Why the Overall Rate Is a Sum
When several mutually exclusive events can end follow-up, the overall first-event rate describes how quickly any of them occurs. The cause-specific hazards separate that total rate into its possible types. The key is to define every rate using the same people still event-free just before time \(t\). With a shared denominator and exhaustive cause labels, the overall first-event hazard is the sum of the cause-specific hazards.
This is an identity about instantaneous hazards, not a claim that cause-specific cumulative incidences add as rates, or that a hazard is itself a probability. The distinction between cause-specific and subdistribution hazards is important here: cause-specific hazards use the event-free risk set, whereas a subdistribution hazard has a different risk-set definition.
For example, if the possible first events are relapse, death before relapse, and treatment discontinuation, each cause-specific hazard concerns individuals still free of all three events immediately before \(t\). It does not use a different risk set for each cause. An individual who has already experienced any first event is no longer at risk for a subsequent event to be counted as the first event.
Deriving the Sum Identity
Assume there are a finite number \(K\) of possible first-event causes, each first event receives exactly one cause label, and the causes cover all first events under consideration. At a given \(t\), the events “cause \(k\) occurs in \([t,t+\Delta t)\)” are mutually exclusive across \(k\). Therefore, conditional on being event-free at \(t\), the probability of any first event in that interval is the sum of the probabilities for its possible causes:
Divide by \(\Delta t\) and take the limit as \(\Delta t\) approaches zero. Since the sum has finitely many terms, the limit of the sum equals the sum of the limits, yielding the result.
The identity does not require the cause-specific hazards to be constant over time, or to be equal to one another. It is a consequence of partitioning the possible first events. Its assumptions matter: an omitted cause means the listed hazards sum only to the rate for the listed causes, while inconsistent denominators or time units prevent a valid comparison. If simultaneous or ambiguous event types are possible, they must be handled consistently before the causes can be treated as a mutually exclusive classification.
Worked Examples
Worked Example: Adding Three Instantaneous Hazards
Suppose, hypothetically, that at month 6 the cause-specific hazards for three first events are \(0.020\), \(0.030\), and \(0.010\) per month. Find the overall first-event hazard and the instantaneous shares attributable to each cause.
Assumption check. Suppose the three causes are mutually exclusive and exhaustive, and all three hazards are defined among the same individuals who are still event-free just before month 6. Their units are all per month.
By the sum identity, $$ \lambda_{\mathrm{all}}(6)=0.020+0.030+0.010=0.060\ \text{per month}. $$
The cause-specific hazard divided by the overall hazard gives each cause’s share of the instantaneous total rate: \(0.020/0.060=0.3333\) for cause 1, \(0.030/0.060=0.5000\) for cause 2, and \(0.010/0.060=0.1667\) for cause 3. These are shares of the total instantaneous hazard, not probabilities that a particular individual will experience each cause over a finite period.
Worked Example: Verifying the Sum With Person-Time Rates
In a hypothetical interval, a group contributes 1,000 person-months while members are still free of any first event. During that event-free follow-up, 12 first events of cause A, 8 of cause B, and 5 of cause C occur. Calculate the cause-specific incidence rates and compare their sum with the pooled rate for any first event.
Assumption check. The reported person-time is the same event-free person-time for all three causes, every first event is assigned to exactly one of A, B, or C, and counts refer only to first events. These conditions make the rates comparable and the total count the sum of the cause counts.
The cause-specific person-time rates are: $$ \widehat{\lambda}_A=\frac{12}{1000}=0.012,\qquad \widehat{\lambda}_B=\frac{8}{1000}=0.008,\qquad \widehat{\lambda}_C=\frac{5}{1000}=0.005 \quad\text{per person-month}. $$
Their sum is \(0.012+0.008+0.005=0.025\) first events per person-month. Directly pooling the events gives the same result: $$ \widehat{\lambda}_{\mathrm{all}}=\frac{12+8+5}{1000} =\frac{25}{1000}=0.025\ \text{per person-month}. $$
Here the equality follows exactly from the shared denominator and the addition of the event counts. These are rates averaged over the observed interval; they need not equal an instantaneous hazard at every time in that interval if the underlying hazards change over time.
