Cumulative Effects for a Linear Marker Trajectory

Why a Linear Trajectory Changes Cumulative Exposure

A cumulative association links the event hazard to marker exposure accumulated over time, rather than only to the marker value at one instant. If the latent marker trajectory is linear, its accumulated exposure is not simply the starting value multiplied by time: the slope contributes an additional term that grows quadratically with elapsed time. This matters when comparing subjects whose marker levels begin at different values, change at different rates, or both.

We use $m_i(t)$ for subject $i$'s latent marker value at time $t$, and $A_i(t)$ for its cumulative exposure from the chosen origin $0$ through $t$. In a shared random-effects model, the trajectory may be represented conditionally on subject-specific random effects. Here the calculations concern the resulting latent trajectory; they do not replace it with noisy observed marker measurements.

Definition: For a linear latent marker trajectory $m_i(t)=b_{0i}+b_{1i}t$, where $b_{0i}$ is the subject-specific level at time zero and $b_{1i}$ is the subject-specific rate of change, cumulative exposure from time zero to $t$ is $$ A_i(t)=\int_0^t m_i(s)\,ds=b_{0i}t+\frac{1}{2}b_{1i}t^2. $$

The level $b_{0i}$ is measured in marker units, and the slope $b_{1i}$ in marker units per time unit. Accordingly, $A_i(t)$ has units of marker-units multiplied by time. The factor $1/2$ arises from integrating the changing part of the trajectory; omitting it gives an incorrect cumulative exposure.

Model and Assumptions for a Hazard Contrast

Suppose the event hazard, conditional on the latent trajectory and baseline covariates $w_i$, follows a proportional-hazards model with a cumulative association:

\lambda_i(t\mid b_{0i},b_{1i},w_i) = \lambda_0(t)\exp\left\{\gamma^\mathsf{T}w_i+\alpha A_i(t)\right\}.

Here $\lambda_0(t)$ is the baseline hazard, $\gamma$ is the covariate coefficient vector, and $\alpha$ is the association parameter for cumulative exposure. This model says that, at time $t$, the hazard depends on the total latent marker exposure accumulated since the specified origin. It does not say that the marker causes the event.

The contrast below assumes that both subjects are evaluated at the same time $t$, use the same time origin and marker scale, and have the same baseline covariates. The trajectories are taken as linear over the interval from 0 to $t$, and the same $\alpha$ applies to both subjects. Under these conditions, the baseline hazard and covariate terms cancel in the hazard ratio. If covariates differ, their contribution must be included as well.

Result: For subjects $i$ and $j$ with the same baseline covariates, the conditional hazard ratio at time $t$ is $$ \frac{\lambda_i(t\mid b_{0i},b_{1i},w)} {\lambda_j(t\mid b_{0j},b_{1j},w)} = \exp\left\{ \alpha\left[ (b_{0i}-b_{0j})t+ \frac{1}{2}(b_{1i}-b_{1j})t^2 \right]\right\}. $$

This is a time-specific hazard ratio conditional on the latent trajectories, not a ratio of event probabilities over an interval. The level difference contributes linearly in $t$; the slope difference contributes quadratically. Thus, even with a constant association parameter $\alpha$, the hazard contrast generally changes over time.

Deriving the Exposure and Contrast

The derivation uses the definition of cumulative exposure and the stated hazard model. First integrate the trajectory. Then take the ratio of the two hazards, so that shared terms cancel.

Integrate the marker trajectory.
With $m_i(s)=b_{0i}+b_{1i}s$, integrate each term from 0 to $t$: $A_i(t)=b_{0i}t+b_{1i}t^2/2$.

Take the hazard ratio.
For shared covariates, the terms $\lambda_0(t)$ and $\gamma^\mathsf{T}w$ are identical in both hazards and cancel. The exponential terms therefore give $\exp\{\alpha[A_i(t)-A_j(t)]\}$.

Substitute the integrated trajectories.
Subtracting the two cumulative exposures gives $A_i(t)-A_j(t)=(b_{0i}-b_{0j})t+(b_{1i}-b_{1j})t^2/2$, yielding the stated hazard ratio.

This also clarifies what $\alpha$ means: a one-unit increase in cumulative exposure, holding the other terms in the hazard model fixed, multiplies the hazard by $\exp(\alpha)$. A one-unit difference in $b_{0i}$ or $b_{1i}$ is not itself a one-unit difference in cumulative exposure at every time. Its cumulative contribution depends on $t$.

