Longitudinal Omics In Competing-Risk Models

Why Competing Events Change the Question

A repeated omics measure may evolve before a participant experiences one of several mutually exclusive first events. For example, a patient may experience disease progression, die before progression, or remain event-free until the study ends. Death is not ordinary loss to follow-up: once it occurs, progression can no longer occur as the first event. A useful analysis must therefore represent both the evolving marker and the alternative event types.

The target in this tutorial is a cause-specific association: how the current underlying omics trajectory is associated with the instantaneous rate of a particular event among people who have not yet experienced any event. This is not the same as asking for the probability of that event by a future time. Competing causes affect that probability, even when they are not the cause being modeled.

Definition: Let $T_i$ be the time to the first event for participant $i$, and let $J_i \in \{1,\ldots,K\}$ identify its cause. The cause-specific hazard for cause $k$, conditional on information $H_i(t)$ available by time $t$, is $$ \lambda_{ik}(t\mid H_i(t))= \lim_{\Delta t\to 0} \frac{P(t\leq T_i<t+\Delta t,\ J_i=k\mid T_i\geq t,\ H_i(t))}{\Delta t}. $$ It is an instantaneous event rate among participants still free of every event at time $t$.

For longitudinal omics, the history $H_i(t)$ should represent the participant's marker trajectory up to time $t$, not measurements that would only become available in the future. In a joint model, the true trajectory is latent: observed assay values are treated as noisy measurements of it. This distinction matters because inserting a noisy last measurement directly into a hazard model can misrepresent the underlying time-varying association.

A Joint Model for Trajectories and Competing Causes

A common specification has two linked parts. For a single marker, the longitudinal submodel can be written as

Y_{ij}=m_i(t_{ij})+\epsilon_{ij}, \qquad m_i(t)=\boldsymbol{x}_i(t)^\top\boldsymbol{\beta} +\boldsymbol{z}_i(t)^\top\boldsymbol{b}_i,

where $Y_{ij}$ is the measured marker for participant $i$ at visit time $t_{ij}$, $m_i(t)$ is the latent trajectory, and $\boldsymbol{b}_i$ represents participant-specific random effects. A standard assumption is that assay errors $\epsilon_{ij}$ have mean zero, are independent of the random effects, and have a specified distribution, often normal with variance $\sigma^2$. The trajectory form and error distribution must be adequate for the data; for example, a linear time trend is inappropriate if the marker has a strong nonlinear pattern.

For each event cause $k$, a cause-specific hazard model can link that trajectory to the event rate:

\lambda_{ik}(t\mid\boldsymbol{b}_i)= \lambda_{0k}(t) \exp\left\{\boldsymbol{\gamma}_k^\top\boldsymbol{w}_i+ \alpha_k m_i(t)\right\}.

Here, $\lambda_{0k}(t)$ is the baseline hazard for cause $k$, $\boldsymbol{w}_i$ contains other covariates, and $\alpha_k$ is the cause-specific association with the latent marker. The coefficients may differ by cause: a rising marker could be associated with a higher rate of one event and little or no change in another. The model assumes that, conditional on the specified covariates and random effects, the hazard depends on the marker through the stated current-value term. A slope or accumulated exposure can be included instead, but it defines a different association and must be specified in advance.

A joint likelihood links the repeated measurements to the observed event history through the shared random effects. Conditional on $\boldsymbol{b}_i$, a participant observed until time $T_i$ contributes a longitudinal density and an event-history contribution. If $\delta_i=1$ indicates an observed event of cause $J_i$, and $\delta_i=0$ indicates right censoring, the event-history contribution is

\left\{\lambda_{iJ_i}(T_i\mid\boldsymbol{b}_i)\right\}^{\delta_i} \exp\left\{-\int_0^{T_i}\sum_{k=1}^{K} \lambda_{ik}(u\mid\boldsymbol{b}_i)\,du\right\}.

The full participant contribution integrates the product of this term and the longitudinal density over the random-effects distribution. This construction accounts for measurement error in the trajectory while using the fact that all event causes remove a participant from the event-free risk set.