Worked Example: From Three Hazards to Short-Interval Probabilities
Suppose, hypothetically, that three cause-specific hazards are constant over the next \(0.1\) year: \(0.040\), \(0.025\), and \(0.015\) per year. Find the overall hazard and the probability of each first-event type during the interval, conditional on being event-free at its start.
Assumption check. The hazards are constant over this interval, use the same event-free risk set, and correspond to mutually exclusive and exhaustive first-event causes. Under a constant overall hazard, the probability of remaining event-free for an interval of length \(\Delta t\) is \(\exp(-\lambda_{\mathrm{all}}\Delta t)\).
First add the cause-specific hazards: $$ \lambda_{\mathrm{all}}=0.040+0.025+0.015=0.080\ \text{per year}. $$ The probability of any first event during \(0.1\) year is $$ 1-\exp(-0.080\times0.1)=1-\exp(-0.008)=0.007968. $$
With constant cause-specific hazards, the probability that the first event is cause \(k\) in this interval is its share of the total hazard multiplied by the probability of any first event. The shares are \(0.040/0.080=0.5\), \(0.025/0.080=0.3125\), and \(0.015/0.080=0.1875\). Thus: $$ P(\text{cause 1 first event})=0.5(0.007968)=0.003984, $$ $$ P(\text{cause 2 first event})=0.3125(0.007968)=0.002490, $$ $$ P(\text{cause 3 first event})=0.1875(0.007968)=0.001494. $$
The probabilities sum to \(0.007968\), the probability of any first event. For a short interval, the first-order approximation for any event is \(\lambda_{\mathrm{all}}\Delta t=0.080(0.1)=0.008\), close to but not identical to the exact probability \(0.007968\). This illustrates why a hazard, measured per unit time, should not be reported as though it were a finite-interval probability.
Diagnostics, Alternatives, and Limits
A practical check is to write the denominator and time unit beside every rate before adding anything. If one cause-specific rate is based on people still free of all first events but another uses a different population, their sum does not verify the overall hazard identity. Likewise, a rate per month cannot be added directly to a rate per year. Convert units first; for example, \(0.06\) per month is \(0.72\) per year when expressed on the same time scale.
For observed data, cause-specific event counts divided by common event-free person-time provide a pooled-rate check, as in the second example. When rates vary over time, however, this pooled calculation averages over the interval and does not establish that the instantaneous hazards sum to a particular constant at every time. The identity itself is pointwise in time.
A related alternative is to describe the probability of a particular cause by a cumulative incidence over a specified follow-up period. That is a probability accumulated over time, whereas a cause-specific hazard is an instantaneous rate among those still event-free. The sum identity applies to hazards at each time; it should not be used to treat a hazard as a cumulative probability. The earlier material on subdistribution hazards uses a different risk-set definition, so the sum identity for cause-specific hazards should not be transferred automatically to subdistribution hazards.
The sum identity is also limited to the event types included in the classification. If a relevant first-event type is omitted, the sum of the listed cause-specific hazards is not the overall hazard for all first events. Before interpreting a total, check that the causes are exhaustive, that event labels are mutually exclusive, and that all rates refer to the same event-free population and time scale.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- At time \(t\), three exhaustive cause-specific hazards are \(0.012\), \(0.018\), and \(0.010\) per year. What is the overall hazard?
- In a hypothetical study, two causes produce 9 and 6 first events over the same 500 person-weeks of event-free follow-up. What are the two rates and the pooled overall rate?
- Why can the cause-specific hazards be added when the causes are mutually exclusive, but not when they use different risk-set denominators?
- With a constant overall hazard of \(0.10\) per year, calculate the probability of any first event within \(0.2\) year, conditional on being event-free at the start.
Solutions. (1) The overall hazard is \(0.012+0.018+0.010=0.040\) per year. (2) The rates are \(9/500=0.018\) and \(6/500=0.012\) events per person-week; their sum is \(0.030\) per person-week, equal to the pooled rate \((9+6)/500=15/500=0.030\). (3) With mutually exclusive causes and a shared event-free risk set, the cause-specific probabilities partition the probability of any first event over a short interval, so their limits add. Different denominators do not describe that same partition and cannot establish the identity. (4) The probability is \(1-\exp(-0.10\times0.2)=1-\exp(-0.02)=0.01980\), or about \(1.98\%\), conditional on being event-free at the interval’s start.