Cause-Specific Hazards With Shared Biomarker Effects

Why Include Shared Biomarker Effects?

In competing-risks data, an individual who experiences one event type is no longer at risk of experiencing another as a first event. A biomarker measured repeatedly before the event may help describe differences in event risk, but its observed measurements can be noisy or incomplete. A joint model connects the longitudinal biomarker process and the event process through latent subject-specific random effects. This tutorial formulates the event part of that model and clarifies what each cause-specific hazard says.

Let $T_i$ be the time to the first event or censoring for individual $i$, and let $J_i$ identify the event cause when an event occurs, with $J_i\in\{1,\ldots,K\}$. Let $b_i$ denote the individual’s random-effects vector in the longitudinal biomarker model, and let $w_i(t)$ denote observed covariates that may vary over time. The random effects are called shared because the same latent vector $b_i$ is used in the longitudinal model and in the cause-specific event models.

Definition: The cause-$k$ hazard at time $t$, conditional on the individual’s random effects and covariates, is the instantaneous rate of experiencing cause $k$ at $t$, among individuals who have had no event before $t$. Formally, $$ h_{ik}(t\mid b_i,w_i)=\lim_{\Delta t\downarrow 0} \frac{P(t\leq T_i<t+\Delta t,\ J_i=k\mid T_i\geq t,\ b_i,\ w_i(t))}{\Delta t}. $$ It is a rate, not a probability. For example, its units are events per year when time is measured in years.

A common shared-effects specification is

h_{ik}(t\mid b_i,w_i)=h_{0k}(t) \exp\!\left\{\gamma_k^\mathsf{T}w_i(t)+\alpha_k^\mathsf{T}b_i\right\}, \qquad k=1,\ldots,K.

Here $h_{0k}(t)$ is the baseline hazard for cause $k$, $\gamma_k$ is the coefficient vector for observed covariates, and $\alpha_k$ is the coefficient vector for the shared random effects. The exponential link keeps the hazard nonnegative. Cause-specific baseline hazards and coefficients allow the event types to have different risk patterns. The $b_i$ appearing in each cause model is the same latent individual effect used by the longitudinal model; the coefficient vectors need not be identical across causes.

Connecting the Hazard to the Biomarker Model

For example, a longitudinal model may represent the biomarker measurement $y_{ij}$ at time $t_{ij}$ as

y_{ij}=m_i(t_{ij})+\epsilon_{ij}, \qquad m_i(t)=x_i(t)^\mathsf{T}\beta+z_i(t)^\mathsf{T}b_i.

In this expression, $m_i(t)$ is the latent biomarker trajectory, $x_i(t)$ and $z_i(t)$ are design vectors, and $\epsilon_{ij}$ is measurement error. The event model uses $b_i$, not the noisy measurement $y_{ij}$ at one visit. This creates a shared-effects connection: information about the biomarker measurements informs the distribution of $b_i$, while $b_i$ also enters the cause-specific hazards. A particular implementation must specify a distribution for $b_i$, commonly a multivariate normal distribution with mean zero and covariance matrix $D$, as well as a model for measurement errors.

The conditional meaning matters. The cause-$k$ hazard describes the instantaneous cause-$k$ event rate for an individual with given $b_i$ and covariate history who remains event-free just before $t$. It does not describe the probability that the individual will ever experience cause $k$. Nor does conditioning on $b_i$ mean that the random effect is directly observed; it is latent and inferred from the longitudinal and event data under the joint model.

Worked Example: A Conditional Hazard Ratio for One Cause

Hypothetical setup. Suppose cause 1 has baseline hazard $h_{01}(t)=0.04$ events per year at a specified time. A binary covariate is coded $1$ for exposed and $0$ for unexposed, with coefficient $\log(1.5)$. For one individual, the random-effects contribution to the linear predictor is $0.15$. Compare the conditional hazard with the baseline hazard.