Worked Example: Interpreting a Cause-Specific Association

Suppose a fitted joint model for progression has $\alpha_1=0.55$, where the marker trajectory is standardized so that one unit is one standard deviation. The cause-specific hazard is proportional to $\exp\{0.55m_i(t)\}$, conditional on the other model terms.

The hazard ratio for a one-standard-deviation higher latent marker value at the same time is

\exp(0.55)=1.733.

Thus, among participants still free of any event and with the same modeled covariates, a one-standard-deviation higher current latent marker value is associated with about 1.73 times the instantaneous progression rate. This is an association in a cause-specific hazard model. It does not mean that the probability of progression is 73.3 percentage points higher, nor does it establish that changing the marker would change progression risk.

Why the Cumulative Incidence Uses Every Cause

The probability of experiencing cause $k$ by time $t$, called its cumulative incidence function, depends both on the cause-$k$ hazard and on remaining event-free long enough for that event to occur. With hazards conditional on a fixed covariate profile, event-free survival is

S(t)=\exp\left\{-\int_0^t\sum_{k=1}^{K}\lambda_k(u)\,du\right\}, \qquad F_k(t)=\int_0^t S(u)\lambda_k(u)\,du.

The sum in $S(t)$ includes every cause. To see why, over a short interval $du$, the probability of remaining event-free is approximately $1-\sum_k\lambda_k(u)du$. Multiplying these conditional survival probabilities over time gives the exponential survival expression. The probability of cause $k$ occurring in a short interval near $u$ is then the probability of reaching that interval event-free, $S(u)$, times its cause-specific event rate, $\lambda_k(u)$, times the interval length. Integrating those increments gives $F_k(t)$.

Worked Example: Constant Competing Hazards

Consider a hypothetical population with constant cause-specific hazards $\lambda_1=0.08$ per year for progression and $\lambda_2=0.12$ per year for death before progression. Assume these hazards are constant over the two-year period for the covariate profile being considered.

The probability of remaining free of either event through time $u$ is $S(u)=\exp\{-(0.08+0.12)u\}=\exp(-0.20u)$. The two-year cumulative incidence of progression is therefore

F_1(2)=\int_0^2 0.08\exp(-0.20u)\,du =\frac{0.08}{0.20}\left(1-\exp(-0.20\times2)\right) =0.1319.

The model gives a 13.19% probability of progression as the first event by two years for this profile. If death were ignored and the calculation instead used $1-\exp(-0.08\times2)=0.1479$, it would overstate that probability: it would count some people as available to progress even though they experienced death first.

When trajectories vary across participants, an individual-specific cumulative incidence can be calculated conditional on their trajectory and covariates. A population-level probability generally requires averaging these individual probabilities over the relevant distribution of trajectories and covariates. Reporting a hazard ratio alone does not provide that probability.

Worked Example: Converting a Predicted Trajectory Into Cause-Specific Rates

Suppose a hypothetical participant has a fitted latent marker trajectory $m_i(t)=0.20+0.40t$, with time in years and the marker in standardized units. At year 2, the predicted marker is

$$ m_i(2)=0.20+0.40(2)=1.00. $$

Assume the cause-specific hazard models at that time have baseline hazards $0.04$ and $0.06$ per year for causes 1 and 2, respectively, with marker coefficients $\alpha_1=0.70$ and $\alpha_2=-0.20$; take other covariate terms to be zero. The predicted cause-specific rates are

\lambda_{i1}(2)=0.04\exp\{0.70(1.00)\}=0.0806 \quad\text{per year},

\lambda_{i2}(2)=0.06\exp\{-0.20(1.00)\}=0.0491 \quad\text{per year}.

Among participants with this modeled profile who are still event-free at year 2, the progression rate is about 0.0806 per year and the competing-event rate is about 0.0491 per year. Their sum, 0.1297 per year, is the instantaneous rate of any first event under these cause-specific hazards. These rates do not by themselves give the chance of either event over a future interval; that requires integrating the hazards while accounting for event-free survival.

