Instantaneous Cause Shares Among Events

From Event Rates to the Cause Mix

A cause-specific hazard measures the instantaneous rate of a particular first event among individuals who are still free of any first event. When several mutually exclusive first-event causes are possible, their hazards determine the rate of any first event. Dividing one cause-specific hazard by that overall rate gives a different, useful quantity: the instantaneous share of events attributable to that cause.

This share answers a narrowly timed question: among the first events occurring just after time $t$, what fraction are of cause $k$? It is not automatically the fraction of all events accumulated since time zero that are of cause $k$. The distinction matters when the relative cause-specific hazards change over time.

Definition: Let $T$ be the time to the first event and $J$ its cause, with possible causes $1,\ldots,K$. For individuals who are event-free just before time $t$, the cause-specific hazard for cause $k$ is $$ \lambda_k(t)=\lim_{h\downarrow 0}\frac{P(t\leq T<t+h,\ J=k\mid T\geq t)}{h}. $$ When these hazards exist, the overall first-event hazard is $\lambda(t)=\sum_{j=1}^{K}\lambda_j(t)$. If $\lambda(t)>0$, the instantaneous cause share is $q_k(t)=\lambda_k(t)/\lambda(t)$.

Why the Ratio Is an Instantaneous Share

The definition conditions on being event-free at $t$, and the cause-specific hazard describes the chance of a particular first event over a very short interval, per unit time. To see why the ratio has the stated interpretation, suppose the hazards are finite and the event-time distribution is continuous at $t$. For a small interval of length $h$, the probability of a cause-$k$ event, conditional on being event-free at $t$, is $\lambda_k(t)h+o(h)$. The probability of any first event in that interval is $\lambda(t)h+o(h)$. Therefore, when $\lambda(t)>0$,

\begin{aligned} &P(J=k\mid t\leq T<t+h,\ T\geq t)\\ &=\frac{P(t\leq T<t+h,\ J=k\mid T\geq t)} {P(t\leq T<t+h\mid T\geq t)}\\ &=\frac{\lambda_k(t)h+o(h)}{\lambda(t)h+o(h)} \longrightarrow \frac{\lambda_k(t)}{\lambda(t)} =q_k(t). \end{aligned}

Thus $q_k(t)$ is the limiting conditional proportion of first events of cause $k$ among events occurring in a shrinking interval after $t$. Because the causes are mutually exclusive and exhaustive for the first event, the shares sum to one wherever the overall hazard is positive:

\sum_{k=1}^{K}q_k(t) =\frac{\sum_{k=1}^{K}\lambda_k(t)}{\lambda(t)} =1.

The ratio is defined only when $\lambda(t)>0$. If no first events can occur at a time, there is no instantaneous event mix to describe. In continuous time the definition applies at times where the hazards and their local approximation are well behaved. If event times have point masses or a study uses discrete time intervals, use conditional cause probabilities for that interval instead of treating the infinitesimal ratio as the appropriate description.

Worked Example: A Constant Cause Mix

Consider a hypothetical population with two possible first-event causes. Suppose the cause-specific hazards are constant over time: $\lambda_1(t)=0.08$ per year and $\lambda_2(t)=0.12$ per year. The causes are mutually exclusive and together account for every first event.

The overall hazard is $0.08+0.12=0.20$ first events per person-year. The instantaneous share for cause 1 is

q_1(t)=\frac{0.08}{0.20}=0.40.

So, among first events occurring in a very short interval at any time, 40% are expected to be cause 1 and 60% cause 2. The hazards have units of events per person-year; their ratio has no units.

Here the cumulative share by one year happens to match the instantaneous share. With constant hazards, survival without either event to time $u$ is $S(u)=e^{-0.20u}$, and the probability of a cause-1 event by one year is

F_1(1)=\int_0^1 e^{-0.20u}(0.08)\,du =\frac{0.08}{0.20}(1-e^{-0.20}) =0.40(0.1813)=0.07251.

Likewise, $F_2(1)=0.12/0.20(1-e^{-0.20})=0.1088$. The probability of either event by one year is $0.07251+0.1088=0.1813$, so the cause-1 cumulative share among those events is $0.07251/0.1813=0.400$. This equality follows from the constant cause mix, not from a general rule.

Instantaneous Shares and Cumulative Shares

To define the cumulative comparison, let $F_k(t)=P(T\leq t,\ J=k)$, the probability that a first event of cause $k$ has occurred by time $t$. When at least one first event has occurred with positive probability by $t$, the cumulative share for cause $k$ is

C_k(t)=\frac{F_k(t)}{\sum_{j=1}^{K}F_j(t)}.

This is the composition of all first events accumulated from the start of follow-up through time $t$. It differs from $q_k(t)$, which describes the composition of events occurring near one particular time. For continuous event times, the cause-specific event probability accumulates according to

F_k(t)=\int_0^t S(u)\lambda_k(u)\,du, \qquad S(u)=P(T>u).

The factor $S(u)$ matters: cause-specific events at time $u$ can occur only among those still event-free then. Consequently, the cumulative share combines instantaneous cause shares across time, weighted by how many first events occur at each time. It is not generally an unweighted average of the values $q_k(u)$.

Worked Example: A Changing Cause Mix

Suppose a hypothetical population has two causes and piecewise-constant hazards, measured per year. During year 1, the hazards are $0.10$ for cause 1 and $0.10$ for cause 2. During year 2, they are $0.02$ and $0.18$, respectively. Assume no other first-event causes and continuous event times.

