A Worked Example With Two Constant Cause-Specific Hazards

From Cause-Specific Rates to an Overall Rate

When a first event can occur for one of two mutually exclusive reasons, each cause has its own cause-specific hazard. The hazards describe rates among people who are still event-free, not probabilities among everyone who began follow-up. This distinction matters when interpreting a reported rate or using it to calculate how quickly first events occur overall.

This tutorial works through a population model in which the two cause-specific hazards are constant over time. The main calculation is simple: add the cause-specific hazards to obtain the overall first-event hazard. The interpretation requires more care, especially when translating a rate into statements about the fraction of people who experience each cause by a particular time.

A hazard has units of inverse time, such as events per person-year. It is not itself a probability. Informally, over a sufficiently short interval of length \(\Delta t\), a cause-specific hazard \(\lambda_A(t)\) corresponds to a conditional probability of a cause-\(A\) event of approximately \(\lambda_A(t)\Delta t\), provided that interval is short enough for the hazard to be nearly constant. The exact conditional probability over a longer interval depends on the hazards throughout that interval.

Assumptions for the Worked Calculations

The examples use a population in which every person starts event-free, and the first event must be either cause \(A\) or cause \(B\). The causes are mutually exclusive: a person can have at most one of them as their first event. Assume continuous event times, so simultaneous first events have probability zero, and suppose both cause-specific hazards are constant during the follow-up period being considered. A constant hazard means the cause-specific rate among those still event-free does not change with time; it does not mean the number of event-free people stays fixed.

These are stipulated population rates for the examples, not estimates from a sample. Therefore no standard errors, confidence intervals, or tests are needed. In an application, the constant-hazard assumption would need to be supported by the study design and data, rather than inferred just because a single rate has been reported.

Why the Two Rates Add

Over a very short interval, a first event can be from \(A\) or from \(B\), but not both. The probability of any first event is therefore the sum of the two disjoint cause-specific event probabilities. Dividing by the interval length and taking the small-interval limit gives the overall hazard.

When the hazards are constant, write \(\lambda_A\) and \(\lambda_B\) for their values. Then \(\lambda=\lambda_A+\lambda_B\) is constant as well. This describes the instantaneous rate of any first event among the people who remain event-free at each time. As events accumulate, the event-free group can become smaller even though its hazard remains constant.

What the Constant-Hazard Model Implies Over Time

The finite-time implication can be derived directly. Let \(S(t)\) be the probability that a person who starts event-free has had no first event by time \(t\). Over a short interval, the conditional probability of any first event is approximately \(\lambda\Delta t\), so the probability of remaining event-free is approximately \(1-\lambda\Delta t\). Thus \(S(t+\Delta t)\) is approximately \(S(t)(1-\lambda\Delta t)\). Taking the limit gives the differential equation \(S'(t)=-\lambda S(t)\), with \(S(0)=1\), whose solution is \(S(t)=\exp(-\lambda t)\).

For cause \(A\), the probability of a cause-\(A\) first event in a short interval after time \(t\) is approximately \(S(t)\lambda_A\Delta t\): the person must still be event-free, and then the cause-\(A\) hazard applies. Integrating this rate over follow-up gives the cumulative incidence for cause \(A\). With constant hazards, the resulting expression is:

The total probability of any first event by time \(\tau\) is \(1-\exp[-(\lambda_A+\lambda_B)\tau]\). Dividing \(F_A(\tau)\) by that total gives \(\lambda_A/(\lambda_A+\lambda_B)\) for \(\tau>0\). Therefore, because the hazards are constant here, 60% of first events by any specified follow-up time are from cause \(A\). The probability that an initially event-free person has cause \(A\) by that time is \(0.6(1-\exp(-0.05\tau))\), not 60%.

Checks, Failure Modes, and Alternatives

A short set of checks catches many errors in this calculation. Confirm that both cause-specific rates use the same time unit before adding them. Confirm that they refer to mutually exclusive first-event causes and to the event-free risk set. Finally, check that the total hazard is no smaller than either component and equals their sum. If either cause-specific hazard varies over time, the hazards still add at each time, but a single constant overall rate cannot represent the whole follow-up.

• Adding probabilities instead of rates. Cause-specific hazards are rates. Exact probabilities over a finite interval are not generally found by simply adding hazards or multiplying a hazard by the entire interval length.
• Using the wrong denominator. Each cause-specific hazard is conditional on being event-free just before the time in question. It is not a rate among all people initially enrolled regardless of whether they have already had an event.
• Mixing units. A rate per month cannot be added directly to a rate per year. Convert both to the same unit first.
• Overextending the cause share. A 60% share of first events does not say that 60% of the initial population experiences cause \(A\) by a specified finite time. The finite-time probability also depends on the total event rate and the length of follow-up.

The overall hazard is useful when the question concerns the instantaneous rate of any first event. Keeping cause-specific hazards separate is useful when the event type matters. Neither rate alone is a finite-time event probability. The constant-hazard formulas above provide a model-based translation between rates and probabilities; when hazards vary with time, finite-time probabilities require accounting for that time variation rather than inserting one average rate without justification.