Biology · Biostatistics

Hazard function

Q: How does hazard differ from risk?

Risk is the probability of an event over a fixed interval. Hazard is the instantaneous conditional event rate among people still event-free. A rate has units of inverse time and is not bounded by zero and one; probability is. With constant hazard λ, risk over duration Δt is 1−e^(−λΔt). A time-varying hazard must be integrated over the interval. State event definition, time scale and risk population. The rate alone is not a probability, and a hazard ratio requires additional model assumptions. Time variation may make a constant-hazard approach unsuitable.

Q: Can a hazard exceed 1?

Yes, depending on the chosen time unit. A rate of 1.5 per day is mathematically possible because hazard is not a probability. It describes instantaneous event intensity among survivors. Risk over a particular duration remains between zero and one and follows from hazard over the entire interval. Always report time scale and units rather than comparing values without their time unit. State event definition, time scale and risk population. The rate alone is not a probability, and a hazard ratio requires additional model assumptions. Time variation may make a constant-hazard approach unsuitable.

Q: How are hazard and the survival function related?

Cumulative hazard H(t) is the integral of instantaneous hazard up to t; for continuous time, S(t)=exp(−H(t)). Conversely, h(t)=−d ln S(t)/dt. With constant hazard, S decays exponentially. A Kaplan–Meier curve estimates S nonparametrically, while hazard models describe the instantaneous rate directly. They are related but answer different interpretive questions and require appropriate handling of censoring. State event definition, time scale and risk population. The rate alone is not a probability, and a hazard ratio requires additional model assumptions. Time variation may make a constant-hazard approach unsuitable.

Q: How do I apply Hazard function to measured data?

Reproduce the example calculation: For an exponential event time with constant rate λ=0.02 per day, h(t)=0.02/day. Hazard function uses h(t) = f(t)/S(t). In plain terms, the hazard is the instantaneous event rate among individuals who remain event-free up to that time. Unit check: h(t) in inverse time · f(t) in inverse time · S(t) dimensionless. For continuous event times and a defined at-risk population; hazards depend on the time scale and event definition. Values are comparable only when their definitions, denominators and periods match. Record these details next to the result so the reference group is clear. If the direction is implausible, check data preparation and the calculation basis before discussing biological causes.

Q: Which model limits should I state for Hazard function?

A key limit of Hazard function is this: For continuous event times and a defined at-risk population; hazards depend on the time scale and event definition. Do not use the metric without its reference; it is useful for Mathematically describing instantaneous event rates in survival analyses. The hazard is the instantaneous event rate among individuals who remain event-free up to that time. When comparing populations or experiments, check whether definitions, denominators and observation periods match. If the assumption is implausible, the result shows only what the simplified model calculates from its inputs; it is not a reliable estimate for the target population. State this limitation in the report, and do not infer a causal effect from the formula alone.

The hazard is the instantaneous event rate among individuals who remain event-free up to that time.

UniversityBachelor foundationsSubject specialization

Create flashcards or a quiz I already have an account

Free to browse · flashcards or a multiple-choice quiz · import from Essential

Formula

h (t) = Δ t \to 0 lim \frac{P ( t \leq T < t + Δ t ∣ T \geq t )}{Δ t} = \frac{f ( t )}{S ( t )}

h(t) in inverse time · f(t) in inverse time · S(t) dimensionless

Variables & units

Variables & units
Symbol	Meaning	Unit
$h(t)$	Instantaneous conditional event rate at time t	time⁻¹
$T$	Random time to event	time
$f(t)$	Event-time density	time⁻¹
$S(t)$	Probability of remaining event-free through t	dimensionless

LaTeX

h(t)=\lim_{\Delta t\to0}\frac{P(t\le T<t+\Delta t\mid T\ge t)}{\Delta t}=\frac{f(t)}{S(t)}

QUANTA STUDYquanta-study.de

Derivation & background

Hazard is a rate, not a probability, and can exceed one per time unit. Risk over a finite interval also depends on how hazard varies over time. The identity h=f/S applies to continuous event times. Censoring must be handled in estimation; a hazard ratio requires additional comparison and model assumptions.

Exam blueprint

Validity range

For continuous event times and a defined at-risk population; hazards depend on the time scale and event definition.

Derivation steps

Density divided by event-free survival gives the conditional instantaneous rate.

1Use f(t) and S(t) for the same time variable.
2Compute h=f/S and note the inverse-time unit.

Rearrangements

Solve for a quantity

S(t)=\exp\left(-int_0^t h(u),du\right)

Relation to cumulative hazard.

Task variant

Constant hazard 0.1 per month: cumulative hazard after 5 months?

H=0.1·5=0.5; S=e^-0.5≈0.607.

Common mistakes

Treating h(t) as the event probability.

Hazard is a conditional rate; interval and integration determine risk.

Exam context

Distinguishing hazard, risk and survival function.

Formula cluster

Biology and quantitative models

Hazard-ratio interpretation requires suitable model assumptions.

Ŝ(t) = Π(1 − dj/nj)

Kaplan–Meier estimator

IR = Ereignisse / Personenzeit

Incidence rate per person-time

RR = [a/(a+c)] / [b/(b+d)]

Relative Risk (Risk Ratio)

Worked example

For an exponential event time with constant rate λ=0.02 per day, h(t)=0.02/day. Over 10 days risk is 1−e^(−0.2)≈0.181, not exactly 0.2. Check: rate multiplied by time is dimensionless.

Applications

Mathematically describing instantaneous event rates in survival analyses

How do you calculate with Hazard function?

