Cumulative Exposure Before an Event

Why the Integration Limit Must Follow Risk Time

A cumulative association links the event process to marker exposure accumulated over time. Its value is therefore not a fixed summary computed once for every subject: as follow-up proceeds, more of the marker trajectory becomes available to accumulate. At any risk time, the relevant exposure is the marker history up to that time—not the trajectory after it.

This boundary matters especially when an event occurs. Once a subject experiences the event, that subject is no longer at risk for a first event in the model. Marker values after the event cannot contribute to the subject’s pre-event exposure. A cumulative quantity that includes such values would use information from the future relative to the risk time and would not represent exposure available to the event process.

Definition: Let $m_i(s)$ be the marker trajectory for subject $i$, measured from a chosen time origin $0$. The cumulative exposure available at risk time $t$ is $$ A_i(t)=\int_0^t m_i(s)\,ds. $$ The integration limit is the current risk time. If the event time is $T_i$, the event-time exposure is $A_i(T_i-)=\int_0^{T_i}m_i(s)\,ds$, using the trajectory only up to the event. For an integrable trajectory, including or excluding the single endpoint does not change the integral.

The notation $T_i-$ emphasizes that the exposure is evaluated immediately before the event. In a continuous-time model, the value of the integrand at one isolated endpoint has no effect on the integral. The essential requirement is that no marker history after $T_i$ is included.

The origin of integration must be stated. The definition above uses time zero as the start of accumulation. If the scientific definition starts at another time $a_i$, the corresponding quantity is $\int_{a_i}^t m_i(s)\,ds$, for $t\geq a_i$. The origin changes what is counted as exposure; it is not a detail that can be changed silently during follow-up.

Assumptions and the Risk-Time Convention

The definition itself requires that the marker trajectory be integrable over the interval being considered. For the resulting quantity to be used as an event-model predictor, the trajectory and time origin must also have a meaningful interpretation, and the value at time $t$ must be determined by marker history no later than $t$. The event model must not use future marker values to explain the hazard at $t$.

For clarity, let $T_i$ denote the event time and $C_i$ the censoring time. The observed follow-up time is $Y_i=\min(T_i,C_i)$, and let $\delta_i=1$ indicate an observed event and $\delta_i=0$ indicate censoring. While subject $i$ is at risk, the cumulative exposure at time $t$ is updated as $A_i(t)$. If the event is observed, the event-time value is evaluated at $T_i=Y_i$. If follow-up is censored first, the observed marker history contributes only through $C_i=Y_i$; censoring does not turn that time into an event time.

One illustrative hazard specification isolates the cumulative component:

h_i(t\mid m_i)=h_0(t)\exp\{\alpha A_i(t)\}, \qquad t<T_i.

Here $h_0(t)$ is the baseline hazard and $\alpha$ is the coefficient for cumulative exposure. The hazard at a risk time $t$ uses $A_i(t)$, not a total accumulated through a later common endpoint. This expression describes an association conditional on the specified trajectory; it does not, by itself, establish a causal effect.

The setup presumes a suitable model for the marker trajectory and a well-defined risk process. In a joint model, the trajectory may be represented through subject-specific model components rather than observed continuously. Regardless of how it is represented, the cumulative predictor’s upper limit remains the current risk time. The time-varying cumulative value is then calculated from the trajectory over the interval available up to that time.

How Cumulative Exposure Updates

The update rule follows directly by splitting an integral at the current time. For $0\leq t_1<t_2$,

A_i(t_2)=A_i(t_1)+\int_{t_1}^{t_2}m_i(s)\,ds.

Thus, as follow-up advances from $t_1$ to $t_2$, only the exposure over the newly elapsed interval is added. If an event occurs at $T_i$, the update stops there. No later interval is appended for that event’s pre-event cumulative exposure.

For an event observed at $T_i$, the event contribution to a time-to-event likelihood is evaluated at the event time. For example, conditional on a specified trajectory, the event-time density has the usual form

f_i(T_i\mid m_i) = h_i(T_i\mid m_i) \exp\left\{-\int_0^{T_i}h_i(u\mid m_i)\,du\right\}.

The hazard factor uses the cumulative predictor evaluated at $T_i-$, while the survival factor integrates the hazard over earlier risk times $u\leq T_i$, using $A_i(u)$ at each such time. This distinction is important: the marker is integrated once to form the predictor $A_i(u)$, and the hazard is then accumulated over time in the survival term. These are different integrations with different roles.

Worked Example: Updating Exposure Before an Event

Suppose a hypothetical subject has marker trajectory $m(t)=2+0.5t$ units from time 0 through an event at $t=4$. Find cumulative exposure at times 2 and 4, and show what is added between those times.

The trajectory is continuous, so it is integrable. Starting from zero,

A(t)=\int_0^t(2+0.5s)\,ds =2t+0.25t^2.

At time 2,

$$ A(2)=2(2)+0.25(2^2)=4+1=5 $$

marker-unit-time. At the event time 4,

$$ A(4-)=2(4)+0.25(4^2)=8+4=12 $$

marker-unit-time. The increment over the interval from 2 to 4 is

\int_2^4(2+0.5s)\,ds = [2s+0.25s^2]_2^4 = 12-5=7.

This agrees with the update identity: $A(4)=A(2)+7=5+7=12$. Since the event occurs at time 4, the event-time exposure is 12 marker-unit-time. A value such as $A(5)$ would include marker history after the event and is not the subject’s pre-event exposure.

Worked Example: Event and Censoring Give Different Endpoints

Consider two hypothetical subjects with the same constant marker value, $m_i(t)=3$ units. Subject A has an observed event at time 5. Subject B is censored at time 5 without an observed event. Calculate each subject’s accumulated exposure through observed follow-up and state how the endpoints are used.

