Why Cause-Specific Hazards Sum To The Overall Hazard

Why an Overall Hazard Has a Cause-Specific Decomposition

When a first event can occur for several mutually exclusive reasons, it is natural to ask how the rate of any first event relates to the rates for its individual causes. The key is that each cause-specific hazard is defined among the same people who are still event-free at time $t$. Over a short interval, the possible causes divide the event “a first event occurs” into disjoint pieces. Their probabilities therefore add; taking the small-interval limit gives the hazard identity.

This is an identity between instantaneous rates under clearly stated conditions. It does not say that cause-specific hazards are event probabilities, nor that hazards can be added across analyses that use different risk sets or different event definitions.

Definition: Let $T$ be the time to the first event, and let $J$ identify its cause, taking values in $\{1,\ldots,K\}$. Suppose these $K$ causes are mutually exclusive and exhaustive for first events. The overall first-event hazard at time $t$ is $$ \lambda(t)=\lim_{\Delta\downarrow 0} \frac{P(t\leq T<t+\Delta\mid T\geq t)}{\Delta}, $$ when the limit exists. The cause-specific hazard for cause $j$ is $$ \lambda_j(t)=\lim_{\Delta\downarrow 0} \frac{P(t\leq T<t+\Delta,\ J=j\mid T\geq t)}{\Delta}. $$ Each hazard has units of inverse time. The conditioning event $T\geq t$ means that the person is in the event-free state at time $t$.

The definitions use a continuous-time hazard: at the times considered, the relevant limits must exist, and the event-time distribution must not have a point mass at $t$ that would require a separate treatment of an instantaneous jump. The causes must refer to the same first-event time $T$ and the same event-free population. In particular, the cause label is assigned to the first event, not to a later event that might occur after the first one.

“Mutually exclusive” means that a first event cannot simultaneously be counted under two causes. “Exhaustive” means that every first event belongs to one of the listed causes. These properties are what let the cause-specific event statements form a partition. If some first events have an unknown or unrecorded cause, then the recorded causes alone are not exhaustive unless that possibility is included as another category.

Deriving the Sum Identity

Fix a time $t$ and a short interval of length $\Delta$. Conditional on being event-free at $t$, the event that a first event occurs in $[t,t+\Delta)$ can happen through exactly one of the $K$ causes. The cause-specific events are disjoint, so their conditional probabilities add exactly:

P(t\leq T<t+\Delta\mid T\geq t) = \sum_{j=1}^{K}P(t\leq T<t+\Delta,\ J=j\mid T\geq t).

Divide both sides by $\Delta$, then let $\Delta$ decrease to zero. The left-hand side converges to $\lambda(t)$; each term on the right converges to its corresponding $\lambda_j(t)$. Since the number of causes $K$ is finite, the limit of this finite sum is the sum of the limits. Thus:

\lambda(t)=\sum_{j=1}^{K}\lambda_j(t).

This is the overall-hazard identity. The common conditioning event matters: every term concerns people event-free at $t$. It does not condition separately on having avoided each cause up to $t$, nor does it place people with a prior first event back at risk for another first event.

Theorem: If the possible causes form a finite, mutually exclusive and exhaustive partition of all first events, and the overall and cause-specific continuous-time hazards exist using the same event-free condition $T\geq t$, then, at times where the hazards are defined, $$ \lambda(t)=\sum_{j=1}^{K}\lambda_j(t). $$ The equality is pointwise in time; it does not require the cause-specific hazards to be constant over time.

A short-interval version makes the limit argument more explicit. When the hazards are well-defined at $t$, each cause-specific conditional probability has first-order behavior

P(t\leq T<t+\Delta,\ J=j\mid T\geq t) =\lambda_j(t)\Delta+o(\Delta).

Here $o(\Delta)$ denotes a remainder that, divided by $\Delta$, tends to zero as $\Delta\downarrow 0$. Adding the $K$ expressions gives $\Delta\sum_j\lambda_j(t)+o(\Delta)$, because a finite sum of terms that are $o(\Delta)$ is still $o(\Delta)$. Dividing by $\Delta$ and taking the limit yields the same identity.

