Why Extend the Model Beyond Two Causes?
A first event can belong to one of several mutually exclusive types. For example, a participant might experience one of three disease events before death or administrative censoring. When the event types compete to be first, analyzing only a single event indicator discards information about which type occurred. A joint model can instead connect the longitudinal marker to each cause-specific hazard while retaining the fact that all causes contribute to the probability of remaining event-free.
Let \(K\geq 3\) be the number of event causes. For subject \(i\), let \(T_i\) be the time to the first event, \(J_i\in\{1,\ldots,K\}\) its cause when an event occurs, and \(C_i\) the censoring time. The observed time is \(\tilde T_i=\min(T_i,C_i)\). Write \(\Delta_i=1\) if an event is observed and \(\Delta_i=0\) if the observation is censored. The event causes are assumed mutually exclusive: at most one cause is recorded as the first event.
The longitudinal component supplies a subject-specific latent trajectory, denoted \(m_i(t;b_i)\), where \(b_i\) is the subject's random-effects vector. A cause-specific association may use this current value, its slope, a cumulative exposure, or a specified combination. Those choices affect the cause-specific linear predictors, but not the way the \(K\) hazards combine in the event-time likelihood.
Cause-Specific Hazards and Parameter Organization
A general cause-specific specification is
Here \(h_{0k}(t)\) is the cause-\(k\) baseline hazard, \(x_i\) is a covariate vector, \(\gamma_k\) is its cause-specific coefficient vector, and \(a_k(t;b_i)\) is the cause-specific association contribution built from the longitudinal trajectory and random effects. For example, a current-value-only association has \(a_k(t;b_i)=\alpha_k m_i(t;b_i)\). The cause index on each parameter matters: \(\gamma_2\) and \(\alpha_2\) describe cause 2, not a generic effect shared automatically across causes.
The parameterization may allow every cause to have its own coefficients, or impose scientifically motivated equality constraints. For instance, a shared current-value association sets \(\alpha_1=\cdots=\alpha_K\); an unconstrained association estimates a separate \(\alpha_k\) for each cause. Such constraints must be specified, not inferred from a common notation. The same principle applies to baseline hazards and covariate effects.
Worked Example: Organizing Four Cause-Specific Associations
Suppose a hypothetical model has four event causes and uses the current marker value, measured in mmol/L. It estimates \(\alpha_1=0.20\), \(\alpha_2=-0.10\), \(\alpha_3=0.35\), and \(\alpha_4=0.00\). The hazard multiplier for a one-unit increase in the current marker, holding the other terms fixed, is \(\exp(\alpha_k)\) for cause \(k\).
For cause 1, \(\exp(0.20)=1.221\), so the cause-1 hazard is multiplied by about 1.22 per mmol/L increase in the marker. For cause 2, \(\exp(-0.10)=0.9048\), a multiplier of about 0.905 per mmol/L. For cause 3, \(\exp(0.35)=1.419\). For cause 4, \(\exp(0)=1\), so this model estimates no current-value association for that cause.
These are conditional hazard associations given the model covariates and random effects; they are not causal effects. A convenient parameter layout is the vector \((\alpha_1,\alpha_2,\alpha_3,\alpha_4)^\mathsf{T}\), alongside a separate \(\gamma_k\) and baseline specification for each cause. The zero estimate for cause 4 does not mean that cause 4 has zero hazard; its baseline and other predictors still determine that hazard.
How the Causes Enter the Joint Likelihood
Assume right censoring is independent of the event process conditional on the modeled covariates and random effects, the event cause is observed whenever an event occurs, and the event hazards are correctly specified. Also assume that, conditional on \(b_i\) and the covariates, the longitudinal measurements and event process are independent. Let \(f(y_i\mid b_i)\) be the longitudinal-data density and \(f_b(b_i)\) the random-effects density. Conditional on \(b_i\), the event-free survival probability through time \(t\) is
The sum appears because any one of the \(K\) causes ends the event-free state. Define \(\delta_{ik}=1\) when subject \(i\) has an observed event of cause \(k\), and 0 otherwise. The conditional event-time contribution is
For a censored subject, every \(\delta_{ik}=0\), leaving only survival through \(\tilde T_i\). For an event of cause \(j\), exactly \(\delta_{ij}=1\), so the contribution includes the cause-\(j\) hazard at the observed time and survival against all causes up to that time. The subject-level joint likelihood contribution integrates over the random effects:
This integral is essential: the latent trajectory and the event hazards are linked through \(b_i\), so replacing the random effects by a fitted average trajectory generally does not give the same likelihood.
