Cause-Specific Hazards Are Not Event Probabilities

Why the Distinction Matters

A cause-specific hazard answers a conditional, time-local question: among individuals who are still event-free at time $t$, how quickly are events of cause $k$ occurring at that time? An event probability answers a different question: what fraction of the original group experiences cause $k$ by a specified time? The first is a rate; the second is a probability. Confusing them can produce incorrect numerical interpretations and comparisons.

For example, a cause-specific hazard of $0.08$ per year is not an 8% probability of the event by the end of the year. It describes an instantaneous rate among people still at risk. To obtain an event probability over time, we must account for both how the rate changes and how many people remain event-free and able to experience the event.

Definition: Let $T$ be the time to the first event and $J$ its cause. For a particular cause $k$, the cause-specific hazard at time $t$ is $$ \lambda_k(t)=\lim_{\Delta t\downarrow 0} \frac{P(t\leq T<t+\Delta t,\ J=k\mid T\geq t)}{\Delta t}, $$ when the limit exists. It is the instantaneous rate of a cause-$k$ event among individuals who are event-free just before $t$. Its units are inverse time, such as events per year; it is not itself a probability.

The risk set in this definition contains individuals who have not yet had any first event by time $t$. People who previously experienced a competing cause are no longer event-free and are not in that risk set. The earlier material on the conditional meaning of a cause-specific hazard gives the interpretation of this conditioning; here the key point is that a conditional rate cannot be read as an unconditional probability for the original cohort.

From a Rate to a Short-Interval Chance

For a sufficiently short interval of length $\Delta t$, the definition gives the approximation

P(t\leq T<t+\Delta t,\ J=k\mid T\geq t) \approx \lambda_k(t)\Delta t.

The units make the conversion visible: inverse years multiplied by years gives a dimensionless probability. The approximation is local. It is most useful when the interval is short and the hazard does not change greatly within it. For a longer interval, simply multiplying the starting hazard by the interval length generally does not give the event probability.

The probability on the left is conditional on being event-free at the start of the interval. It is not the probability that a randomly selected member of the original cohort experiences cause $k$ during that interval. If only a fraction of the original cohort is still event-free at $t$, the cohort-level probability also depends on that fraction.

Worked Example: Converting a Short-Interval Rate

In a hypothetical follow-up, 800 people are event-free at year 2. During the next $0.1$ year, 6 experience a first event of cause $k$. Use these counts to estimate the cause-specific hazard over this short interval, and distinguish the rate from the observed conditional probability.

Conditional interval probability. Among the 800 people at risk at year 2, the observed proportion with cause $k$ in the interval is

\frac{6}{800}=0.0075=0.75\%.

Rate estimate. Dividing the interval probability by the interval length gives the short-interval rate estimate

\widehat{\lambda}_k \approx \frac{0.0075}{0.1\ \text{year}} =0.075\ \text{per year}.

The estimate is about $0.075$ events per person-year among those event-free at the start of the interval. The $0.75\%$ is the observed conditional probability over the $0.1$-year interval; $0.075$ per year is a rate. They are related by the interval length, not interchangeable. This calculation is a local approximation because the interval has positive length and the data alone do not establish that the hazard was constant within it.

Why the Probability by a Time Point Is Different

Let $S(t)=P(T\geq t)$ be the probability of remaining event-free through time $t$, and let $F_k(\tau)=P(T\leq \tau,\ J=k)$ be the probability that a member of the original cohort has a first event of cause $k$ by time $\tau$. In a small interval beginning at $t$, the cohort-level probability of a cause-$k$ event is the probability of still being event-free at $t$, multiplied by the conditional probability of cause $k$ in that interval. Consequently, for a continuous event-time distribution,

F_k(\tau)=\int_0^\tau S(u)\lambda_k(u)\,du.

This identity expresses the difference between a rate and an accumulated probability. The hazard $\lambda_k(u)$ is conditional on being event-free at $u$; the factor $S(u)$ converts that conditional rate into a contribution on the original-cohort scale. The integral accumulates those contributions over time. Thus, a cause-specific hazard by itself is not generally enough to determine the event probability by $\tau$: the event-free probability over that period also matters.

A Short Derivation

The identity follows by considering a small time interval. The probability that an individual is event-free at $u$ is $S(u)$. Conditional on being event-free then, the probability of a cause-$k$ event between $u$ and $u+\Delta u$ is $\lambda_k(u)\Delta u+o(\Delta u)$, where $o(\Delta u)/\Delta u$ tends to zero as the interval shrinks. Multiplying gives the unconditional interval contribution:

P(u\leq T<u+\Delta u,\ J=k) =S(u)\{\lambda_k(u)\Delta u+o(\Delta u)\}.

Divide the interval into small pieces and add these probabilities. In the limit, the sum becomes the integral of $S(u)\lambda_k(u)$. This argument assumes that event times have no point masses over the interval and that the hazards and event-free probability are sufficiently well behaved for the integral to exist. The result does not assert that the cause-specific hazard is itself an event probability; it shows how the rate contributes to one.

