Opposing Associations and Cumulative Incidence

Why Opposing Hazard Associations Need Not Cancel

A longitudinal biomarker may be associated with greater risk of one event cause and lower risk of another. For example, a higher latent biomarker level could accompany a higher instantaneous hazard of cause 1 but a lower instantaneous hazard of cause 2. Those directions alone do not tell us how the probabilities of experiencing each cause accumulate over time: each cause competes with the other, because the first event prevents a later event of the other type from being observed.

The relevant quantities are the cause-specific cumulative incidence functions. Each one combines the hazard for its particular cause with the probability of remaining event-free up to each time. In a joint model, predictions can be conditional on a subject’s latent biomarker trajectory and random effects, or averaged over their distribution. This tutorial develops the conditional calculation first, then explains the distinction.

Hazards, Survival, and Cumulative Incidence

Write \(m_i(t)\) for the latent underlying biomarker value at time \(t\) from the longitudinal model, and \(w_i\) for covariates in the event model. A two-cause current-value association model can be written as

Here \(\lambda_{0k}(t)\) is the baseline hazard for cause \(k\), \(\gamma_k\) is its covariate coefficient vector, and \(\alpha_k\) is the association parameter for the biomarker. Holding the other terms fixed, \(\alpha_1>0\) means that a one-unit increase in \(m_i(t)\) multiplies the cause-1 hazard by \(\exp(\alpha_1)\); \(\alpha_2<0\) means it multiplies the cause-2 hazard by a factor below one. Hazards have units of inverse time, while the \(\alpha_k\) coefficients are on the log-hazard scale per biomarker unit.

The two cause-specific hazards add to give the hazard of any first event. Conditional event-free survival therefore depends on their sum, not on either cause in isolation:

The cause-specific cumulative incidence is the accumulated chance of having cause \(k\) at time \(t\). At each time \(u\), the cause-\(k\) hazard contributes only for subjects still event-free at \(u\):

Why the Formula Has This Form

Over a short interval from \(u\) to \(u+du\), the conditional probability of cause \(k\) occurring is approximately \(\lambda_{ik}(u)\,du\), given event-free survival just before \(u\). The probability of reaching that interval event-free is \(S_i(u)\). Their product is thus the approximate probability of a cause-\(k\) event in that interval, and integrating these contributions from 0 to \(t\) gives \(F_{ik}(t)\).

This argument also explains why integrating the cause-specific hazard alone is not sufficient. A subject who has already had cause 1 cannot later have cause 2 as their first event. Accordingly, the survival factor in the cause-2 integral must reflect removal by both causes. Differentiating the survival function gives \(S_i'(u)=-S_i(u)\{\lambda_{i1}(u)+\lambda_{i2}(u)\}\). Summing the two cumulative-incidence derivatives and integrating therefore yields

when there are exactly two possible first-event causes and no additional event type. This identity is a useful calculation check: the two cause-specific probabilities must sum to the probability of any event by \(t\).

When the Total Hazard Also Changes

Opposing associations do not generally leave overall survival unchanged. If the two hazard changes differ in size, their sum can increase or decrease. That changes how much time subjects remain at risk for either cause and therefore changes both cumulative-incidence curves.

Piecewise Hazards and Accumulation Over Time

When hazards vary with time, the constant-hazard shortcut does not apply across the full follow-up period. One can still calculate cumulative incidence by integrating over time, or by summing interval contributions when hazards are piecewise constant. On an interval of length \(\Delta\) with constant hazards \(\lambda_1,\lambda_2\), let \(q=\lambda_1+\lambda_2\). If event-free survival at the start of the interval is \(S_{\text{start}}\), the interval contributes

to the cumulative incidence of cause \(k\). This follows from integrating \(S_{\text{start}}\exp(-qs)\lambda_k\) from \(s=0\) to \(s=\Delta\).

Conditional Predictions, Population Averaging, and Checks

In a joint model, a subject’s \(m_i(t)\) is typically estimated through a longitudinal model, and may depend on subject-specific random effects. A cumulative-incidence curve calculated for one specified trajectory and random-effect value is conditional on those values. A population-averaged prediction instead averages the conditional curve over the relevant distribution of random effects and, if required, covariates. Because the cumulative-incidence formula is nonlinear, evaluating it at an average trajectory generally does not equal averaging the curves:

in general, where \(b_i\) denotes the random effects. State which prediction is being reported; otherwise, a conditional risk for a typical specified trajectory can be mistaken for a population risk.

The calculation assumes a correctly specified pair of cause-specific hazards and a well-defined event-free survival process. For a joint-model analysis, the longitudinal model and its link to the hazards must also be adequate for the prediction being made. In estimation, the event cause must be observed as modeled, and censoring must be handled under the study’s stated censoring assumptions; for example, likelihood-based inference commonly requires censoring to be independent of the event process conditional on the modeled information. The numerical examples assume exact event times and causes and no censoring in the calculation.

Several checks help identify errors. Each cause-specific cumulative incidence should be nondecreasing and between 0 and 1. The sum \(F_{i1}(t)+F_{i2}(t)\) should not exceed 1 and, with only these two first-event causes, should equal \(1-S_i(t)\). Under constant hazards, the cause-specific shares of the event probability should equal \(\lambda_k/(\lambda_1+\lambda_2)\). Large discrepancies may indicate that a cause was omitted from the total hazard, the survival factor was not updated across intervals, or a hazard ratio was applied in the wrong direction.

A common mistake is to treat the cause-1 cumulative incidence as though it were \(1-\exp\{-\int_0^t\lambda_{i1}(u)\,du\}\). That expression ignores removal by cause 2 and is not the probability of cause 1 as the first event. The correct integrand is \(S_i(u)\lambda_{i1}(u)\), where \(S_i\) accounts for all modeled causes.

How This Prediction Differs From Alternatives

Cause-specific hazards describe instantaneous rates among subjects who remain event-free. Their fitted values can be combined as above to obtain cause-specific cumulative incidence. A subdistribution-hazard model instead parameterizes a different hazard linked directly to a cause-specific cumulative-incidence function; its coefficients are not interchangeable with cause-specific hazard coefficients. A net-risk calculation that treats competing events as if they could not occur answers a different, hypothetical question and can overstate the observed probability of a particular cause. Choose the approach according to the target quantity: for the probability of each first-event cause by a time point, calculate or estimate cumulative incidence while accounting for all competing causes.

These curves also depend on the model, the covariates or trajectories being conditioned on, and the time horizon. Sparse events for one cause, misspecified baseline hazards, or poorly estimated association parameters can make a curve unstable. A smooth curve is not evidence that its cause-specific hazards are well estimated; inspect cause-specific event counts and uncertainty in the fitted parameters, and report uncertainty intervals alongside predictions when available.