Here is how to work through a typical Hazard function (h(t) = f(t)/S(t)) task step by step:

1
Task
Constant hazard 0.1 per month: cumulative hazard after 5 months?
Solution path
H=0.1·5=0.5; S=e^-0.5≈0.607.

Frequently asked questions about Hazard function

How does hazard differ from risk?+

Risk is the probability of an event over a fixed interval. Hazard is the instantaneous conditional event rate among people still event-free. A rate has units of inverse time and is not bounded by zero and one; probability is. With constant hazard λ, risk over duration Δt is 1−e^(−λΔt). A time-varying hazard must be integrated over the interval. State event definition, time scale and risk population. The rate alone is not a probability, and a hazard ratio requires additional model assumptions. Time variation may make a constant-hazard approach unsuitable.

Can a hazard exceed 1?+

Yes, depending on the chosen time unit. A rate of 1.5 per day is mathematically possible because hazard is not a probability. It describes instantaneous event intensity among survivors. Risk over a particular duration remains between zero and one and follows from hazard over the entire interval. Always report time scale and units rather than comparing values without their time unit. State event definition, time scale and risk population. The rate alone is not a probability, and a hazard ratio requires additional model assumptions. Time variation may make a constant-hazard approach unsuitable.

How are hazard and the survival function related?+

Cumulative hazard H(t) is the integral of instantaneous hazard up to t; for continuous time, S(t)=exp(−H(t)). Conversely, h(t)=−d ln S(t)/dt. With constant hazard, S decays exponentially. A Kaplan–Meier curve estimates S nonparametrically, while hazard models describe the instantaneous rate directly. They are related but answer different interpretive questions and require appropriate handling of censoring. State event definition, time scale and risk population. The rate alone is not a probability, and a hazard ratio requires additional model assumptions. Time variation may make a constant-hazard approach unsuitable.

How do I apply Hazard function to measured data?+

Reproduce the example calculation: For an exponential event time with constant rate λ=0.02 per day, h(t)=0.02/day. Hazard function uses h(t) = f(t)/S(t). In plain terms, the hazard is the instantaneous event rate among individuals who remain event-free up to that time. Unit check: h(t) in inverse time · f(t) in inverse time · S(t) dimensionless. For continuous event times and a defined at-risk population; hazards depend on the time scale and event definition. Values are comparable only when their definitions, denominators and periods match. Record these details next to the result so the reference group is clear. If the direction is implausible, check data preparation and the calculation basis before discussing biological causes.

Which model limits should I state for Hazard function?+

A key limit of Hazard function is this: For continuous event times and a defined at-risk population; hazards depend on the time scale and event definition. Do not use the metric without its reference; it is useful for Mathematically describing instantaneous event rates in survival analyses. The hazard is the instantaneous event rate among individuals who remain event-free up to that time. When comparing populations or experiments, check whether definitions, denominators and observation periods match. If the assumption is implausible, the result shows only what the simplified model calculates from its inputs; it is not a reliable estimate for the target population. State this limitation in the report, and do not infer a causal effect from the formula alone.

Quanta exam set

Curated exam set for "Hazard function":

Question (front)

Which formula describes Hazard function?

Answer in your set

Question (front)

How do you rearrange Hazard function for Solve for a quantity?

Answer in your set

Question (front)

Which common mistake happens with Hazard function?

Answer in your set

+ 7 more cards: units, variables, derivation, example, exam task

The formula set contains 10 cards. After importing, review them and start practicing; your ratings determine later review intervals.

Create flashcards or a quiz I already have an account

Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]Lin, X. & Halabi, S. (2022). Finding the right hazard function for time-to-event modeling. Cancer Research.

Related formulas

OR = \frac{a/c}{b/d} = \frac{ad}{bc}

Odds Ratio

More formulas: Biostatistics and epidemiology

\mathrm{Sensitivitaet}=\frac{TP}{TP+FN}, \qquad \mathrm{Spezifitaet}=\frac{TN}{TN+FP}

Diagnostic Test Sensitivity and Specificity

\hat{p}=\frac{x}{n},\quad p_{L,U}=\frac{\hat{p}+\frac{z^2}{2n}\mp z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}+\frac{z^2}{4n^2}}}{1+\frac{z^2}{n}}

Wilson Score Confidence Interval for a Proportion

d = \frac{\bar{x}_1-\bar{x}_2}{s_p}, \quad s_p=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}

Cohen's d Effect Size

AUC=\int_0^1 \text{Sensitivität}(FPR)\,d(FPR)

ROC area under the curve

Spaced-repetition apps

Compare scheduling workflows

Study evidence

Read methods, results and limits

Compare flashcard apps

FSRS support and study features

Create a formula exam set

From Essential in Quanta

Retain Hazard function for exams

Create a curated FSRS exam set for h(t) = f(t)/S(t): formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Create flashcards or a quiz Already registered? Continue directly

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential

Quanta Study editorial team

Symbol

Meaning

Unit

h(t)

Instantaneous conditional event rate at time t

time⁻¹

T

Random time to event

time

f(t)

Event-time density

time⁻¹

S(t)

Probability of remaining event-free through t

dimensionless

Derivation & background

Exam blueprint

Validity range

For continuous event times and a defined at-risk population; hazards depend on the time scale and event definition.

Derivation steps

Density divided by event-free survival gives the conditional instantaneous rate.

1Use f(t) and S(t) for the same time variable.
2Compute h=f/S and note the inverse-time unit.

Frequently asked questions about Hazard function

How does hazard differ from risk?+

Can a hazard exceed 1?+

How are hazard and the survival function related?+

How do I apply Hazard function to measured data?+

Which model limits should I state for Hazard function?+