Worked Calculations

Worked Example: Different Starting Levels and Slopes

Consider two hypothetical subjects observed on a time scale measured in months. Subject A has $b_{0A}=4$ marker units and $b_{1A}=0.3$ marker units per month. Subject B has $b_{0B}=3$ marker units and $b_{1B}=0.1$ marker units per month. At month 6, suppose the cumulative association parameter is $\alpha=0.04$ per marker-month, and the subjects have the same baseline covariates. Find their conditional hazard ratio.

First calculate each cumulative exposure:

A_A(6)=4(6)+\frac{1}{2}(0.3)(6^2)=24+5.4=29.4\text{ marker-months},

A_B(6)=3(6)+\frac{1}{2}(0.1)(6^2)=18+1.8=19.8\text{ marker-months}.

The exposure difference is $29.4-19.8=9.6$ marker-months. Therefore,

\frac{\lambda_A(6)}{\lambda_B(6)} =\exp\{0.04(9.6)\} =\exp(0.384) \approx 1.468.

At month 6, subject A's conditional hazard is about 1.47 times subject B's, or about 47% higher, under the specified model and equal-covariate comparison. The calculation is about instantaneous hazards at that time, not the probability that either subject experiences the event.

Worked Example: Same Starting Level, Different Rates of Change

Two hypothetical subjects both start at $b_0=5$ marker units. Subject C has slope $0.2$ marker units per month, while subject D has slope $-0.1$ marker units per month. At month 8, let $\alpha=-0.025$ per marker-month. Assume equal baseline covariates. Calculate the hazard ratio for C relative to D.

Because the starting levels are equal, their level contributions cancel. The slope difference is $0.2-(-0.1)=0.3$ marker units per month. The cumulative exposure difference at month 8 is

A_C(8)-A_D(8) =(5-5)(8)+\frac{1}{2}(0.3)(8^2) =0+9.6 =9.6\text{ marker-months}.

The hazard ratio is

\frac{\lambda_C(8)}{\lambda_D(8)} =\exp\{-0.025(9.6)\} =\exp(-0.24) \approx 0.7866.

At month 8, C's conditional hazard is about 0.787 times D's, or about 21% lower, under this model. The negative association parameter means greater cumulative exposure corresponds to a lower modeled hazard, all else equal. It does not reverse the fact that C has greater cumulative exposure in this comparison.

Worked Example: A Contrast That Changes With Time

Suppose hypothetical subject E has $b_{0E}=3$ marker units and $b_{1E}=0.4$ marker units per month, while subject F has $b_{0F}=2$ marker units and $b_{1F}=0.2$ marker units per month. Let $\alpha=0.02$ per marker-month and assume equal baseline covariates. Compare their hazard ratios at months 2 and 8.

The differences are $b_{0E}-b_{0F}=1$ marker unit and $b_{1E}-b_{1F}=0.2$ marker units per month. At month 2, their cumulative exposure difference is

\Delta A(2)=1(2)+\frac{1}{2}(0.2)(2^2)=2+0.4=2.4\text{ marker-months}, \qquad \exp\{0.02(2.4)\}=\exp(0.048)\approx1.049.

At month 8, the corresponding calculation is

\Delta A(8)=1(8)+\frac{1}{2}(0.2)(8^2)=8+6.4=14.4\text{ marker-months}, \qquad \exp\{0.02(14.4)\}=\exp(0.288)\approx1.334.

The conditional hazard ratio for E relative to F is about 1.05 at month 2 and 1.33 at month 8. Both subjects have the same fixed association parameter, but their cumulative exposure difference grows with time, so their hazard contrast grows too.

How to Check the Calculation and Model

A numerical calculation can be correct while the model is a poor description of the scientific question or data. Before interpreting a contrast, check that the origin and time units are clear, that the linear trajectory is a reasonable representation over the integration interval, and that the cumulative association is the intended link to the hazard. In a fitted joint model, also consider whether the estimated latent trajectories are adequately supported by the longitudinal measurements and whether the assumed hazard structure is plausible.