The assumptions for this calculation are that the specified baseline hazard and covariate value apply at this time, the individual is event-free immediately beforehand, and the random-effects contribution is held fixed at $0.15$. The linear predictor is

\eta=\log(1.5)(1)+0.15=0.4055+0.15=0.5555.

Thus the conditional hazard is

h_{i1}=0.04\exp(0.5555) \approx0.04(1.7428) =0.06971\ \text{events per year}.

At these covariate and random-effects values, the modeled cause-1 hazard is about $0.06971$ events per year. The covariate coefficient corresponds to a hazard ratio of $1.5$ for exposed versus unexposed individuals at the same random-effects value, with other model terms held fixed. This rate calculation is not a statement that the individual has a $6.971\%$ probability of an event in a year; a hazard is an instantaneous rate, and a probability over an interval depends on the hazard over that interval.

What the Shared-Effects Structure Implies

The model is conditional on $b_i$. Holding the covariate history and random effects fixed, changing a covariate by one unit changes the cause-$k$ hazard by a multiplicative factor determined by its coefficient. Similarly, comparing two random-effects values $b_i$ and $b_i^\ast$ at the same time and covariates gives

\frac{h_{ik}(t\mid b_i,w_i)} {h_{ik}(t\mid b_i^\ast,w_i)} = \exp\!\left\{\alpha_k^\mathsf{T}(b_i-b_i^\ast)\right\}.

This is a conditional comparison, not a causal effect. The shared random effects allow the latent biomarker trajectory and event risk to be associated through subject-specific features represented by $b_i$. They do not, by themselves, establish that changing the biomarker would change event risk. The choice of which components enter the hazard, and how they are parameterized, must follow a scientifically justified model.

Because only one first event can occur, the cause-specific hazards describe competing ways to leave the event-free state. Conditional on $b_i$ and covariates, the instantaneous rate of any first event is the sum of the cause-specific hazards:

h_{i\cdot}(t\mid b_i,w_i)=\sum_{k=1}^{K}h_{ik}(t\mid b_i,w_i).

To see why, over a short interval $[t,t+\Delta t)$, the probability of a cause-$k$ event for someone event-free at $t$ is approximately $h_{ik}(t)\Delta t$. The cause-specific events are mutually exclusive as first events, so the probability of any event over that interval is approximately $\sum_k h_{ik}(t)\Delta t$. Dividing by $\Delta t$ and taking the limit gives the sum. This is a local statement about rates; it does not give the probability of a particular cause by a later time.

Worked Example: Summing Two Cause-Specific Hazards

Hypothetical setup. At a specified time, suppose two causes have baseline hazards $0.03$ and $0.02$ events per year. For a particular event-free individual with fixed random effects and covariates, the cause-1 linear predictor is $\log(1.2)+0.10$, and the cause-2 linear predictor is $\log(0.8)-0.05$. Assume the cause-specific hazards follow the log-linear model above.

For cause 1,

\eta_{i1}=\log(1.2)+0.10=0.2823, \qquad h_{i1}=0.03\exp(0.2823) \approx0.03(1.3262) =0.03979\ \text{events per year}.

For cause 2,

\eta_{i2}=\log(0.8)-0.05=-0.2731, \qquad h_{i2}=0.02\exp(-0.2731) =0.02(0.7610) =0.01522\ \text{events per year}.

The total instantaneous event rate is therefore

h_{i\cdot}=0.03979+0.01522 =0.05501\ \text{events per year}.

For a sufficiently short interval of $0.01$ year, the corresponding first-order approximations to the conditional event probabilities are $0.03979(0.01)=0.0003979$ for cause 1 and $0.01522(0.01)=0.0001522$ for cause 2. The total is $0.0005501$. These are short-interval approximations, not exact one-year probabilities, and all are conditional on the stated covariates, random effects, and being event-free at the start of the interval.

