Cumulative-Exposure Associations by Event Cause

Why cumulative exposure?

A biomarker measured near an event may not capture the biological history most relevant to that event. A hazard might instead depend on the biomarker burden accumulated over a recent period. In a joint model, that burden is calculated from the subject’s latent biomarker trajectory, rather than by summing noisy measurements as if they were exact. The window must be specified: a total over the last two years and a total over the last five years are different exposures, with different scales and potentially different associations.

In competing-risks data, the association can also differ by event cause. The same cumulative exposure may be associated with a higher cause-specific hazard for one event and a lower hazard for another. These are conditional hazard associations among subjects still event-free at time $t$, not direct causal effects.

Define the exposure and the cause-specific model

Definition: Let $m_i(s)$ be the latent biomarker level for subject $i$ at time $s$, as specified by the longitudinal model. For a prespecified window length $L>0$, define the rolling cumulative exposure at time $t$ by $$ A_i(t;L)=\int_{\max(0,t-L)}^t m_i(s)\,ds. $$ This is the area under the latent biomarker trajectory over the available part of the preceding $L$ time units. If $t\geq L$, the window is $[t-L,t]$; if $t<L$, it is $[0,t]$.

For example, if $m_i(s)$ is measured in milligrams per litre and time is measured in years, $A_i(t;L)$ has units of milligram-years per litre. It is a total over time, not a biomarker level at a single instant. The definition uses only the trajectory up to $t$, so the exposure is available for predicting an event at $t$ without using future biomarker values.

Let $T_i$ denote the time to the first event and $D_i$ its cause, with possible causes $k=1,\ldots,K$. While subject $i$ remains free of every event, the cause-specific hazard for cause $k$, conditional on the latent trajectory and covariates $w_i$, can be written as

\lambda_{ik}(t\mid b_i,w_i) = \lambda_{0k}(t)\exp\!\left\{\gamma_k^\mathsf{T}w_i+\alpha_k A_i(t;L)\right\}.

Here $b_i$ denotes the subject-specific effects in the longitudinal model, and $m_i(t)$, hence $A_i(t;L)$, is determined by that model conditional on $b_i$. The cause-specific hazard is the instantaneous event rate for cause $k$ among subjects who have not yet experienced any event. The baseline hazard $\lambda_{0k}(t)$, covariate coefficients $\gamma_k$, and exposure coefficient $\alpha_k$ may vary by cause. A competing event removes a subject from the event-free risk set.

Interpretation: At a fixed time $t$, for two otherwise comparable subjects whose rolling cumulative exposures differ by $\Delta A$, the conditional cause-$k$ hazard ratio is $\exp(\alpha_k\Delta A)$, under the stated proportional-hazards model. The coefficient $\alpha_k$ is a log-hazard change per one unit of cumulative exposure. Its units are the reciprocal of the units of $A_i(t;L)$.

This interpretation is specific to the chosen window length and exposure scale. If exposure is instead defined as the window average $A_i(t;L)/L$, its units are biomarker units and its coefficient has a different numerical scale. A coefficient for total exposure cannot be transferred unchanged to an average-exposure model.

Calculating the rolling exposure

The integral is over the latent trajectory, not over the observed measurements alone. For a linear trajectory $m_i(s)=c+ds$, integrating over a window $[a,t]$ gives

A_i(t;L) = \int_a^t(c+ds)\,ds = c(t-a)+\frac{d}{2}(t^2-a^2), \qquad a=\max(0,t-L).

When $t\geq L$, this can also be written $A_i(t;L)=Lc+dL(t-L/2)$, or $L$ times the biomarker level at the window midpoint. For nonlinear trajectories, integrate the fitted latent function over the same bounds; numerical integration may be needed. The exposure definition and bounds should be kept consistent for all subjects and causes.

Worked Example: A Linear Trajectory and a Cause-Specific Hazard Ratio

Suppose a hypothetical biomarker trajectory is $m_i(s)=4+0.5s$ milligrams per litre, with time in years. At $t=6$ years, use a two-year window. A second subject has a cumulative exposure of $10$ milligram-years per litre at the same time. For cause 1, suppose $\alpha_1=0.06$ per milligram-year per litre. Calculate the first subject’s exposure and the hazard ratio comparing the two subjects.