Check units. If time is in months, the slope must be expressed per month and $A_i(t)$ in marker-months. The units of $\alpha$ must be the reciprocal of the cumulative-exposure units so that $\alpha A_i(t)$ is dimensionless.
Check the integration interval. The formula here accumulates exposure from time zero. It is not automatically the correct expression if the scientific exposure period begins at another time.
Check the linearity assumption. Plot or otherwise examine the longitudinal data and fitted latent trajectories. Systematic curvature over the relevant interval warns that a linear integral may not represent accumulated exposure adequately.
Check the scale and sign. A positive $\alpha$ makes the hazard increase with cumulative exposure; a negative $\alpha$ makes it decrease. Verify that the marker scale and fitted coefficient use the same units and transformation.
Check the interpretation. The result is a conditional hazard contrast at a specified time. It is not an event probability, a survival probability, or a causal effect without additional assumptions.

Comparison and Limitations

A cumulative link answers a different question from an association with the marker's current value or instantaneous slope. A current-value link compares subjects using $m_i(t)$ at the evaluation time; a slope link uses the rate of change at that time. The cumulative link instead compares integrals over the chosen history. For a linear trajectory, the three quantities are related but not interchangeable: current value is $b_{0i}+b_{1i}t$, slope is $b_{1i}$, and cumulative exposure is $b_{0i}t+b_{1i}t^2/2$.

Association component	Quantity entering the hazard	Question it represents
Current value	$m_i(t)$	How does the marker level at this time relate to the hazard?
Instantaneous slope	$b_{1i}$ for a linear trajectory	How does the rate of change relate to the hazard?
Cumulative exposure	$A_i(t)=\int_0^t m_i(s)\,ds$	How does accumulated marker exposure up to this time relate to the hazard?

The cumulative specification is useful when total exposure over time is scientifically relevant, but it has limitations. Its value depends on the chosen origin and interval; a long interval can make small trajectory differences accumulate substantially. A linear trajectory may be too restrictive when the marker changes nonlinearly, and a constant $\alpha$ assumes the same log-hazard change per unit cumulative exposure throughout follow-up. Finally, cumulative exposure can be strongly related to elapsed time and to other trajectory summaries, so estimates may be sensitive to the model specification and available information.

Common pitfall: Treating the slope contribution as $b_{1i}t$, or as $\frac{1}{2}b_{1i}t$, confuses the instantaneous marker level with its integral. The correct contribution is $\frac{1}{2}b_{1i}t^2$, because $\int_0^t b_{1i}s\,ds=\frac{1}{2}b_{1i}t^2$. Another error is to report $\exp(\alpha)$ as the hazard ratio for any two subjects: it is the multiplier for a one-unit difference in cumulative exposure, while the subject-specific contrast is $\exp\{\alpha[A_i(t)-A_j(t)]\}$.

Check Your Understanding

Use the linear trajectory and cumulative hazard link defined above. Each problem includes its solution.

A subject has $b_0=2$ marker units and $b_1=0.5$ marker units per week. What is cumulative exposure after 4 weeks?
Solution. $A(4)=2(4)+\frac{1}{2}(0.5)(4^2)=8+4=12$ marker-weeks.
Two subjects have the same starting level, and their slope difference is $0.1$ marker units per month. What is their cumulative exposure difference after 6 months?
Solution. The starting-level term is zero, so $\Delta A(6)=\frac{1}{2}(0.1)(6^2)=1.8$ marker-months.
If the cumulative exposure difference is 5 marker-months and $\alpha=0.03$ per marker-month, what is the conditional hazard ratio, assuming equal covariates?
Solution. The ratio is $\exp\{0.03(5)\}=\exp(0.15)\approx1.162$. The first subject's hazard is about 1.16 times the second subject's at the evaluated time.
Why does a constant $\alpha$ not imply a constant hazard ratio between two subjects with different linear trajectories?
Solution. Their cumulative exposure difference is $\Delta b_0t+\frac{1}{2}\Delta b_1t^2$, which generally varies with $t$. Since the hazard ratio is the exponential of $\alpha$ times that difference, it generally varies with time as well.

Association component	Quantity entering the hazard	Question it represents
Current value	\(m_i(t)\)	How does the marker level at this time relate to the hazard?
Instantaneous slope	\(b_{1i}\) for a linear trajectory	How does the rate of change relate to the hazard?
Cumulative exposure	\(A_i(t)=\int_0^t m_i(s)\,ds\)	How does accumulated marker exposure up to this time relate to the hazard?