Predicting Under a Cumulative-Exposure Association

Why Cumulative Exposure Changes a Landmark Prediction

A cumulative-exposure association makes the event hazard depend on the marker’s accumulated history, not only its current level or rate of change. At a prediction landmark, some of that history has already accrued. It remains part of the exposure used to calculate risk after the landmark: resetting exposure to zero at the landmark changes the model.

The calculations below use a latent marker trajectory $m_i(t;b_i)$, where $b_i$ denotes the patient-specific random effects. The trajectory is latent because observed biomarker measurements may be noisy; it is the modeled underlying level, not the raw measurement, that is integrated. The integration begins at the model’s specified time origin.

Definition: The cumulative latent exposure for patient $i$ by time $t$ is $$ A_i(t;b_i)=\int_0^t m_i(u;b_i)\,du. $$ A cumulative-exposure association specifies a hazard such as $$ \lambda_i(t\mid b_i)=\lambda_0(t)\exp\{\gamma^\mathsf{T}x_i+\alpha A_i(t;b_i)\}, $$ where $\lambda_0(t)$ is the baseline hazard, $x_i$ is a vector of baseline covariates, and $\gamma$ and $\alpha$ are association parameters. If $m_i(t)$ is measured in biomarker units and time in months, then $A_i(t)$ is measured in biomarker-months and $\alpha$ has units of inverse biomarker-months, making $\alpha A_i(t)$ dimensionless.

This specification assumes that the hazard at time $t$ depends on the accumulated latent exposure through $t$, as defined from time zero. It does not, by itself, assume that the marker is constant or that its cumulative effect is causal. The joint model must also specify a valid longitudinal model and event-time model, and its assumptions must be adequate for the intended prediction. For the calculations here, assume the hazard is conditionally specified given the latent trajectory and covariates, the baseline hazard is nonnegative, and the patient is event-free at landmark $s$. A right-censored patient can contribute event-free follow-up through $s$ if censoring is conditionally noninformative under the model.

Conditional Survival After the Landmark

For a fixed trajectory $b_i$, survival from the landmark $s$ to a later time $t>s$ is determined by the hazard over that interval. Exposure inside the hazard is still $A_i(v;b_i)$, which includes the contribution from times before $s$. It is not just the integral from $s$ to $v$.

P(T_i>t\mid T_i>s,b_i,x_i) =\exp\left\{-\int_s^t \lambda_0(v) \exp\bigl(\gamma^\mathsf{T}x_i+\alpha A_i(v;b_i)\bigr)\,dv\right\}.

For a patient with observed longitudinal history $Y_i(s)$, the random effects are not known. Let $p(b_i\mid Y_i(s),T_i>s,x_i)$ denote their conditional distribution given the measurements and the fact that the patient survived event-free to $s$. The dynamic survival prediction averages the fixed-trajectory survival over this distribution:

P(T_i>t\mid T_i>s,Y_i(s),x_i) =\int P(T_i>t\mid T_i>s,b_i,x_i)\, p(b_i\mid Y_i(s),T_i>s,x_i)\,db_i.

The conditioning on event-free survival matters. In a joint model, trajectories associated with greater hazard are less likely to belong to someone who has survived to the landmark. In a typical formulation, the conditional random-effects density is proportional to the prior density times the longitudinal likelihood and the probability of surviving to $s$:

p(b_i\mid Y_i(s),T_i>s,x_i)\ \propto\ p(Y_i(s)\mid b_i,x_i)\,P(T_i>s\mid b_i,x_i)\,p(b_i\mid x_i).

This expression presumes the specified conditional-independence structure of the joint model; any additional modeled observation or censoring processes would need to be handled according to that model. Once the patient-specific conditional distribution is obtained, each trajectory’s survival is calculated and then averaged. Averaging the trajectories first and inserting the mean trajectory into the hazard is generally not equivalent, because the hazard contains an exponential.

Why Past Exposure Must Remain in the Calculation

For a fixed trajectory, the conditional survival formula follows directly from the definition of hazard. Over a small interval $dv$, conditional survival changes at rate approximately $-\lambda_i(v\mid b_i)\,dv$. Thus the survival function $S_i(v\mid b_i)$ satisfies $dS_i(v\mid b_i)/dv=-\lambda_i(v\mid b_i)S_i(v\mid b_i)$. Integrating this differential equation from $s$ to $t$, and dividing by survival through $s$, gives:

\frac{S_i(t\mid b_i,x_i)}{S_i(s\mid b_i,x_i)} =\exp\left\{-\int_s^t\lambda_i(v\mid b_i)\,dv\right\}.