The window is $[6-2,6]=[4,6]$. Integrating the trajectory,

A_i(6;2) = \int_4^6(4+0.5s)\,ds = 4(6-4)+\frac{0.5}{2}(6^2-4^2) = 8+5 = 13

The exposure difference is $13-10=3$ milligram-years per litre. Thus the conditional cause-1 hazard ratio is

\exp\{\alpha_1(13-10)\} = \exp(0.06\times3) = \exp(0.18) \approx 1.197.

Under this model, at year 6 and for the same covariates, the subject with the higher two-year cumulative exposure has an estimated cause-1 hazard about 1.20 times that of the comparison subject. This is a conditional hazard comparison, not a comparison of event probabilities.

Worked Example: Integrating Across a Change in Trajectory

Consider a hypothetical latent trajectory in milligrams per litre: $m_i(s)=3+s$ for $0\leq s\leq2$, then $m_i(s)=5+0.2(s-2)$ for $2<s\leq4$. At $t=4$, calculate exposure over the preceding three years. For cause 2, let $\alpha_2=-0.05$ per milligram-year per litre; compare this subject with one whose exposure is $12.4$ milligram-years per litre.

The exposure window is $[1,4]$, which crosses the change in trajectory at year 2. Integrate the two segments separately:

\begin{aligned} A_i(4;3) &=\int_1^2(3+s)\,ds+\int_2^4\{5+0.2(s-2)\}\,ds\\ &=\left[3s+\frac{s^2}{2}\right]_1^2 +\left[5s+0.1(s-2)^2\right]_2^4\\ &=4.5+10.4=14.9. \end{aligned}

The exposure difference is $14.9-12.4=2.5$ milligram-years per litre. The cause-2 hazard ratio is

\exp\{-0.05(2.5)\} = \exp(-0.125) \approx 0.8825.

Conditional on the model and the same covariates at year 4, the higher-exposure subject has an estimated cause-2 hazard about 0.88 times that of the comparison subject. The negative coefficient describes an inverse association with this cause-specific hazard; it does not establish that increasing the biomarker would prevent the event.

Estimating and reporting cause-specific coefficients

In a joint model, the longitudinal component estimates a latent trajectory, while the event component relates that trajectory-derived exposure to each cause-specific hazard. The observed biomarker values are measurements of the underlying process and can contain measurement error. Treating each observed value as the true biomarker level, or calculating a crude sum from irregularly timed measurements without accounting for the trajectory, generally does not implement the model above.

The following hypothetical estimates illustrate coefficient interpretation. Suppose the two-year exposure is measured in milligram-years per litre, and fitted coefficients and standard errors are $\hat{\alpha}_1=0.080$ with $SE=0.030$, and $\hat{\alpha}_2=-0.040$ with $SE=0.025$. For a five-unit exposure difference, the estimated cause-1 hazard ratio is

\exp(5\hat{\alpha}_1)=\exp(0.400)\approx1.492.

Using a normal-approximation 95% Wald interval for the coefficient, the cause-1 interval is $0.080\pm1.96(0.030)=[0.0212,0.1388]$. Scaling both endpoints by five and exponentiating gives

\left[\exp\{5(0.0212)\},\exp\{5(0.1388)\}\right] \approx [1.112,2.002].

If the Wald approximation is adequate, this interval is consistent with cause-1 hazards roughly 1.11 to 2.00 times as high for a five milligram-year-per-litre exposure difference, conditional on the model and covariates. It is a confidence interval for the hazard ratio, not a 95% probability statement about a fixed parameter.

For cause 2, the estimate per five units is $\exp\{5(-0.040)\}=\exp(-0.200)\approx0.819$. Its coefficient interval is $-0.040\pm1.96(0.025)=[-0.0890,0.0090]$, yielding a hazard-ratio interval $[\exp(-0.445),\exp(0.045)]\approx[0.641,1.046]$. Because this interval includes 1, the estimate does not provide clear evidence of an association at the 5% two-sided level using this Wald interval. The estimates have opposite signs, but comparing the two coefficients formally requires their covariance; separate standard errors alone are insufficient.

