Reading Cause-Specific Hazard Statements Carefully

What a Cause-Specific Hazard Statement Says

A statement such as “the treatment group had a cause-specific hazard of 8 events per 100 person-years” is easy to read as an event probability. That reading is incorrect. The statement describes an instantaneous rate, conditional on being event-free up to the time in question. It does not, by itself, tell us the proportion of the original group who will experience that cause over a year or any other fixed period.

To read a cause-specific hazard carefully, keep three features in view: which event cause is being considered, when the rate applies, and who is in the risk set at that time. The risk set consists of people who have not yet experienced any of the competing first events. This conditional language matters because people leave that risk set when any first event occurs, not only when the focal cause occurs.

Definition: Let $T$ be the time to the first event and $J$ its cause, with causes indexed by $k=1,\ldots,K$. The cause-specific hazard for cause $k$ at time $t$ is $$ \lambda_k(t)=\lim_{\Delta t\downarrow 0} \frac{P(t\leq T<t+\Delta t,\ J=k\mid T\geq t)}{\Delta t}. $$ It is the instantaneous rate of a first event of cause $k$ at time $t$, conditional on having had no first event by $t$. Its units are inverse time, such as events per person-year.

The conditional probability in the numerator refers to a short interval just after $t$, given that the person is still event-free at $t$. For a sufficiently small interval, it is approximately $\lambda_k(t)\Delta t$. More precisely, the probability equals $\lambda_k(t)\Delta t+o(\Delta t)$, under the usual regularity conditions for the hazard. The rate is therefore not itself a probability: it has units of inverse time, while a probability is dimensionless and refers to a specified event and period.

The definition concerns the event process, not the observation process. In observed data, people may be censored before an event is observed. Estimating hazards from such data requires appropriate assumptions about censoring; independent censoring is a separate assumption from the definition of the cause-specific hazard. Also, assume first-event types are mutually exclusive and simultaneous first events occur with probability zero, or specify a rule for classifying ties. Continuity of event times alone does not guarantee that simultaneous events have probability zero.

Translate the Statement Before Interpreting It

A precise verbal translation should say “among those still event-free at time $t$, the instantaneous rate of a first event from cause $k$ at that time is $\lambda_k(t)$.” If a hazard is constant over time, the same rate applies at each time while people remain event-free. If it varies over time, a rate at one time does not automatically describe the rate at another.

A cause-specific hazard ratio compares these conditional instantaneous rates between groups at a specified time or under a stated model. For example, a ratio of 1.5 means that the cause-specific hazard in one group is 1.5 times that in the reference group, among those event-free at the relevant time. It does not say that the group has 1.5 times the event probability by a particular follow-up time. The risk sets can change over time and between groups, and competing causes affect how many people remain event-free.

Worked Example: Translating a Rate per Person-Time

In a hypothetical clinical cohort, suppose the cause-specific hazard for a particular first event at time $t$ is 8 events per 100 person-years. Translate the rate, and estimate the conditional probability of that event over the next 0.1 year under stated assumptions.

First convert the rate to inverse years: $$ \lambda_k(t)=\frac{8}{100}=0.08\ \text{per year}. $$

The correct translation is: among people still event-free at time $t$, the instantaneous rate of a first event from this cause is 0.08 per year. It is not a statement that 8% of the original cohort will experience the event in one year.

For a short-interval approximation, $0.08\times 0.1=0.008$, or about 0.8%, conditional on being event-free at the start of the interval. To check the approximation with competing causes, suppose, hypothetically, that over this interval the focal cause has constant hazard 0.08 per year and all competing causes together have constant hazard 0.12 per year. The total first-event rate is then 0.20 per year. Under constant rates, the conditional probability of the focal cause in 0.1 year is $$ \frac{0.08}{0.20}\left(1-e^{-0.20(0.1)}\right) =0.4(1-e^{-0.02}) \approx 0.007921. $$

That is about 0.792% among people event-free at the start of the interval. It is close to the short-interval approximation of 0.8%, but it is not identical. The calculation assumes constant cause-specific rates over the interval and mutually exclusive first-event causes with no simultaneous first events; it also makes explicit that a competing event can happen first.

Why a Hazard Does Not Determine Absolute Risk

Absolute risk asks for a probability over a specified period in a population defined at a starting time. A cause-specific hazard instead describes the instantaneous rate among the people who have avoided every first event up to the current time. The difference is consequential: people who experience a competing event can no longer experience the focal event as their first event.

A simple constant-rate calculation makes this distinction clear. Suppose there are two causes, with constant cause-specific hazards $\lambda_1$ and $\lambda_2$, and no censoring in the hypothetical event process. At time $u$, the total rate of any first event is $\lambda_1+\lambda_2$. The probability of remaining event-free through time $u$ is $e^{-(\lambda_1+\lambda_2)u}$. For a short interval after $u$, the probability of cause 1 is approximately the probability of still being event-free multiplied by $\lambda_1\,du$. Summing these increments over a follow-up period from 0 to $\tau$ gives: $$ P(T\leq \tau,\ J=1) =\int_0^\tau e^{-(\lambda_1+\lambda_2)u}\lambda_1\,du =\frac{\lambda_1}{\lambda_1+\lambda_2} \left(1-e^{-(\lambda_1+\lambda_2)\tau}\right). $$

The derivation shows why $\lambda_1$ alone cannot determine the probability of cause 1 by $\tau$: the competing hazard $\lambda_2$ also affects how many remain event-free long enough to experience cause 1. This formula is a simple constant-rate illustration, not a general method for estimating absolute risk from data.

