Choosing the Time Origin for Cumulative Effects

Why the Integration Origin Matters

A cumulative association links the event hazard to marker history accumulated over a period of time. The upper integration limit is usually the current time, but the lower limit is a modeling choice with a direct interpretation: it specifies when accumulation begins. Integrating from enrollment asks about exposure accumulated since enrollment; integrating from a later clinical landmark asks about exposure accumulated only after that landmark.

These are not merely two ways to write the same quantity. If subjects differ in their marker histories before the landmark, including or excluding that history can change their cumulative exposures and their modeled relative hazards. The lower limit should therefore be chosen to match the scientific question and the population at risk.

Definition: Let $m_i(s)$ denote the marker trajectory for subject $i$ at time $s$, and let $a$ be the chosen accumulation origin. For $t\geq a$, the cumulative marker exposure is $$ A_i(t;a)=\int_a^t m_i(s)\,ds. $$ The interval $[a,t]$ is the exposure period represented by this cumulative quantity. If marker values are measured in marker units and time in years, $A_i(t;a)$ has units of marker-units times years.

A common cumulative-link specification for the subject-specific hazard is $$ h_i(t\mid b_i)=h_0(t)\exp\left\{\gamma^\mathsf{T}w_i+\alpha A_i(t;a)\right\},\qquad t\geq a. $$ Here $h_0(t)$ is the baseline hazard on the chosen survival-time scale, $w_i$ is a vector of covariates, $\gamma$ is its coefficient vector, and $b_i$ denotes any subject-specific latent trajectory parameters on which $m_i$ depends. The parameter $\alpha$ measures the association with cumulative exposure. This model assumes the specified trajectory and exposure period are appropriate, the hazard is positive, and the log hazard is linear in the included covariates and cumulative exposure. A hazard ratio comparing two exposure values at the same time and covariate values is $\exp\{\alpha\Delta A\}$, where $\Delta A$ is their difference in cumulative exposure.

Enrollment as the Accumulation Origin

Suppose time $t=0$ is enrollment and follow-up starts at enrollment. With $a=0$, the exposure in the hazard at time $t$ includes the marker trajectory from enrollment through $t$. This definition is appropriate when the target is the association of accumulated marker history since enrollment with the event hazard during follow-up.

The trajectory need not be observed continuously to define the integral, but it must be estimated or otherwise specified over the interval. The examples below use explicit piecewise-linear trajectories so that the integrals can be calculated exactly. In an applied joint model, uncertainty in the estimated latent trajectory must be accounted for rather than treating noisy measurements as the trajectory itself.

Worked Example: Accumulation From Enrollment

Consider a hypothetical subject enrolled at $t=0$, with marker values assumed to follow straight-line segments between the points $(0,2)$, $(2,4)$, and $(5,3)$. Time is measured in years and the marker in units. Find cumulative exposure at $t=5$ when accumulation starts at enrollment.

On each linear segment, the area under the trajectory is the interval length times the average of its endpoint marker values. Therefore, $$ A_i(5;0)=2\left(\frac{2+4}{2}\right)+3\left(\frac{4+3}{2}\right)=6+10.5=16.5\ \text{marker-units}\cdot\text{years}. $$

Suppose the fitted cumulative coefficient is $\alpha=0.08$ per marker-unit-year. Holding the baseline hazard and covariates fixed, this subject’s cumulative-exposure contribution to the hazard multiplier at $t=5$ is $$ \exp\{0.08(16.5)\}=\exp(1.32)\approx 3.743. $$ This is a multiplier relative to a subject with cumulative exposure zero under the same model conditions; it is not an absolute event probability. For a one-unit difference in cumulative exposure, the hazard ratio is $\exp(0.08)\approx1.083$, or about an 8.33% higher hazard, at that time and with other terms held fixed.

Landmark as the Accumulation Origin

A clinical landmark $L$ is a specified time at which the analysis focuses on subjects who are alive and event-free. If accumulation begins at that landmark, define $$ A_i(t;L)=\int_L^t m_i(s)\,ds,\qquad t\geq L. $$ This exposure excludes all marker history before $L$. It answers a post-landmark question about accumulation after the landmark, not a question about the effect of the subject’s full history since enrollment.