Assumptions, diagnostics, and alternatives

The model’s interpretation relies on several substantive and statistical conditions. The latent trajectory must be adequately specified over the exposure window; the window length and its placement must reflect the intended scientific exposure; and the log hazard must be linear in cumulative exposure unless a nonlinear association is explicitly modeled. The proportional-hazards form assumes that, for a given exposure difference and covariates, the hazard ratio does not itself vary with time. The joint model must also represent the dependence between the repeated biomarker measurements and event process well enough for latent exposure estimation. Event causes and censoring must be recorded and handled according to the study design; in particular, a competing event ends observation of the first-event risk process.

Useful checks include comparing fitted and observed biomarker trajectories over the time ranges that contribute to the integral, examining whether influential subjects or sparse measurements dominate exposure estimates, and checking the event-model fit for time patterns inconsistent with proportional hazards. Investigators can also examine sensitivity to scientifically plausible window lengths and to reasonable alternatives for the longitudinal trajectory. Such checks do not prove the assumptions, but can reveal when results depend heavily on them.

A rolling cumulative exposure is not the only possible association structure. A current-value association uses the latent biomarker level $m_i(t)$; a slope association uses its rate of change. These capture distinct features of the trajectory: level, change, and accumulated burden. A full-history integral from time zero to $t$ is another cumulative-exposure definition, but it grows over time and weights remote measurements differently from a rolling window. These alternatives answer different modeling questions and should not be treated as interchangeable versions of the same coefficient.

Look-ahead bias: integrating beyond $t$ uses biomarker values unavailable at the event time and can leak future information into the hazard model.
Window ambiguity: omitting the window length or start-time rule makes the exposure and coefficient difficult to reproduce or interpret.
Scale confusion: reporting a coefficient without exposure units hides whether it applies per biomarker unit, per unit-time total, or per window average.
Overinterpretation: cause-specific hazard associations are not causal effects and do not alone give cumulative event probabilities.

A practical limitation is that results may be sensitive to how the latent trajectory is modeled, especially when the biomarker is measured infrequently or the chosen window extends into poorly observed periods. A single coefficient also imposes a particular functional form: equal increases in cumulative exposure are assumed to have equal effects on log hazard, regardless of how that exposure was distributed within the window. The fitted model therefore summarizes a specified exposure history, not every possible biomarker pattern.

Key takeaway: Define the past window, integrate the latent trajectory over exactly that window, and interpret each cause-specific coefficient using the resulting exposure units. The cause-specific hazard ratio is conditional on remaining event-free and on the model assumptions; it is not by itself a causal effect or a cumulative-incidence comparison.

Check Your Understanding

Use the definitions and model above to answer the questions. Brief solutions follow.

A trajectory is $m_i(s)=2+0.5s$ units, with time in years. What is the cumulative exposure at $t=5$ using $L=2$?
If two subjects differ by 4 exposure units and $\alpha_k=0.10$ per exposure unit, what is their modeled cause-$k$ hazard ratio for the higher-exposure subject?
Why is it invalid to define an exposure for the hazard at time $t$ by integrating the biomarker trajectory from $t-2$ to $t+1$?
What additional information is needed to test whether two cause-specific exposure coefficients differ, when estimates and their separate standard errors are already available?

Solutions. (1) The window is $[3,5]$, so $A_i(5;2)=\int_3^5(2+0.5s)\,ds=2(2)+0.25(25-9)=8$ unit-years. (2) The ratio is $\exp(0.10\times4)=\exp(0.4)\approx1.492$. (3) The interval includes future biomarker values after $t$, so it uses information unavailable at the event time and creates look-ahead. (4) The covariance between the coefficient estimates is needed to calculate the standard error of their difference; separate standard errors do not determine it.

Cumulative-Exposure Associations by Event Cause

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