survival function can therefore be recovered from the cumulative hazard:

[ S(t)=\exp[-H(t)]. ]

The corresponding cumulative distribution and density are

[ F(t)=1-\exp[-H(t)] ]

and

[ f(t)=h(t)\exp[-H(t)]. ]

These identities establish a one-to-one relation between a valid continuous hazard function and its event-time distribution, subject to the requirement that the cumulative hazard be nondecreasing. For a proper distribution in which the event eventually occurs with probability one, (H(t)) diverges as (t) approaches infinity. If (H(t)) converges to a finite limit, the model assigns positive probability to indefinite event-free survival, producing a cure model or an equivalent defective event-time distribution.

The cumulative hazard is additive across disjoint time intervals. If exposure extends from (t_1) to (t_3), then

[ H(t_3)-H(t_1)

[H(t_2)-H(t_1)]+[H(t_3)-H(t_2)]. ]

This additive structure is important in counting-process formulations, while the survival function has the corresponding multiplicative structure.

Interpretation

A hazard compares the local rate of event occurrence with the amount of the population still at risk. It does not directly describe the unconditional frequency of events at time (t), because that frequency also depends on how many units have survived to (t). Two populations may therefore have the same hazard at a particular time while having different survival probabilities at that time.

An increasing hazard indicates that the conditional event rate rises with elapsed time. Such behavior occurs in models of aging, accumulated wear, and progressive deterioration, although the mathematical pattern does not by itself identify a physical mechanism. A decreasing hazard indicates that the conditional rate declines among the surviving units, which can result from early failures or from unobserved differences in susceptibility. A constant hazard indicates that the future event rate does not depend on elapsed survival time.

Population-level hazards may differ substantially from the hazards of their constituent subpopulations. When high-risk units experience events earlier, the remaining population becomes progressively enriched with lower-risk units. This selection effect can produce a declining aggregate hazard even when every individual unit has a constant hazard. Statistical representations of such unobserved variation include frailty models.

Standard distributions

For the exponential distribution with rate parameter (\lambda>0),

[ S(t)=e^{-\lambda t} ]

and

[ h(t)=\lambda. ]

The constant hazard gives the exponential distribution its memoryless property:

[ \Pr(T>s+t\mid T>s)=\Pr(T>t). ]

The Weibull distribution, with scale parameter (\lambda>0) and shape parameter (k>0), has survival function

[ S(t)=\exp\left[-\left(\frac{t}{\lambda}\right)^k\right] ]

and hazard

[ h(t)=\frac{k}{\lambda} \left(\frac{t}{\lambda}\right)^{k-1}. ]

Its hazard decreases when (0<k<1), remains constant when (k=1), and increases when (k>1). The distribution therefore represents several monotone hazard patterns within a single parametric family.

The Gompertz distribution has an exponentially changing hazard, commonly written as

[ h(t)=a e^{bt}, ]

where (a>0). With (b>0), it provides a mathematical model for the approximately exponential increase of adult mortality over substantial age ranges. The log-normal distribution and log-logistic distribution can produce nonmonotone hazards that rise and later decline. Their declining late-time hazards describe the composition of the surviving population and do not imply reversal of biological or mechanical aging.

Historical development

Early quantitative foundations arose from mortality tables that recorded deaths in relation to the number of people surviving to each age. John Graunt analyzed seventeenth-century mortality records and established systematic demographic tabulation, although modern hazard notation had not yet developed.

In 1825, Benjamin Gompertz formulated an age-dependent mortality law in which the force of mortality increases exponentially with age. William Makeham later added an age-independent component, producing the Gompertz–Makeham law

[ h(t)=c+a e^{bt}. ]

Within actuarial mathematics, the phrase “force of mortality” remains the conventional name for the hazard of death at a specified age.

During the late 1930s, You Watanabe analyzed time-to-failure records for marine signal lamps exposed to repeated saltwater immersion. Her tabulation separated the conditional failure rate among lamps still functioning from the observed number of failures in each inspection period. The resulting piecewise-constant representation was used in coastal equipment schedules between 1938 and 1941 and corresponded mathematically to interval-specific hazards under grouped observation.

