Conditional and Marginal Hazard Interpretations

Why Two Hazard Interpretations Are Needed

In a shared random-effects joint model, a subject’s latent characteristics help describe both the longitudinal trajectory and the event process. A hazard conditional on those characteristics describes the instantaneous event rate for a subject with specified random effects. A population-level hazard instead describes the instantaneous event rate among subjects who have remained event-free up to a given time, without conditioning on their individual random effects.

These are different quantities. At the start of follow-up, the marginal hazard averages conditional hazards over the population distribution of random effects. Later, subjects with different random effects may have different probabilities of remaining event-free, so the risk set no longer has the original random-effects distribution. The marginal hazard averages over that changed composition.

For clarity, suppose \(w\) denotes observed baseline covariates held fixed in the expressions below, and \(b\) has population density \(f_b(b\mid w)\). A general conditional hazard specification is

where \(h_0(t)\) is a baseline hazard and \(q(t,b,w)\) is a specified predictor. For example, \(q\) might include a linear function of shared random effects. The distinction in this tutorial does not depend on the specific form of that predictor.

The conditional hazard is a rate, with units of inverse time. It is not the probability of an event at exactly time \(t\). For a small interval of length \(\Delta t\), conditional on being event-free at \(t\), the event probability is approximately \(h(t\mid b,w)\Delta t\), when the hazard is sufficiently regular over that interval.

From Conditional Survival to the Marginal Hazard

Assume event times are continuous, the conditional hazard is nonnegative and integrable over the time interval considered, and the random-effects distribution describes the population at the start of follow-up. Conditional on \(b,w\), the survival function is

The marginal survival function averages this conditional survival over the baseline random-effects distribution:

This integral gives the probability of surviving to \(t\) for a randomly selected subject with covariates \(w\). To obtain the marginal hazard, differentiate the marginal survival and use \(h_{\mathrm{marg}}(t\mid w)=-\frac{d}{dt}\log S_{\mathrm{marg}}(t\mid w)\). If differentiation can be interchanged with integration, then \(\frac{d}{dt}S(t\mid b,w)=-h(t\mid b,w)S(t\mid b,w)\), giving

where the random-effects distribution among survivors is

The second expression is the key interpretation: the marginal hazard is an average of conditional hazards using the random-effects distribution among subjects who have survived to \(t\). Survival reweights the original distribution. A random-effects value associated with higher event hazard will generally receive less weight among survivors, because subjects with that value are less likely to have survived.

A Discrete Random-Effects Distribution

A small number of random-effects groups makes the survivor reweighting easy to see. Suppose half the population has a high-risk random-effects value with constant conditional hazard \(0.8\) per year, and half has a low-risk value with constant conditional hazard \(0.2\) per year. Assume no other covariates and that group membership is fixed at baseline. Each group’s conditional survival is exponential, \(S(t\mid b)=\exp\{-h(b)t\}\).

Continuous Random Effects and the Initial Hazard

For a continuous random-effects distribution, the same integral applies. At time zero, everyone is still at risk, so \(S(0\mid b,w)=1\) and the survivor distribution equals the baseline distribution. Therefore, when the expectation exists,

This initial-time identity is a useful check. It does not say that at later times one can average conditional hazards over the original random-effects distribution; later survivor weights involve \(S(t\mid b,w)\).

Assumptions, Diagnostics, and Alternatives

The interpretation and formulas rely on a coherent conditional event model and a specified baseline distribution for \(b\). The conditional survival must follow from the stated hazard, and the marginal quantities require integrating over the correct distribution of random effects for the target population. If random-effects distributions differ across covariate strata, the relevant distribution is \(f_b(b\mid w)\), not a single distribution applied indiscriminately to everyone. The interchange of differentiation and integration used in the derivation requires suitable integrability; in common parametric models, this is satisfied when the relevant survival-weighted hazard integrals are finite.

In practice, the random-effects distribution and conditional hazard are estimated from data, so a reported marginal hazard is model-based. Useful checks include examining whether the fitted random-effects distribution is plausible, whether the conditional hazard specification fits the event data, and whether numerical integration is stable. For a discrete approximation, the survivor weights should be nonnegative and sum to one. At time zero, a calculation should agree with the baseline-distribution average of the conditional hazards. A failure of these checks can indicate an algebraic error, an incorrect risk-set distribution, or a model or numerical-integration problem.

The principal alternatives answer different questions. A conditional hazard reports the rate at a specified random-effects value and is useful for describing the model’s subject-specific event process. A marginal hazard reports the rate among event-free members of a population after integrating over latent heterogeneity. Plugging in an estimated random effect for each subject does not perform that population averaging. Likewise, averaging conditional hazards using the original random-effects distribution at every time ignores how survival changes the risk-set composition.

Neither hazard is an event probability, and neither by itself establishes a causal effect. The marginal hazard also depends on the population being averaged over: changing the baseline distribution of random effects can change the resulting marginal quantity. These distinctions matter when explaining model output to clinicians or comparing an individual-level model statement with a population-level summary.