A Checklist For Reporting Hazard Definitions

Why a Hazard Statement Needs More Than a Number

A hazard value is not a complete result by itself. To interpret it, a reader needs to know which event is being counted, when follow-up time begins, who contributes to the denominator at each time, and the units of the rate. Leaving out any one of these details can make two different quantities sound identical. For example, “the hazard of hospitalization was 0.04” does not say whether the number is a cause-specific hazard or a subdistribution hazard, whether time is measured from randomization or diagnosis, or whether 0.04 is per month or per year.

A compact reporting checklist helps keep these details visible without turning every result into a long technical explanation. The checklist is especially useful when presenting estimates, comparing groups, or summarizing model coefficients. It also guards against treating a hazard as though it were a fixed-time event probability.

Define the Event, Clock, and Denominator

Let \(T\) be the time from a stated origin to the first event, and let \(J\) identify the cause of that event. Causes refer to distinct first-event types. For a focal cause \(k\), a cause-specific hazard and a subdistribution hazard use different denominators, and therefore answer different questions.

For the cause-specific hazard, the conditioning set \(T\geq t\) contains people still free of every first-event type at time \(t\). A competing event removes a person from this event-free risk set. For the subdistribution hazard, a prior competing event does not count as the focal cause and the person remains represented in the denominator \(1-F_k(t)\). This construction supports a different estimand; it is not the ordinary instantaneous rate among people who are still physically able to experience any first event.

The time origin is the point from which \(T\) is measured, such as randomization, diagnosis, or surgery. The time scale is the unit used to measure elapsed time, such as days, months, or years. The origin determines what “at time \(t\)” means; the unit determines how to express the rate. State both when either could be unclear.

A Reporting Checklist

Before writing a hazard interpretation, check each of the following items. A short statement can include them all, either in one sentence or in a sentence followed by a brief definition.

A useful sentence pattern is: “From [time origin], the [cause-specific/subdistribution] hazard of [focal cause] at time \(t\) is [interpretation of rate], measured per [time unit], with a denominator of [conditioning set or subdistribution risk set].” A model-based comparison can be added after this definition, but it should not replace it.

Worked Interpretations

Why the Definitions Lead to These Denominators

The cause-specific denominator follows directly from a conditional probability. In a short interval after \(t\), first ask whether a person is still free of any first event at \(t\); among those who are, count the chance that cause \(k\) occurs in the interval. Dividing that conditional probability by the interval length gives a rate in the limit:

A person who has already had a competing first event cannot later have a different first event in this setup, so that person is not in the event-free risk set. This is why “among those still free of any first event” is the correct verbal denominator for the cause-specific hazard.

For the subdistribution hazard, the rate is built from the rate of increase in the focal cause's cumulative incidence, \(f_k(t)\), divided by the probability \(1-F_k(t)\) of not yet having had that focal cause. Thus the two hazard definitions use different denominators even when they refer to the same cause and time origin. A report must identify which definition is intended rather than relying on the word “hazard” alone.

Diagnostics, Alternatives, and Limits

A practical diagnostic is to ask whether each phrase in the report matches the actual analysis. Check the event coding, the stated time origin, the time scale in the data and model, and the risk-set definition used for the reported hazard. If the analysis uses the cause-specific risk set but the sentence says “including those with competing events,” the interpretation is inconsistent. If the analysis uses a subdistribution hazard but the sentence says “among people who remain free of every event,” it is also inconsistent.

The main alternative here is not a different wording for the same quantity: cause-specific and subdistribution hazards have distinct denominators. Choose the cause-specific hazard when the target is the instantaneous rate of the focal first event among people still free of all first events. Choose the subdistribution hazard when the target is the rate defined through the focal cause's cumulative incidence and its \(1-F_k(t)\) denominator. Neither statement alone is a fixed-time probability of the event.

The definitions above use continuous time and a differentiable cumulative incidence for the subdistribution formula. With genuinely discrete event times, a conditional event probability over a time interval is often the relevant quantity; it should be labeled as a probability for that interval rather than given continuous-time rate units. Tied events, cause misclassification, or an unclear event origin can also undermine the meaning of the reported quantity. Censoring assumptions affect estimation from incomplete follow-up, but censoring is not a competing event and should not be inserted into the event-cause description.

Hazard statements are descriptive or model-based rate interpretations, not automatically causal claims. Even in a randomized study, a hazard comparison should be interpreted as the stated comparison of hazards under the model, not as a direct statement about individual outcomes or a fixed-time risk difference. Reporting the denominator explicitly makes this limitation easier to see.

Common Mistake

A second warning is to avoid turning a rate into a probability by changing the wording alone. For example, \(0.05\) per person-year is not automatically a 5% probability of hospitalization over one year. The probability depends on how the hazard behaves over time and on competing events; report the rate as a rate unless a probability has been separately defined and estimated.