Assumptions, Diagnostics, and Alternatives

A shared-effects cause-specific model is meaningful only with a clearly defined event process and a plausible longitudinal model. In particular, the analysis should make the following assumptions explicit:

Well-defined first-event causes. Event types are classified consistently, and an event of one cause removes the individual from the risk set for the other first-event causes.
Appropriate hazard form. Conditional on the modeled covariates and random effects, the log hazard has the specified form. If the model is proportional in a covariate, its hazard ratio is constant over time unless time interactions or other time-varying effects are included.
Adequate longitudinal and random-effects models. The trajectory, measurement-error structure, and distribution of $b_i$ are plausible enough to represent the latent subject-specific heterogeneity relevant to the analysis.
Suitable censoring assumptions. For likelihood-based inference, censoring must be handled by the specified analysis and be independent of the event process conditional on the modeled information, or have its dependence represented appropriately.
Correct use of time and covariates. Covariates used at $t$ must be available according to the model at that time; future biomarker measurements cannot be treated as if they were already observed.

Diagnostics should assess both parts of the joint model. Check whether the longitudinal model reproduces the observed biomarker patterns and whether residuals suggest unmodeled structure or changing measurement variability. For the event component, examine cause-specific fit over time, covariate functional forms, and whether observed event patterns are systematically missed. Also check whether the random-effects distribution and estimated variability are credible; sparse events for a cause or limited repeated measurements can leave the association with $b_i$ weakly identified. A good-looking fit for one component does not guarantee that the shared structure is adequate.

The main modeling alternative is an ordinary cause-specific regression that uses observed covariates but omits the longitudinal random effects. That can be appropriate if the latent biomarker heterogeneity is not part of the scientific target or the available data cannot support a joint model. It does not, however, represent the same conditional hazard structure. A joint shared-effects model instead connects the event hazard to latent subject-specific features estimated using the longitudinal data, at the cost of additional distributional and model-form assumptions. The choice should reflect the question and the data, not a presumption that the more complex model is always preferable.

Common pitfall: It is incorrect to describe $h_{ik}(t\mid b_i,w_i)$ as the probability that individual $i$ will eventually experience cause $k$, or to interpret $\alpha_k$ as a causal effect of the biomarker. The hazard is a conditional instantaneous rate among individuals still event-free; the random effects are latent model components. State the conditioning set and the rate units whenever interpreting it.

Putting the Formulation to Work

A disciplined formulation keeps the event type, risk set, and conditioning set visible. For each cause, specify a baseline hazard, covariate terms, and the random-effects contribution. Then interpret the result as a rate among people who remain event-free at the time in question, conditional on their latent effects and modeled covariates. This prevents a cause-specific hazard from being confused with a probability of eventual event type or with an unconditional population rate.

The shared-effects approach is useful when longitudinal measurements are noisy and subject-specific biomarker patterns may carry information about event risk. Its limitations are equally important: results depend on the assumed longitudinal trajectory, random-effects distribution, and event-hazard form, and latent effects are not observed directly. Sensitivity analysis and diagnostics should therefore accompany substantive interpretation.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

If $h_{ik}(t\mid b_i,w_i)=0.06$ per year, what does this number mean, and what does it not mean?
Two individuals have the same covariates but random effects $b_i$ and $b_i^\ast$. Write their cause-$k$ hazard ratio.
At a fixed time, two cause-specific hazards are $0.012$ and $0.008$ per year. What is the total instantaneous hazard of any first event?
Why does using the same $b_i$ in the longitudinal and event models not make the biomarker effect causal?

Solutions. (1) It is the conditional instantaneous rate of cause $k$ among individuals still event-free, for the specified $b_i$ and covariates. It is not a $6\%$ probability of eventual cause $k$. (2) The ratio is $\exp\{\alpha_k^\mathsf{T}(b_i-b_i^\ast)\}$, holding time and covariates fixed. (3) The total instantaneous hazard is $0.012+0.008=0.020$ per year. (4) The random effects are latent subject-specific model components; their shared use represents conditional association within the model, not an intervention on the biomarker or proof of causation.

Cause-Specific Hazards With Shared Biomarker Effects

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