During year 1, the overall hazard is $0.20$, so $q_1=0.10/0.20=0.50$. The probability of remaining event-free through year 1 is $S(1)=e^{-0.20}=0.8187$. The probability of a cause-1 event during that year is

F_1(1)=\frac{0.10}{0.20}(1-e^{-0.20}) =0.5(0.1813)=0.09063.

The cause-2 probability during year 1 is also $0.09063$. In year 2, the overall hazard is again $0.20$, but the instantaneous cause-1 share is now $0.02/0.20=0.10$. Conditional on being event-free at year 1, the probability of any event during year 2 is $1-e^{-0.20}=0.1813$. The probability of cause 1 during year 2, accounting for the $0.8187$ who remain event-free at year 1, is

S(1)\frac{0.02}{0.20}(1-e^{-0.20}) =0.8187(0.10)(0.1813)=0.01484.

For cause 2 in year 2, the corresponding probability is $0.8187(0.90)(0.1813)=0.1336$. By the end of year 2, the cumulative cause-1 probability is $0.09063+0.01484=0.1055$, and the cumulative probability of either event is $0.1055+0.09063+0.1336=0.3297$. Thus the cumulative share for cause 1 is $0.1055/0.3297=0.3200$.

At the end of year 2 the instantaneous cause-1 share is 0.10, but the cumulative share is 0.32. The earlier year, when half of events were cause 1, contributes substantially to the cumulative composition.

Reading and Checking a Cause Share

A useful interpretation should state the cause, the time, and the conditioning population. For example: “At time $t$, among first events occurring just after $t$ in individuals still event-free at $t$, the instantaneous share attributable to cause 1 is 40%.” This does not say that 40% of all individuals will experience cause 1, nor that 40% of the original population will experience it by a specified time.

The ratio can be calculated from cause-specific hazards on the same time scale and in the same population. A quick diagnostic is to check that the hazards have matching time units, the overall hazard is their sum, and the shares are between zero and one and sum to one. A share outside that range or shares that do not sum to one usually signal mismatched causes, denominators, or time points.

In data, hazards are not observed directly at every instant. Estimation requires appropriate methods for the observed event times and risk sets. Right censoring is distinct from a competing event: censoring removes follow-up information, while a competing event is a first-event outcome. The population identity defining $q_k(t)$ does not itself require an independent-censoring assumption, but estimating it from censored data does require suitable conditions on censoring and the estimation method.

Worked Example: Calculating the Local Share From Rates

At a particular time in a hypothetical setting, suppose the cause-specific hazards among event-free individuals are $0.04$ per month for cause A and $0.06$ per month for cause B. Assume these hazards are locally constant over the short interval considered, and that A and B are the only possible first-event causes.

The overall hazard is $0.04+0.06=0.10$ per month, and the instantaneous share for A is $0.04/0.10=0.40$. To check the conditional-probability interpretation over a short interval of $0.1$ month, the probability of any event, conditional on being event-free at its start, is $1-e^{-0.10(0.1)}=1-e^{-0.01}=0.009950$. The probability of event A in the interval is

\frac{0.04}{0.10}(1-e^{-0.01}) =0.40(0.009950)=0.003980.

For B it is $0.60(0.009950)=0.005970$. Conditional on an event occurring in the interval, the proportion due to A is $0.003980/0.009950=0.400$. Under constant hazards over this interval, the ratio is exact for the interval; in general, the hazard ratio gives the limiting share as the interval shrinks.

Common Mistakes and Limitations

Confusing local and cumulative composition. A student might report $q_k(t)$ as the percentage of all events by $t$ due to cause $k$. The correction is to use $F_k(t)/\sum_jF_j(t)$ for that cumulative question.
Using the wrong denominator. The denominator is the overall first-event hazard among individuals still event-free at $t$, not the number of all individuals originally enrolled or a cause-specific hazard from a different time or population.
Treating the ratio as a risk or causal effect. A hazard share is an instantaneous event composition, not a probability of eventual disease and not, by itself, a causal comparison between interventions.
Reporting a share when no event can occur. If $\lambda(t)=0$, the ratio is undefined; there is no event composition at that time.
Ignoring cause definitions. The causes must form mutually exclusive first-event categories for the shares to sum to one. Misclassified, overlapping, or omitted causes can make the calculation misleading.

The principal alternative depends on the question. Use a cause-specific hazard to describe a cause’s instantaneous rate among those still event-free; use the instantaneous cause share to describe the composition of events occurring now; use a cumulative share to describe the composition of events already accumulated by a specified time. These quantities answer related but distinct questions.

Key takeaway: The ratio $\lambda_k(t)/\lambda(t)$ is the instantaneous share of first events due to cause $k$, provided the overall hazard is positive and the causes are well-defined. The cumulative share depends on the event mix over the entire period and generally need not equal the instantaneous share at its endpoint.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

At a time when the hazards for three causes are $0.03$, $0.02$, and $0.05$ per year, what is the instantaneous share for cause 2?
Why can the instantaneous share at time $t$ differ from the cumulative share by time $t$?
If the overall hazard is zero at a particular time, what is the cause share, and why?
Give one condition needed for the cause shares to sum to one.

Solutions. (1) The overall hazard is $0.03+0.02+0.05=0.10$ per year, so the cause-2 share is $0.02/0.10=0.20$, or 20%. (2) The instantaneous share describes events near $t$, while the cumulative share includes events from all earlier times, weighted by when they occurred and how many people remained event-free. (3) It is undefined: the ratio has a zero denominator, and no first events are occurring at that time to have a cause composition. (4) The causes must be mutually exclusive and exhaustive categories for the first event, and the overall hazard must be positive at the time in question.