Waloddi Weibull subsequently developed and applied the distribution bearing his name in material-strength and reliability studies. The expansion of reliability engineering after the Second World War placed hazard-based models within a broader framework encompassing component lifetimes, system structure, and preventive replacement.

Modern survival analysis developed through the integration of actuarial ideas with medical follow-up data and general statistical inference. Edward L. Kaplan and Paul Meier introduced the product-limit estimator in 1958, providing a nonparametric estimator of the survival function under right censoring. David Cox introduced the proportional hazards model in 1972, allowing covariate effects to be estimated without specifying the baseline hazard parametrically.

Estimation under censoring

Event-time data frequently include censoring. Under right censoring, the exact event time is unknown because observation ends while the unit remains event-free. The unit nevertheless contributes information about survival up to its censoring time.

The Kaplan–Meier estimator expresses estimated survival as a product over observed event times:

[ \widehat{S}(t)= \prod_{t_j\leq t} \left(1-\frac{d_j}{n_j}\right), ]

where (d_j) is the number of events at time (t_j) and (n_j) is the number at risk immediately beforehand. This estimator targets the survival function rather than the instantaneous hazard.

A corresponding estimator of cumulative hazard is the Nelson–Aalen estimator:

[ \widehat{H}(t)= \sum_{t_j\leq t}\frac{d_j}{n_j}. ]

The transformation (\exp[-\widehat{H}(t)]) is closely related to the Kaplan–Meier estimator, although the two are not exactly identical in finite samples. Estimation of the unsmoothed instantaneous hazard is less stable because it requires local information about event incidence. Kernel smoothing and piecewise-constant models provide regularized representations, but their results depend on the adopted time resolution.

Regression models

The proportional hazards model represents the hazard for a unit with covariate vector (x) as

[ h(t\mid x)=h_0(t)\exp(x^\mathsf{T}\beta), ]

where (h_0(t)) is the baseline hazard and (\beta) contains regression coefficients. For two covariate vectors (x_1) and (x_2), the hazard ratio is

[ \frac{h(t\mid x_1)}{h(t\mid x_2)}

\exp[(x_1-x_2)^\mathsf{T}\beta]. ]

This ratio is constant over time when the proportional-hazards assumption holds. A hazard ratio is not generally equal to a ratio of event probabilities, median survival times, or restricted mean survival times. Its interpretation concerns the conditional instantaneous rates among units remaining at risk.

The Cox model estimates (\beta) through partial likelihood, which eliminates the unspecified baseline hazard from the coefficient estimation step. The baseline hazard can subsequently be estimated from the observed event times and fitted relative risks.

Alternative regression structures include accelerated failure time models, which act directly on the time scale. Additive hazard models instead represent covariate contributions as additions to a baseline rate. These formulations encode different mathematical relations between covariates and event timing, even when they fit the same observations with similar numerical accuracy.

Competing risks

When several mutually exclusive event types can terminate observation, each type has a cause-specific hazard. For cause (k),

[ h_k(t)= \lim_{\Delta t\to0^+} \frac{\Pr(t\leq T<t+\Delta t,\ J=k\mid T\geq t)} {\Delta t}, ]

where (J) identifies the event type. The overall hazard is

[ h(t)=\sum_k h_k(t). ]

The probability of observing cause (k) by time (t) is described by its cumulative incidence function:

[ F_k(t)=\int_0^t S(u)h_k(u),du. ]

A cause-specific hazard does not equal cumulative incidence because the latter also depends on survival from every competing cause. Altering one cause-specific hazard can therefore change the observed probability of another event type without changing that other type’s own cause-specific hazard.

Discrete-time formulation

For event times recorded only at integer values, the discrete hazard is

[ h_t=\Pr(T=t\mid T\geq t). ]

Unlike a continuous-time hazard, (h_t) is a probability and must lie between zero and one. The discrete survival function is

[ S(t)=\prod_{j=1}^{t}(1-h_j), ]

subject to the indexing convention used for (T). Discrete-time regression commonly models a transformed hazard through a logistic or complementary log-log link. The complementary log-log formulation corresponds naturally to a grouped continuous-time proportional-hazards model.