A landmark analysis must also define its risk set: subjects enter the post-landmark risk period only if they have survived event-free to $L$. The resulting hazard concerns this selected, event-free population. If the survival-time clock is reset at the landmark, write $u=t-L$; the post-landmark trajectory is then $m_i(L+v)$, and exposure accumulated by follow-up time $u$ is $\int_0^u m_i(L+v)\,dv$. Resetting the clock does not itself determine whether pre-landmark exposure is included. That is determined by the integral’s lower limit.

Worked Example: Comparing Enrollment and Landmark Exposure

Use the hypothetical trajectory from the previous example and take the clinical landmark to be $L=2$ years. At $t=5$, enrollment-based exposure is $16.5$ marker-units-years. The exposure accumulated only after the landmark is the area over the segment from year 2 to year 5: $$ A_i(5;2)=3\left(\frac{4+3}{2}\right)=10.5\ \text{marker-units}\cdot\text{years}. $$

The difference, $16.5-10.5=6.0$ marker-units-years, is precisely the exposure from enrollment to the landmark: $2(2+4)/2=6.0$. Thus changing the lower limit removes, rather than re-estimates, that pre-landmark area for this subject.

For $\alpha=0.08$ per marker-unit-year, the hazard multiplier associated with enrollment-based exposure is $\exp(0.08\times16.5)\approx3.743$; for landmark-based exposure it is $\exp(0.08\times10.5)=\exp(0.84)\approx2.316$. Their ratio is $$ \frac{\exp\{0.08A_i(5;0)\}}{\exp\{0.08A_i(5;2)\}}=\exp\{0.08(6.0)\}\approx1.616. $$ Under these model terms, the enrollment-origin contribution is 1.616 times the landmark-origin contribution for this subject at $t=5$. This comparison concerns the cumulative-exposure terms, not necessarily the full hazards: baseline hazards, covariates, and the risk populations also matter.

Changing the Clock Without Changing the Exposure Definition

Time can be recorded as elapsed time since enrollment or elapsed time since a landmark. These are alternative coordinates for describing the same calendar point, but they do not automatically imply the same accumulation period. If $u=t-L$, then enrollment-origin exposure expressed on the landmark clock is $$ A_i(L+u;0)=\int_0^L m_i(s)\,ds+\int_0^u m_i(L+v)\,dv. $$ The first term is the accumulated pre-landmark exposure; the second is post-landmark exposure. In contrast, landmark-origin exposure on that clock is only $\int_0^u m_i(L+v)\,dv$.

The baseline hazard clock can also be represented in either coordinate. A model with baseline $h_0(t)$ on the enrollment clock has baseline $h_0(L+u)$ when expressed using landmark time $u$. It should not be silently replaced by an unrelated function of $u$ without recognizing that this changes the baseline-hazard specification. Likewise, resetting the baseline-hazard clock does not justify resetting the cumulative exposure origin.

Worked Example: Re-expressing Exposure on a Landmark Clock

Suppose the landmark is $L=2$ years and post-landmark follow-up is $u=3$ years, so the same endpoint is $t=L+u=5$. In the hypothetical trajectory above, exposure from enrollment to year 2 is $6.0$ marker-units-years, and exposure from year 2 to year 5 is $10.5$ marker-units-years.

The enrollment-origin exposure written using $u$ is $$ A_i(2+3;0)=6.0+10.5=16.5\ \text{marker-units}\cdot\text{years}. $$ The landmark-origin exposure is $$ A_i(2+3;2)=10.5\ \text{marker-units}\cdot\text{years}. $$ Both calculations refer to the same endpoint, year 5. The difference is not caused by using $u=3$ instead of $t=5$; it is caused by including or excluding the first two years of marker history.

Justification: What Changing the Lower Limit Does

For any two candidate origins $a_1<a_2\leq t$, additivity of integration gives $$ A_i(t;a_1)=\int_{a_1}^{a_2}m_i(s)\,ds+\int_{a_2}^{t}m_i(s)\,ds =A_i(a_2;a_1)+A_i(t;a_2). $$ Thus the change in cumulative exposure is exactly the area accumulated between the two candidate origins. With the same $\alpha$, covariates, and baseline hazard, the ratio of the two modeled hazards’ cumulative-exposure multipliers is $$ \frac{\exp\{\alpha A_i(t;a_1)\}}{\exp\{\alpha A_i(t;a_2)\}} =\exp\{\alpha A_i(a_2;a_1)\}. $$ This identity explains why the lower limit matters. It also shows why the impact can differ across subjects: their pre-$a_2$ areas need not be equal. A common additive amount for every subject would contribute a common factor at a given time, but heterogeneous pre-origin histories can change relative hazards across subjects.