Assumptions, Diagnostics, and Alternatives

The model is useful only to the extent that its assumptions fit the study. In addition to adequate trajectory and hazard specifications, common assumptions include correctly classified first-event causes, an appropriately specified random-effects distribution, and censoring that is independent of the event process conditional on modeled information. If measurement visits depend on a participant's unobserved health or marker level, the visit process may also need to be modeled; otherwise, the observed trajectory can be selectively sampled. The earlier material on informative omics visit processes addresses this distinct source of selection.

Check that assay residuals and fitted trajectories do not show systematic time patterns, and examine whether the assumed random-effects distribution is plausible. For each cause, assess whether the marker association and other covariate effects have the assumed time pattern; for a proportional-hazards specification, an association that changes markedly over time is a warning. Check event counts and risk-set sizes, cause coding, influential observations, and whether predictions are being made beyond the range of observed marker histories. Predicted cause-specific cumulative incidences can also be compared with observed event proportions over time using an appropriate competing-risk summary.

A cause-specific Cox model is a common alternative when the goal is a cause-specific association and a time-updated covariate is adequately observed. It models one cause's hazard among those still event-free; competing events end follow-up for that cause's risk set. It does not, by itself, model a latent noisy trajectory. A joint model is attractive when repeated measurements are error-prone or their observation is linked to the event process, but it adds distributional and computational assumptions.

A Fine–Gray subdistribution-hazard model is another alternative, often used when the main target is a covariate association with the cumulative incidence function. Its risk-set construction retains people with competing events in a modified way, so its coefficient is not a cause-specific hazard ratio. Neither model is universally preferable: choose the approach according to whether the scientific target is an instantaneous cause-specific rate or the probability of a particular event by a specified time.

Common Pitfalls and Practical Limits

Treating a competing event as ordinary censoring for event probability. A cause-specific hazard can be estimated from a risk set that ends at any first event, but a cumulative incidence calculation must include the hazards of all competing causes.
Reading a hazard ratio as a risk ratio. A cause-specific hazard ratio compares instantaneous rates among those still event-free; it is not a ratio of cumulative probabilities.
Using the last observed assay as if it were the true current marker. This can introduce measurement-error and timing problems. A joint model instead estimates the latent trajectory, subject to its assumptions.
Interpreting an association causally. A marker coefficient describes a conditional association under the model. Confounding, selection, and the distinction between a biomarker and an intervention remain relevant.

Joint competing-risk models can be demanding to fit, particularly with several markers, causes, or flexible trajectories. Sparse events for one cause can make its association imprecise, and results can be sensitive to the baseline-hazard and random-effects specifications. Report cause-specific associations with their uncertainty and, when the clinical question concerns event probabilities, provide cumulative incidence estimates that account for all causes.

Check Your Understanding

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

In a cause-specific hazard model, who is in the risk set for cause 1 at time $t$?
With constant hazards $\lambda_1=0.05$ and $\lambda_2=0.05$ per year, what is the two-year cumulative incidence for cause 1?
What does a cause-1 marker coefficient of $\log(2)$ mean if the marker is measured in one-unit increments?
Why can a cause-specific Cox model treat a competing event as ending follow-up for that cause, while a cumulative incidence calculation cannot simply ignore it?
Name one advantage and one limitation of using a joint model for a noisy repeated omics marker.

Solutions. (1) The risk set consists of participants who have not yet experienced any event by $t$ and are still under observation. (2) The total hazard is $0.10$ per year, so $F_1(2)=\frac{0.05}{0.10}\{1-\exp(-0.10\times2)\}=0.0906$, or 9.06%. (3) Conditional on the model terms and being event-free, a one-unit higher current latent marker value is associated with twice the instantaneous cause-1 hazard; it does not imply twice the cumulative incidence. (4) Ending the cause-specific risk set at the competing event correctly defines the instantaneous rate among people still event-free. But the event removes a person from the possibility of later experiencing the event of interest, so its hazard must contribute to event-free survival when calculating cumulative incidence. (5) A joint model can account for measurement error by linking the latent trajectory to the event process. Its limitation is that results depend on assumptions about the trajectory, error, random effects, and hazard structure.