Can a hazard exceed one?

Yes, because its units are inverse time; it is not a probability.

Does a hazard of zero at one instant mean the event can never happen?

No. The hazard may be positive at other times.

Is cumulative hazard the same as cumulative risk?

No. In the simple continuous setting, risk is 1 − exp(−H), not H itself.

AI Glossary / Statistics and probability / Intermediate

Hazard rate

Definition

A hazard rate is the instantaneous rate at which an event occurs at time t among units that have not yet experienced it. It is a central quantity in survival and reliability analysis, where the event might be a component's first failure, a customer's first cancellation, or another defined transition. A hazard is a rate per unit of time, not the probability of an event by a specified date. It can exceed one without violating probability rules.

What it means

The hazard asks a conditional question: among units still event-free at this time, how quickly are events occurring now? Units that already experienced the event are no longer part of that risk set in a first-event analysis.

This differs from cumulative risk, which asks what fraction of an initial population has experienced the event by time t. Cumulative risk can keep increasing even when the hazard is constant or declining, because events continue to accumulate.

The event definition, time origin, and time unit are essential. A rate per day is numerically different from a rate per month even when both describe the same underlying process.

How it works

For an event time T, define h(t) as the limit of P(t ≤ T < t + delta | T ≥ t)/delta as the positive interval delta approaches zero. For a continuous event-time distribution with density f(t) and survival probability S(t), h(t) = f(t)/S(t) wherever S(t) is positive.

The cumulative hazard is H(t) = integral from zero to t of h(u) du. In the standard continuous setting, S(t) = exp[−H(t)] and cumulative event probability is 1 − exp[−H(t)]. For a short interval with slowly changing hazard, event probability conditional on survival is approximately h(t)*delta.

Estimating hazards from data requires handling the changing risk set and censoring appropriately. A first-event hazard should also be distinguished from recurrent-event rates or failure rates for repairable systems, which involve different counting and modeling structures.

Three practical examples

A reliability engineer studies whether first-failure rates increase as components age.
A subscription analyst models cancellation timing among customers who remain subscribed at each duration.
An AI maintenance system translates a fitted event-time model into conditional near-term failure probabilities for equipment still operating.

Think of it this way

Think of watching people leave a waiting room. The hazard describes the pace of departures among those still inside at a particular moment. The fraction who have already left answers a different question about the original group. This analogy emphasizes the changing denominator; it does not assume that real departures have a constant rate.

Worked example

Assume an illustrative component lifetime has constant hazard 0.2 per year. Its cumulative hazard at five years is H(5) = 0.2*5 = 1. Survival to five years is exp(−1) = 0.3679, so the probability of failure by then is 0.6321, or 63.21%.

For a component that has already survived five years, the conditional probability of failure in the next year is 1 − exp(−0.2*1) = 0.1813. The value 0.2 is an instantaneous rate, so calling it an exact 20% annual probability would be incorrect.

Over the next 0.1 year, the exact conditional probability is 1 − exp(−0.02) = 0.01980. The short-interval approximation gives 0.2*0.1 = 0.02, which is close. Independent exponential calculations confirm all values. These results use a constant-hazard exponential model; they are not a claim that actual components have age-independent failure behavior.

Illustrative case study

In a fictional equipment-rental business, a dashboard labels a model output of 0.2 per year as “20% failure probability.” The maintenance analyst identifies the unit and interpretation error.

The team changes the display to show a probability over a clearly stated planning horizon, calculated from the model's integrated hazard. It records the equipment's current event-free age and checks whether the fitted hazard changes with usage. Units removed from observation are handled according to the analysis's censoring assumptions rather than counted as failures or guaranteed survivors. The revised display supports clearer scheduling discussions, but the fictional example does not establish the model's calibration or maintenance benefit.

Common misconceptions and mistakes

Do not confuse hazard with cumulative probability or cumulative hazard. H(t) is not generally a probability and can exceed one. A constant hazard does not mean the cumulative event probability grows linearly forever.

Do not read a continuous-time hazard as the probability of an event at an exact instant. Under a continuous distribution, that exact-time probability is zero. Nor does a declining observed hazard necessarily mean every individual unit becomes safer; changing composition among survivors can contribute. Competing events and informative censoring require care because they affect what remains observable and which risk is being estimated.

Business use

Hazard models help businesses reason about timing, but most operational decisions need probabilities over meaningful intervals. Define the event and horizon before converting a model output into a decision. For a unit event-free at t, the interval probability depends on the hazard accumulated from t to the end of that interval.

Evaluate predictions against representative follow-up data, including censoring and changes in operating conditions. If several mutually exclusive event types matter, distinguish a cause-specific hazard from the probability of that event occurring before its competitors. A single hazard number is rarely a complete account of operational risk.

Check your understanding

What is five-year cumulative hazard in the example?

Answer: One, while five-year event probability is approximately 0.6321.

Why is the next-year probability about 18.13% rather than 20%?

Answer: It is calculated by integrating the rate and applying the exponential survival relationship.

Related terms

Survival function
Risk set
Censoring
Hazard ratio
cumulative hazard
event probability
exponential distribution
competing risks

Learn next

Next, plot hazard, cumulative hazard, and survival for a constant-hazard model. Then compare a time-varying model and learn how censoring and competing events change the estimation and interpretation of event probabilities.

Censoring

Sources

Sources: Frank Harrell, Glossary of Statistical Terms, https://hbiostat.org/glossary/; NIST/SEMATECH, “Failure (or hazard) rate,” https://www.itl.nist.gov/div898/handbook/apr/section1/apr123.htm, and “Exponential,” https://www.itl.nist.gov/div898/handbook/apr/section1/apr161.htm. Primary teaching materials were opened and definitions and exponential identities checked. The negative sign in exp(−H) is retained consistently; one line on NIST's hazard page omits it. Original numerical examples were independently calculated.

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