The lower limit $s$ determines the interval over which future hazard is accumulated. It does not change the model’s definition of $A_i(v)$. If the marker is constant at level $c$ from time zero, then $A_i(v)=cv$; at landmark $s$, the cumulative exposure is $cs$, not zero. More generally, for $v>s$,

A_i(v)=A_i(s)+\int_s^v m_i(u)\,du.

Worked Example: Constant Marker Level After a Landmark

Suppose a hypothetical patient has a latent marker level of 2 biomarker units from time 0 through month 4. The landmark is $s=2$ months, prediction is to $t=4$ months, the baseline hazard is constant at $0.02$ per month, there are no baseline covariates, and $\alpha=0.10$ per biomarker-month. Compute the patient’s conditional survival, treating this trajectory as known.

The cumulative exposure at time $v$ is $A_i(v)=2v$ biomarker-months. Therefore the conditional cumulative hazard from month 2 to month 4 is $$ \int_2^4 0.02\exp\{0.10(2v)\}\,dv =\frac{0.02}{0.20}\left(e^{0.20(4)}-e^{0.20(2)}\right) =0.1(e^{0.8}-e^{0.4}) \approx 0.07337. $$ The conditional survival is $\exp(-0.07337)\approx 0.9293$. Under the stated model and trajectory, the estimated probability of remaining event-free through month 4, conditional on being event-free at month 2, is about 92.93%.

The exponent uses the full cumulative exposure $2v$, including the 4 biomarker-months accumulated by month 2. The hazard is measured per month, so integrating it over months gives a dimensionless cumulative hazard.

Changing Levels and Exposure Accrued Before the Landmark

When the trajectory changes over time, calculate its integral piece by piece and carry the accumulated value forward. The value at the landmark is the starting accumulated exposure for the post-landmark segment, not a new zero.

Worked Example: A Marker Increase After the Landmark

Consider a hypothetical patient whose latent marker level is 2 biomarker units from month 0 through month 2, then 3 biomarker units from month 2 through month 4. Let $s=2$, $t=4$, $\lambda_0(v)=0.02$ per month, $\alpha=0.10$ per biomarker-month, and no baseline covariates. Assume this piecewise trajectory is known.

At the landmark, $A_i(2)=2(2)=4$ biomarker-months. For $2\leq v\leq4$, the marker adds $3(v-2)$ biomarker-months, so $$ A_i(v)=4+3(v-2)=3v-2. $$ The post-landmark cumulative hazard is $$ \int_2^4 0.02e^{0.10(3v-2)}\,dv =\frac{0.02}{0.30}\left(e^{1.0}-e^{0.4}\right) \approx 0.08176. $$ Thus the conditional survival is $e^{-0.08176}\approx0.9215$, or about 92.15% through month 4 given survival through month 2.

For comparison, incorrectly resetting exposure to zero at month 2 would use $A_i(v)=3(v-2)$. That gives a cumulative hazard of $$ \int_2^4 0.02e^{0.30(v-2)}\,dv =\frac{0.02}{0.30}(e^{0.6}-1) \approx 0.05481, $$ and survival of about $e^{-0.05481}=0.9467$. This is not the prediction under the stated model: it discards the 4 biomarker-months accrued before the landmark and therefore understates risk here.

Averaging Over Plausible Latent Trajectories

In practice, a patient’s trajectory is not known exactly. The posterior or conditional distribution of random effects represents the plausible trajectories after accounting for the biomarker history and survival to the landmark. The integral over that distribution is a weighted average of trajectory-specific survival probabilities, not a single calculation using an assumed known trajectory.

Worked Example: Two Trajectories With Different Conditional Weights

Suppose a hypothetical patient has reached month 2 event-free. After accounting for the patient’s biomarker history and survival to month 2, the conditional distribution assigns weight 0.70 to a trajectory with constant level 1 biomarker unit and weight 0.30 to one with constant level 2 biomarker units. Predict survival to month 4 with $\lambda_0=0.02$ per month, $\alpha=0.10$ per biomarker-month, and no baseline covariates.

