Conditional and Marginal Cause-Specific Hazards

Why Conditioning Changes the Hazard

A joint model can describe each cause-specific event hazard as depending on a subject’s latent random effects. Conditional on those effects, the hazard describes the instantaneous event rate for a subject with a specified latent profile. A population-level question is different: among subjects who have remained free of every event up to time \(t\), what is the instantaneous rate of a particular cause when the latent effects are not observed? The answer is generally not the average of the conditional hazards under the original random-effects distribution.

The difference arises because remaining event-free is informative about latent risk. If subjects with some random-effect values tend to experience events earlier, fewer of them remain in the risk set at later times. Thus, the distribution of random effects among current survivors may differ from the distribution at study entry. The distinction matters when describing individual-level model parameters, reporting population-level rates, or interpreting an effect averaged over a population.

For a shared-random-effects model, one possible specification is \(\lambda_k(t\mid\mathbf b)=\lambda_{0k}(t)\exp(\boldsymbol\alpha_k^\mathsf{T}\mathbf b)\), where \(\lambda_{0k}(t)\) is the cause-specific baseline hazard and \(\boldsymbol\alpha_k\) describes how the latent effects enter that cause’s hazard. “Shared” means the same latent vector can contribute to multiple cause-specific hazards; its coefficients need not be identical across causes. The competing events are treated as mutually exclusive first events, and, conditional on \(\mathbf b\), the total event hazard is the sum of the cause-specific hazards: \[ \lambda_{\mathrm{all}}(t\mid\mathbf b)=\sum_{k=1}^K\lambda_k(t\mid\mathbf b). \]

Deriving the Marginal Hazard

Write \(S(t\mid\mathbf b)=P(T\geq t\mid\mathbf b)\) for conditional event-free survival. Under the cause-specific hazard model, if the hazards are integrable, \(S(t\mid\mathbf b)=\exp\{-\int_0^t\sum_{j=1}^K\lambda_j(u\mid\mathbf b)\,du\}\). The marginal survival probability is the population average of this conditional survival: \(\bar S(t)=E_{\mathbf b}\{S(t\mid\mathbf b)\}\). The crucial step is to calculate the cause-\(k\) event rate in the population still at risk, not in the population at entry.

The first line follows because, conditional on \(\mathbf b\) and being event-free at \(t\), the probability of a cause-\(k\) event in a short interval is \(\lambda_k(t\mid\mathbf b)\Delta t+o(\Delta t)\). Multiplying by the probability \(S(t\mid\mathbf b)\) of reaching that interval event-free and averaging over \(\mathbf b\) gives the population event probability. Divide the first line by \(\Delta t\,P(T\geq t)\) and take the limit as \(\Delta t\) decreases to zero. This yields

The last expectation is over the random-effect distribution among subjects who have survived event-free to \(t\). In fact, this distribution is proportional to \(S(t\mid\mathbf b)dF_{\mathbf b}(\mathbf b)\). This survivor weighting is the reason that \(\bar\lambda_k(t)\) generally differs from \(E_{\mathbf b}\{\lambda_k(t\mid\mathbf b)\}\). At entry, when \(S(0\mid\mathbf b)=1\), the two agree if the target distribution is the population distribution at entry. Later, they need not.

Survivor Weighting in a Two-Group Population

A finite mixture makes the distinction transparent. Suppose half the population has latent type L and half type H. Their conditional cause-specific hazards are constant over time: type L has hazards \(0.2\) and \(0.6\) per year for causes 1 and 2, while type H has hazards \(0.8\) and \(1.0\) per year. Conditional survival to time \(t\) is therefore \(e^{-0.8t}\) for type L and \(e^{-1.8t}\) for type H. The higher-risk type H is depleted from the event-free group more quickly.

Which Distribution Is Being Averaged?

The target distribution must be stated. A population marginal hazard uses the random-effects distribution at entry, updated by event-free survival as time passes. When a subject’s longitudinal measurements \(Y\) are available, prediction for subjects with those measurements instead starts from a distribution such as \(F_{\mathbf b\mid Y}\). If the prediction also conditions on being event-free to time \(t\), survival updates that distribution as well: \[ \bar\lambda_k(t\mid Y,T\geq t) = \frac{E_{\mathbf b\mid Y}\{S(t\mid\mathbf b)\lambda_k(t\mid\mathbf b)\}} {E_{\mathbf b\mid Y}\{S(t\mid\mathbf b)\}}. \] This is a marker-informed, survivor-specific rate. It is not generally the same as a population rate or a conditional hazard evaluated at a single estimated random effect.

Interpretation, Diagnostics, and Limitations

Conditional hazards answer a subject-specific, latent-profile question: what is the instantaneous cause-\(k\) rate at a specified \(\mathbf b\), given that the subject remains event-free? Marginal hazards answer a risk-set question: what is the instantaneous cause-\(k\) rate among the target group still event-free, after integrating over its current distribution of latent effects? A coefficient in a conditional hazard model therefore need not describe the change in a marginal hazard by the same amount. Even when conditional hazards have a simple proportional form, changing composition of the event-free group can make marginal hazards vary over time.

The formula relies on a coherent distribution for \(\mathbf b\), valid conditional cause-specific hazards, and a survival function generated by their sum for the mutually exclusive first events. For a marker-informed calculation, the conditional distribution of \(\mathbf b\) given the observed longitudinal information must also be appropriate. These are model-based quantities: misspecification of the random-effects distribution, hazard forms, or longitudinal model can distort the resulting marginal rates. Numerical integration may be necessary for continuous random effects; it should be checked by increasing integration accuracy and confirming that the marginal hazard is stable.

A useful diagnostic is to compare the original random-effects distribution with the implied distribution among survivors at several times. If high-risk latent profiles become rare among modeled survivors, marginal hazards should reflect that depletion. Unexpectedly unchanged marginal hazards in a strongly heterogeneous model, or numerical estimates that vary substantially with quadrature settings, warrant investigation. Also check that cause-specific hazards are nonnegative and that the summed hazard and survival calculations use all competing causes.

A common mistake is to calculate \(E_{\mathbf b}\{\lambda_k(t\mid\mathbf b)\}\) at every time and call it the marginal cause-specific hazard. That averages over the entry distribution, not the distribution among subjects still event-free at \(t\). The correction is to weight each conditional hazard by \(S(t\mid\mathbf b)\) and divide by \(E_{\mathbf b}\{S(t\mid\mathbf b)\}\). Another mistake is to read the cause-specific hazard share as a probability of eventually experiencing that cause. It is a share of instantaneous rates among those currently event-free, not a cumulative event probability. Cause-specific and subdistribution hazards are distinct quantities, as established in the earlier material on cause-specific and subdistribution hazards.

Marginalization does not remove dependence on modeling choices: it depends on the target population, the random-effects distribution, and—when relevant—the longitudinal information used to condition that distribution. Marginal hazards are also not interchangeable with conditional hazards for estimating subject-level associations. The appropriate quantity depends on the scientific question; neither is universally preferable.