For a constant trajectory, $A_i(t)=\int_0^t3\,ds=3t$. Subject A has $\delta_A=1$, $Y_A=T_A=5$, and therefore

A_A(T_A-)=\int_0^5 3\,ds=3(5)=15

marker-unit-time. This is the cumulative exposure evaluated at the observed event.

Subject B has $\delta_B=0$, $Y_B=C_B=5$, and

A_B(C_B)=\int_0^5 3\,ds=3(5)=15

marker-unit-time observed by censoring. The numerical integral is the same, but its role differs: subject B contributes observed at-risk follow-up through time 5, not an event at time 5. The censoring endpoint must not be mislabeled as an event-time exposure.

Evaluating a Cumulative Hazard Component

The cumulative exposure’s changing limit has a direct consequence for the hazard component. Suppose the model contains $\exp\{\alpha A_i(t)\}$, and compare two risk times for the same subject while holding the baseline hazard and other model components fixed. The ratio of the cumulative components is

\frac{\exp\{\alpha A_i(t_2)\}}{\exp\{\alpha A_i(t_1)\}} = \exp\left\{\alpha\int_{t_1}^{t_2}m_i(s)\,ds\right\}.

This follows by applying the update identity and the rule $\exp(x)/\exp(y)=\exp(x-y)$. It compares the cumulative association component at the two times, not necessarily the full hazard ratio if other time-varying components also change.

Worked Example: The Component’s Change Before the Event

Use $m(t)=2+0.5t$ from the first example and suppose, hypothetically, that $\alpha=0.08$ per marker-unit-time. Compare the cumulative components at times 2 and 4, where the event occurs at time 4.

The exposures are $A(2)=5$ and $A(4-)=12$, so the exposure increment is $12-5=7$ marker-unit-time. The ratio is

\exp\{\alpha[A(4-)-A(2)]\} = \exp\{0.08(7)\} = \exp(0.56) \approx 1.751.

Thus, holding the baseline hazard and other components fixed, the cumulative-exposure component at time 4 is about 1.751 times its value at time 2. This comparison is conditional on the stated model and the same subject-specific trajectory. It does not say that the full hazard must have risen by that factor if other components changed, and it does not imply causation.

Diagnostics, Alternatives, and Limitations

A practical check is to write the integration bounds explicitly for each risk time, event time, and censoring time before calculating any exposure. Then verify that the cumulative quantity at a later risk time equals its earlier value plus the integral over only the intervening interval. For a nonnegative marker, the cumulative total should not decrease; if it does, check the trajectory, time units, and arithmetic. If markers can be negative because of centering or another transformation, monotonicity is not guaranteed, so the update identity—not monotonicity—is the general check.

Future-data error: calculating exposure through a common study end for someone who had an earlier event uses post-event history. Stop at that subject’s event time.
Event/censoring confusion: a censored subject has accumulated exposure through censoring, but censoring is not an observed event. Keep the event indicator and endpoint together.
Origin or unit error: changing the start of accumulation or mixing time units changes the numerical integral. State the origin and use consistent units for time and marker values.
Trajectory error: a numerical calculation is only as meaningful as the trajectory supplied to it. Check the fitted or assumed trajectory over the entire interval being integrated, not only its value at the endpoint.

Cumulative exposure is not the only possible association structure. A current-value association uses $m_i(t)$ at the risk time rather than the history integral. A fixed-horizon total computed after follow-up would not be a valid time-updated predictor for earlier risk times unless it were redefined using information available at each time. Windowed exposure restricts the accumulation interval to a specified recent period; its bounds must still respect the current risk time. These quantities answer different questions, so they are not interchangeable by algebraic rescaling alone.

Cumulative exposure also has limitations. It can be sensitive to the chosen time origin, marker scale, and trajectory model. If early trajectory information is sparse, its contribution to the integral may be uncertain; errors in the estimated trajectory can carry into the cumulative predictor. Moreover, a cumulative total can conceal whether the same total arose from high recent values or lower values over a longer interval. The interpretation should therefore remain tied to the exact integration interval and the model in which the quantity appears.

A common mistake is to say, “The subject’s cumulative exposure at risk time 3 is the area under the marker curve through the end of the study, because the whole curve is available for analysis.” That reasoning confuses information available to the analyst after follow-up with information available to the event process at time 3. The correction is to use $A_i(3)=\int_0^3m_i(s)\,ds$ at risk time 3, and to update it only as additional follow-up occurs.

Key takeaway: At each risk time $t$, cumulative exposure ends at $t$. If an event is observed, evaluate the pre-event total through $T_i$; if follow-up is censored, the observed trajectory ends at $C_i$, which is not an event time.

Check Your Understanding

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

If $m(t)=4$ units from time 0 and a subject experiences an event at time 3, what is the event-time cumulative exposure, with units?
A subject is censored at time 6. Should the integral through time 6 be called the subject’s observed event-time exposure? Explain.
For an integrable marker trajectory, write the update identity from $t_1$ to $t_2$, and state the condition on their order.
Why is an integral through the end of follow-up unsuitable as the cumulative predictor at an earlier risk time?

Solutions. (1) $A(3)=\int_0^3 4\,ds=4(3)=12$ marker-unit-time. (2) No. The integral $\int_0^6m(s)\,ds$ is exposure accumulated through censoring, but censoring is not an observed event. (3) For $0\leq t_1<t_2$, $A(t_2)=A(t_1)+\int_{t_1}^{t_2}m(s)\,ds$. (4) It includes marker values after the earlier risk time, so it uses future information rather than the history available at that time.