Name the target period.
Specify whether the scientific question concerns accumulation since enrollment or only accumulation after a clinical landmark.

Define the at-risk population and clock.
For a landmark analysis, identify who is event-free at the landmark and state whether the survival-time coordinate is elapsed since enrollment or since the landmark.

Write the exposure integral explicitly.
Give its lower limit, upper limit, marker trajectory, and units before interpreting the cumulative coefficient.

Check the fitted model matches that definition.
Verify that trajectory coverage and risk-set construction support the chosen interval, and interpret any hazard ratio as conditional on the specified model terms.

Diagnostics, Alternatives, and Limitations

A practical diagnostic is to tabulate, for representative subjects, the start and end of the exposure interval and the resulting cumulative value. Check that follow-up times have been aligned to the same clock, that the trajectory is defined over every interval being integrated, and that landmark eligibility is applied consistently. Implausible jumps in cumulative values around a landmark can indicate that the clock was reset in one part of the analysis but not another. Also compare the spread of pre-landmark and post-landmark exposure: if pre-landmark histories vary substantially, omitting them may materially alter subject rankings and estimated associations.

The main alternatives answer different questions. A current-value association links hazard to the marker at the current time; a slope association links hazard to its rate of change. A cumulative association instead represents accumulated history over a defined interval. Choosing between enrollment and landmark origins is a further choice within the cumulative approach, not a replacement for choosing between current-value, slope, and cumulative links. These approaches should not be described as interchangeable merely because they use the same marker.

Landmark restriction has an important limitation: it conditions on being event-free at $L$. The resulting association need not describe the original enrolled population, because subjects with early events are excluded and the surviving risk set can differ systematically. Conversely, an enrollment-origin integral used for post-landmark risk retains pre-landmark history, but its interpretation must account for the fact that the analysis is now among survivors to $L$. Neither choice by itself removes confounding, corrects a misspecified longitudinal trajectory, or establishes a causal effect.

Common Mistakes

Assuming a new clock automatically resets exposure. Re-labeling time as $u=t-L$ changes the coordinate, not the exposure definition. The lower integral limit must be changed explicitly to exclude pre-landmark history.
Calling a landmark-origin integral “total exposure.” It is total only over the interval from the landmark to the current time. State the interval and units.
Comparing hazard multipliers as if they were full hazards. The exponential cumulative term is only one component of the hazard; the baseline hazard and other covariates also enter.
Ignoring landmark selection. A post-landmark hazard is conditional on entering the landmark risk set, not a hazard for everyone originally enrolled.
Interpreting $\alpha$ without an exposure unit. Its units are the reciprocal of the units of $A_i(t;a)$. A coefficient tied to marker-units-years cannot be interpreted as an effect per marker unit alone.

Key takeaway: The lower limit in a cumulative link defines which marker history contributes to risk. State the origin, time scale, exposure units, and risk set together; a landmark clock alone does not determine the exposure period.

Check Your Understanding

Answer each question, then compare with the solutions below.

A landmark occurs at year 4. In one sentence, distinguish the meaning of $\int_0^t m_i(s)\,ds$ from $\int_4^t m_i(s)\,ds$ for $t\geq4$.
If time is reset to $u=t-4$, write enrollment-origin exposure in terms of $u$ and the marker trajectory.
A subject has pre-landmark exposure of 7 marker-units-years and post-landmark exposure of 12 marker-units-years. With $\alpha=0.05$ per marker-unit-year, what is the ratio of the enrollment-origin to landmark-origin cumulative-exposure hazard multipliers?
Why does a post-landmark analysis not describe the hazard for all subjects enrolled at baseline?

Solutions. (1) The first integral includes all exposure since enrollment; the second includes only exposure from year 4 onward. (2) With $u=t-4$, it is $\int_0^4m_i(s)\,ds+\int_0^u m_i(4+v)\,dv$. (3) The ratio is $\exp\{0.05(7)\}=\exp(0.35)\approx1.419$. (4) It restricts the risk set to subjects who survived event-free to year 4, so early events are excluded and the target population is conditional on landmark survival.