Worked Examples

Worked Example: Adding Three Cause-Specific Rates

Suppose, hypothetically, that at a particular time $t$, three exhaustive first-event causes have cause-specific hazards of $0.012$, $0.008$, and $0.005$ per person-year. Assume all three hazards are defined among the same individuals who are event-free at $t$. Find the overall first-event hazard.

The causes are stated to be exhaustive and mutually exclusive, and the risk-set condition is shared. The sum identity therefore applies:

\lambda(t)=\lambda_1(t)+\lambda_2(t)+\lambda_3(t) =0.012+0.008+0.005 =0.025\ \text{per person-year}.

Thus, at time $t$, the overall instantaneous first-event hazard is $0.025$ per person-year. This is a rate, not a statement that $2.5\%$ of people will have an event during a year. Converting hazards into event probabilities over a non-infinitesimal period requires additional calculation; the number alone is not that probability.

Worked Example: Summing Conditional Probabilities Over a Short Interval

In hypothetical data, consider people event-free at time $t$. Over the next $0.05$ years, suppose the conditional probability of a first event from cause 1 is $0.014$, from cause 2 is $0.009$, and from cause 3 is $0.002$. Assume these causes are mutually exclusive and include every first event in the interval. Find the total conditional event probability and the corresponding interval rates.

Because the cause-specific events partition first events, the conditional probability of any first event in this interval is

$$ 0.014+0.009+0.002=0.025. $$

Dividing each cause-specific interval probability by the interval length gives rates for this particular interval:

\frac{0.014}{0.05}=0.28,\qquad \frac{0.009}{0.05}=0.18,\qquad \frac{0.002}{0.05}=0.04 \quad\text{per year}.

Their sum is $0.28+0.18+0.04=0.50$ per year, which also equals $0.025/0.05=0.50$ per year. The arithmetic is exact for these interval probabilities. However, $0.50$ per year is an interval probability divided by interval length, not necessarily the instantaneous hazard at $t$. The equality with the hazard identity is obtained in the limit as the interval shrinks, provided the hazards exist.

Worked Example: Finding a Missing Cause Category

Suppose, hypothetically, an analyst reports an overall first-event hazard of $0.25$ per year at time $t$, and two cause-specific hazards of $0.12$ and $0.08$ per year. All quantities are supposed to use the same event-free risk set. If the listed causes were exhaustive, would the values be consistent?

The two listed hazards sum to

0.12+0.08=0.20\ \text{per year}.

This is $0.05$ per year less than the reported overall hazard of $0.25$ per year. Under the stated common-risk-set condition, the discrepancy means that at least one premise needs checking: perhaps a first-event cause was omitted, the overall and cause-specific quantities were estimated on different risk sets, or the reported values use different definitions or time scales.

If the causes are mutually exclusive and the only issue is an omitted category, its hazard would have to be $0.25-0.20=0.05$ per year. Including that category gives $0.12+0.08+0.05=0.25$ per year. This arithmetic identifies what would reconcile the figures; it does not establish that an omitted cause is actually the explanation.

What the Identity Does—and Does Not—Say

The identity is a decomposition of an instantaneous rate among those who have not yet experienced a first event. It is not a claim that separate cause-specific hazards describe independent processes, and no such independence condition is part of the derivation. The argument uses only a partition of first events and a shared conditioning event.

It is also useful to distinguish two related statements. Over any fixed interval, mutually exclusive and exhaustive cause-specific conditional event probabilities add exactly to the conditional probability of any event in that interval. Dividing those probabilities by the interval length preserves that arithmetic. But a fixed-interval probability divided by interval length is not, in general, the instantaneous hazard at the interval’s start. The hazard is the limiting rate as the interval length approaches zero.

For a small $\Delta$, the cause-specific conditional event probabilities are approximately $\lambda_j(t)\Delta$. Their sum is approximately $\lambda(t)\Delta$. The approximation improves as the interval shrinks under the regularity conditions for the hazard. Treating the approximation as an exact probability for a sizeable interval is a common source of error.