For the level-1 trajectory, $A_i(v)=v$. Its cumulative hazard from 2 to 4 is $$ \frac{0.02}{0.10}(e^{0.10(4)}-e^{0.10(2)}) =0.2(e^{0.4}-e^{0.2}) \approx0.05408, $$ so its conditional survival is $e^{-0.05408}\approx0.9474$. For the level-2 trajectory, the calculation from the first example gives cumulative hazard $0.07337$ and conditional survival $0.9293$.

Average the two survival probabilities using their conditional weights: $$ 0.70(0.9474)+0.30(0.9293)\approx0.9419. $$ The resulting prediction is about 94.19% survival through month 4, conditional on the patient’s history and being event-free at month 2. The weights are conditional on that history and survival; using unupdated population weights would in general target a different prediction.

Model Checks, Alternatives, and Limitations

Before trusting a prediction, check that the association and integration match the model. In particular, verify the time origin used for cumulative exposure, the units of the marker and time, and whether the exposure is defined from baseline or another specified origin. Check that the latent trajectory is evaluated over the entire interval from its origin through each hazard time. For piecewise trajectories, confirm continuity of the accumulated integral at each change point. Numerical integration should be refined or checked with a second method if results change materially with the integration grid or tolerance.

The trajectory beyond the landmark is unobserved. A prediction therefore depends on the longitudinal model’s extrapolation as well as on the event model. Sparse measurements, a poorly fitting trajectory, influential random-effects estimates, or a baseline hazard that is inadequately represented can make the prediction unstable. A sensible diagnostic review includes examining longitudinal fit and residual patterns, checking the event-model fit and estimated association, and comparing predictions under reasonable changes to integration accuracy and model specification. These checks do not prove the model is correct; they can reveal consequential failures.

A current-value association uses the latent marker level at time $t$; a slope association uses its rate of change. A cumulative-exposure association instead uses the integral of the marker over its modeled history. It can represent a persistent contribution of earlier marker levels, but the basic form above weights exposure at all past times equally. It does not represent a decaying effect unless the model explicitly defines a different weighted integral. Nor does an estimated association establish that changing the marker would causally change event risk.

The exponential link requires a correctly scaled, dimensionless linear predictor; the exposure coefficient’s units must match the definition of the integral. If the marker has been centered or transformed, that definition must be used consistently in both the longitudinal trajectory and cumulative exposure. Predictions also rely on the joint model’s assumptions about the latent trajectory, event hazard, and dependence between them. A model that is mathematically computable can still yield misleading predictions if those assumptions are implausible.

Common pitfall: “At the landmark, future exposure starts at zero, so I integrate only the marker values after $s$.” That reasoning confuses the interval of future hazard accumulation with the definition of cumulative exposure. Under an exposure defined from time zero, use $A_i(v)=A_i(s)+\int_s^v m_i(u)\,du$ inside the hazard for every $v>s$.

Check Your Understanding

Try each question, then compare your reasoning with the solutions.

A marker is constant at 1.5 biomarker units from time zero. What is its cumulative exposure at month 4, and what are the units?
With $\lambda_0=0.01$ per month, no covariates, and $\alpha=0$, what is conditional survival from month 2 to month 5?
A marker is 2 biomarker units from month 0 to month 1 and 1 biomarker unit from month 1 onward. Write $A_i(v)$ for $1\leq v\leq3$.
Why is the average of trajectory-specific conditional survival probabilities generally preferable to putting an average trajectory directly into the hazard?
What information must the random-effects weights reflect for a prediction conditional on a patient’s biomarker history and survival to the landmark?

Solutions. (1) $A_i(4)=1.5(4)=6$ biomarker-months. (2) When $\alpha=0$, the hazard is $0.01$ per month, so survival is $\exp[-0.01(5-2)]=e^{-0.03}\approx0.9704$. (3) Exposure accumulated by month 1 is 2 biomarker-months; thereafter $A_i(v)=2+(v-1)=v+1$ biomarker-months. (4) The hazard includes an exponential of exposure, and averaging and exponentiating do not generally commute; the conditional prediction averages survival over trajectory uncertainty. (5) They must reflect both the longitudinal history and event-free survival through the landmark, as well as the model’s prior distribution for random effects.