Worked Example: A Constant Rate in a One-Cause Setting

Suppose, hypothetically, that there is only one possible event cause, the event-free hazard is constant at $0.4$ per year, and follow-up begins with everyone event-free. Find the probability of the event by 2 years. The one-cause assumption matters: this calculation does not determine a competing-risks event probability from one cause-specific hazard alone.

With a constant hazard and no competing cause, the event-free probability at time $t$ is $S(t)=e^{-0.4t}$. Therefore,

F(2)=\int_0^2 0.4e^{-0.4u}\,du =1-e^{-0.8} \approx 0.5507.

The event probability by 2 years is about $55.1\%$, not $0.4$, and not $0.4\times 2=0.8$. The hazard remains $0.4$ per year, while the event probability accumulates as the event-free group gets smaller. The multiplication $0.4\times2$ is not a valid finite-time probability calculation.

Worked Example: Same Rate, Different Cohort-Level Interval Probabilities

Consider two hypothetical cohorts at year 1. In each cohort, the cause-$k$ hazard over the next $0.1$ year is approximately $0.10$ per year. In cohort A, $90\%$ of the original group remains event-free at year 1; in cohort B, $50\%$ remains event-free. Estimate the probability, measured from each original cohort, of a cause-$k$ event during the next interval.

First convert the conditional rate into the short-interval conditional probability:

0.10\ \text{per year}\times 0.1\ \text{year}=0.01.

This is approximately a $1\%$ chance among individuals still event-free at year 1. For cohort A, the original-cohort probability is approximately

0.90\times 0.01=0.009=0.9\%.

For cohort B, it is approximately

0.50\times 0.01=0.005=0.5\%.

The conditional cause-specific rate is the same in both cohorts, but the approximate numbers of cause-$k$ events per original cohort differ because different fractions remain event-free at the start of the interval. The calculation is local: it uses the stated short-interval hazard approximation and the event-free fractions at year 1.

Common Mistakes and Diagnostics

Calling a hazard a percentage. A hazard of $0.10$ per year is not automatically a $10\%$ chance by one year. State the inverse-time units and specify the time interval and conditioning set when giving a probability.
Multiplying a rate by a long follow-up time. The product may exceed 1, which is impossible for a probability. Even if the product is below 1, it may still be a poor probability estimate because the hazard and event-free group can change over time.
Dropping the conditioning. A cause-specific hazard describes people still event-free at the time in question. It is not directly the chance for every person originally enrolled.
Assuming one cause-specific hazard determines a competing-risks event probability. The probability by a given time also depends on how many people remain event-free. In particular, the rate for cause $k$ does not tell us, by itself, how much of the original cohort is still at risk.
Ignoring interval width in a numerical estimate. Dividing an observed interval proportion by the interval length estimates a local rate, but the approximation is less defensible when the interval is wide or the hazard changes within it.

A useful diagnostic is to write down the target before calculating: is it a conditional rate among the event-free risk set, a conditional probability over a short interval, or an unconditional probability from the original cohort by a specified time? Then check the units. A probability is dimensionless and lies between 0 and 1; a hazard has inverse-time units and need not lie below 1.

Choosing a Quantity for the Question

Cause-specific hazards are useful for describing how event rates vary over time among those still event-free. They are not a substitute for reporting the probability of experiencing a particular cause by a clinically meaningful time. When the question concerns absolute event probability, report a probability such as $F_k(\tau)$, with its time horizon and population clearly stated. Estimating that probability requires accounting for the event-free experience as well as the cause-specific rate.

One tempting alternative is to treat competing events as though they were ordinary loss to follow-up and interpret a cause-specific rate as if it directly gave the observed probability of cause $k$. That changes the target: the observed cause-specific event probability concerns the real cohort, including the fact that a competing first event removes someone from being event-free. The appropriate summary depends on the scientific question, but a rate and a probability should never be labelled as though they were the same quantity.

Key takeaway: A cause-specific hazard is an instantaneous rate among individuals still event-free. Multiplying it by a short interval gives an approximate conditional interval probability. A probability by a specified time is accumulated on the original-cohort scale and must account for the probability of remaining event-free over time.

Check Your Understanding

For each question, identify whether it asks about a rate or a probability, then compare your reasoning with the solutions.

A cause-specific hazard is $0.06$ per month. What is the approximate conditional probability of that cause during the next $0.2$ month, assuming the interval is short and the hazard is roughly stable?
Why is $0.06$ per month not the probability of the cause by the end of one month?
Two cohorts have the same cause-specific hazard over a short interval, but different proportions remain event-free at the interval's start. Can their probabilities of a cause-specific event in the original cohorts differ? Explain.
In a one-cause setting with constant hazard $0.3$ per year and no competing cause, calculate the event probability by 1 year.

Solutions. (1) The approximation is $0.06\ \text{per month}\times0.2\ \text{month}=0.012$, or $1.2\%$, conditional on being event-free at the interval's start. (2) The hazard is an instantaneous rate with units of inverse time; a one-month probability requires accounting for how the rate and event-free group behave throughout that month. (3) Yes. The original-cohort interval probability is approximately the event-free proportion at the start multiplied by the conditional interval probability, so different event-free proportions can produce different cohort-level probabilities. (4) Under the stated one-cause constant-hazard assumptions, $F(1)=1-e^{-0.3}\approx0.2592$, or $25.9\%$ by 1 year.