Worked Example: Same Hazard Ratio, Different Absolute-Risk Contrast

Consider two hypothetical groups followed for one year from a common starting point. Assume constant cause-specific hazards and two mutually exclusive first-event causes, with simultaneous first events having probability zero. In group A, cause 1 has hazard 0.10 per year and cause 2 has hazard 0.10 per year. In group B, the corresponding hazards are 0.15 and 0.85 per year. Compare the cause 1 hazard and the one-year probability of cause 1.

The cause 1 hazard ratio, group B relative to group A, is $$ \frac{0.15}{0.10}=1.5. $$ Thus, while people remain event-free, group B's instantaneous cause 1 rate is 50% higher under these constant-rate assumptions.

For group A, the one-year probability of cause 1 is $$ \frac{0.10}{0.10+0.10}\left(1-e^{-(0.10+0.10)(1)}\right) =0.5(1-e^{-0.20}) \approx 0.09063. $$ That is about 9.06%.

For group B, it is $$ \frac{0.15}{0.15+0.85}\left(1-e^{-(0.15+0.85)(1)}\right) =0.15(1-e^{-1}) \approx 0.09482. $$ That is about 9.48%. The absolute difference is $0.15(1-e^{-1})-0.5(1-e^{-0.20})\approx 0.00418$, or about 0.418 percentage points. In this example a 50% higher cause-specific hazard corresponds to a much smaller relative difference in one-year cause 1 probability, because the competing hazard is much higher in group B. The calculation is hypothetical and depends on the stated constant-rate model; it does not establish a causal effect.

Check the Risk Set, Time, and Claim

A hazard statement can be technically correct and still be misleading if its risk set or time scale is left unstated. At a later time, the people still event-free may differ from those at baseline. The hazard is conditional on this changing group; it is not a rate calculated over everyone who entered the study regardless of what subsequently happened.

Worked Example: Reading a Hazard Ratio at a Specific Time

Suppose a hypothetical model reports a cause-specific hazard ratio of 1.5 for a first event in group B versus group A at two years. At that time, assume the modeled cause-specific hazards are 0.06 and 0.04 per year, respectively. Translate the result without making an absolute-risk claim.

The ratio is $0.06/0.04=1.5$. The appropriate interpretation is: at two years, among individuals in each group who have remained free of all first events until then, the modeled instantaneous rate of a first event from this cause is 50% higher in group B than in group A. The units of each hazard are per year, and the ratio itself is unitless.

This does not imply that 50% more people in group B experience the event by two years, nor does it give the probability of an event during the next year. Those claims would require an interval or follow-up horizon and information about the hazards over time, including competing causes. If the model assumes a constant hazard ratio, its interpretation applies across the modeled times; the example's numerical rates at two years alone do not establish that assumption.

Common Misreadings and Diagnostics

Turning a rate into a probability. “0.08 per year” does not mean “8% by one year.” A short-interval approximation requires a specified interval and conditions; an absolute event probability also depends on the event process over that period.
Using the wrong risk set. “Among everyone originally enrolled” is not the cause-specific hazard's conditioning statement. It is conditional on having had no first event by time $t$.
Omitting the cause. A cause-specific hazard concerns one specified cause. It is not automatically the rate of any event. With mutually exclusive causes and no simultaneous first events, the overall first-event hazard is the sum of the cause-specific hazards.
Ignoring units or the time scale. A rate per day cannot be compared numerically with a rate per year without conversion. A hazard at time since enrollment may differ from one indexed by age or another time scale.
Overreading a hazard ratio. A relative rate comparison is not an absolute risk difference, a risk ratio, or a causal effect. Causal claims require an appropriate design and assumptions beyond hazard interpretation.
Trusting an unexplained tie rule or cause label. If simultaneous first events can occur, their classification must be defined. If causes are misclassified, the hazard for the recorded cause may not represent the intended event type.

As diagnostics, check that the report identifies the focal cause, the time scale and time point or model, the units, and the risk-set conditioning. If a numerical hazard is presented as a probability, ask for the time horizon and how competing events were handled. If an estimated hazard comes from censored observations, check whether the censoring assumptions are plausible for the analysis.

The main alternative summaries answer different questions. A cumulative incidence is an absolute probability of a particular first event by a stated time; it accounts for competing first events. A subdistribution hazard uses a different risk-set construction and should not be given the cause-specific hazard's “among those still event-free” interpretation. Neither summary can be substituted for a cause-specific hazard without changing the question being answered.

Key takeaway: Read a cause-specific hazard as an instantaneous rate for one cause, conditional on no first event by the time under discussion. A rate or hazard ratio alone does not state absolute risk over a follow-up period.

Check Your Understanding

For each item, identify the correct interpretation or calculation. Solutions follow the questions.

A cause-specific hazard is 0.03 per month at time $t$. What does this say about the risk set and units?
A hazard ratio of 0.80 is reported for cause 1. Does this mean 20% fewer people experience cause 1 by the end of follow-up?
With constant cause-specific hazards of 0.04 per year for cause 1 and 0.06 per year for cause 2, what is the probability of cause 1 within one year under the two-cause model?
Why must a report specify how simultaneous first events are handled when they may occur?

Solutions. (1) At time $t$, among people who have had no first event by $t$, the instantaneous rate of a first event from cause 1 is 0.03 per month. It is not itself a probability. (2) No. It means the cause-specific instantaneous rate is 20% lower in the compared group under the stated model and time conditions; it does not directly compare absolute event probabilities. (3) The total rate is $0.04+0.06=0.10$ per year, so the probability is $$ \frac{0.04}{0.10}(1-e^{-0.10(1)}) =0.4(1-e^{-0.10}) \approx 0.03807. $$ Under the stated assumptions, this is about 3.81% within one year. (4) The cause-specific definition assigns a first event to one cause. If simultaneous first events can occur, an explicit classification rule is needed so that the event type and cause-specific rates are defined consistently.