Checking the Assumptions and Diagnosing Discrepancies

When a reported set of hazards does not add up, first check the definition and data structure rather than changing the identity. The relevant questions are:

Are the causes exhaustive? Look for unclassified, unknown, or otherwise omitted first events. If those events are part of the overall event definition, they must be represented in the cause partition for the sum to cover them.
Are the causes mutually exclusive? A first event counted in two cause categories violates the partition and can lead to double-counting.
Is the conditioning risk set the same? Every cause-specific hazard in this identity conditions on being event-free at the same time $t$. Quantities calculated among different populations are not terms in this sum.
Do the units and time scales match? A rate per year cannot be added directly to a rate per month. Convert units before comparing values.
Are the reported values exact or rounded estimates? Rounded numbers may not add exactly even when the underlying quantities do. With estimated hazards, sampling variability and estimation choices can also produce discrepancies.
Does the continuous-time definition apply? If event times have point masses or are only recorded in discrete intervals, the continuous-time limiting argument may not describe the data without modification.

In practice, observed data may be subject to censoring and may contain few events in some categories. Those features affect estimation and precision; they do not alter the mathematical identity when the hazards are defined under the stated conditions. Estimates should be compared with their definitions, time scale, risk set, and uncertainty in view.

Comparison With Other Summaries

Quantity	What it describes	Relation to the identity
Overall first-event hazard	The instantaneous rate of any first event among people event-free at $t$	Equals the sum of the cause-specific hazards under the partition conditions
Cause-specific hazard	The instantaneous rate of a particular first-event cause among people event-free at $t$	Each cause contributes one term to the sum
Conditional event probability over a fixed interval	The probability of an event within that interval, given event-free status at its start	Cause-specific probabilities add exactly if they partition events; dividing by interval length does not generally produce the instantaneous hazard

The cause-specific decomposition answers a local rate question. It does not, by itself, report the probability of experiencing a particular cause by a later time. That is a different summary and should not be inferred by treating a cause-specific hazard as a probability.

Common Mistake

Wrong reasoning: “The cause-specific hazards add to the overall hazard, so the probability of each cause over the next year is just its hazard, and the cause probabilities add to the overall hazard.”

Correction: Hazards have units of inverse time; probabilities do not. The sum identity concerns instantaneous rates at a given time and a shared event-free risk set. Over a fixed interval, cause-specific conditional event probabilities add to the probability of any first event, but those probabilities must not be confused with the instantaneous hazards.

Key takeaway: If first-event causes form a finite, mutually exclusive and exhaustive partition, then the probability of any first event over a short interval is the sum of the cause-specific probabilities over that interval. With a common event-free risk set and well-defined continuous-time limits, this gives $\lambda(t)=\sum_{j=1}^{K}\lambda_j(t)$.

Check Your Understanding

Answer each question, then compare with the solutions below.

At time $t$, two exhaustive cause-specific hazards are $0.03$ and $0.07$ per year, both defined among the same event-free population. What is the overall hazard?
Why must the causes be mutually exclusive as well as exhaustive for the partition argument?
Over an interval of $0.02$ years, the conditional probabilities of first events from two exhaustive causes are $0.004$ and $0.002$. What is the conditional probability of any first event? What is the sum of the interval rates obtained by dividing each probability by $0.02$? Is that sum necessarily the instantaneous hazard at the interval’s start?
A reported overall hazard is $0.16$ per year, while listed cause-specific hazards sum to $0.16$ per month. Give one necessary step before comparing these values.

Solutions. (1) The overall hazard is $0.03+0.07=0.10$ per year. (2) Exhaustiveness ensures every first event is represented; mutual exclusivity ensures no event is counted in more than one category. Both are needed for a partition. (3) The probability is $0.004+0.002=0.006$. The interval rates sum to $0.004/0.02+0.002/0.02=0.2+0.1=0.3$ per year. This is an interval probability divided by interval length, so it is not necessarily the instantaneous hazard at the start. (4) Convert both quantities to the same time unit before comparing them; for example, a rate per month can be expressed per year by multiplying by 12.