Worked Example: Event and Censoring Contributions With Three Causes
For a hypothetical subject, suppose the conditional cause-specific hazards are constant over time: \(h_{i1}=0.10\), \(h_{i2}=0.06\), and \(h_{i3}=0.04\) per year. For an event at \(t=2\) years, the total hazard is \(0.10+0.06+0.04=0.20\) per year and the integrated total hazard is \(0.20(2)=0.40\).
If the observed event is cause 2, the conditional contribution is
The cause-2 hazard supplies the event factor, while all three causes supply the survival factor. If instead the subject is censored at 2 years, the contribution is
This is the event-process contribution conditional on the random effects; the full joint contribution also includes the longitudinal density and integrates over \(b_i\). In this constant-hazard illustration, the cause-specific assumptions are that each hazard stays fixed over the two-year interval and the three causes are mutually exclusive first events.
Cumulative Incidence: Cause-Specific Risk in the Presence of Other Causes
A cause-specific hazard is an instantaneous rate among subjects who remain event-free. The corresponding cause-specific cumulative incidence is the probability of experiencing that cause by time \(t\), before any competing cause. Conditional on \(b_i\), it is
The survival term inside the integral includes all causes. Consequently, a cause can have a large hazard but a modest cumulative incidence if competing causes remove many subjects from the event-free state first. To obtain a population probability for a covariate profile, average the conditional incidence over the random-effects distribution. This averaging is distinct from evaluating the incidence at \(b_i=0\).
Worked Example: Three Cumulative Incidences Under Constant Hazards
Use the hypothetical constant hazards \(0.10\), \(0.06\), and \(0.04\) per year from the preceding example. The total hazard is \(0.20\) per year. With constant hazards, the cause-\(k\) cumulative incidence at \(t=2\) years is \((h_{ik}/0.20)\{1-\exp(-0.20(2))\}\). The common event probability by 2 years is \(1-\exp(-0.40)=0.3297\).
For cause 1, \(F_{i1}(2)=(0.10/0.20)(0.3297)=0.1648\). For cause 2, \(F_{i2}(2)=(0.06/0.20)(0.3297)=0.09890\). For cause 3, \(F_{i3}(2)=(0.04/0.20)(0.3297)=0.06594\). Their sum is \(0.3296\), with the small discrepancy from \(0.3297\) due to rounding. The event-free probability is \(\exp(-0.40)=0.6703\), so event-free probability plus the three incidences is approximately 1.
These are conditional probabilities under the specified hazards and random effects. A population-level prediction would average over the random-effects distribution; an empirical estimate would also need to account for the fitted model's parameter uncertainty.
Comparison, Diagnostics, and Limits
| Approach | What it represents | Important distinction |
|---|---|---|
| Cause-specific hazard model | Instantaneous cause-\(k\) rate among subjects still event-free | Other causes enter through the event-free risk set and the total survival term |
| Cause-specific cumulative incidence | Probability of cause \(k\) occurring first by time \(t\) | Depends on every cause-specific hazard, not just \(h_{ik}\) |
| Shared versus cause-specific association parameters | A constraint choice for how the longitudinal trajectory relates to causes | Shared coefficients reduce dimension but may conceal differing associations |
With three or more causes, the number of cause-specific parameters and hazards grows quickly. Before interpreting estimates, verify that each cause is defined consistently, that observed event labels are mutually exclusive, and that the event indicator and cause indicators agree: an observed event has exactly one cause indicator equal to 1, while censoring has none. Check convergence and numerical integration accuracy, especially for the random-effects integral, and inspect whether a cause has so few events that its baseline or association parameters are unstable.
A useful model-based diagnostic is to compare predicted cause-specific cumulative incidences with observed cause-specific event patterns over time, using a method appropriate for censoring. Also check that predicted incidences are nonnegative and that their sum does not exceed 1; under a coherent model, their sum is \(1-S_i(t)\). Large discrepancies may indicate misspecified hazards, an unsuitable association structure, poor numerical integration, or incorrect cause coding. Sparse-cause instability and identifiability of cause-specific associations are especially relevant; the earlier material on these subjects provides further detail.
The cause-specific hazard formulation is not the only way to summarize competing events. Cumulative incidence directly targets event probability, while cause-specific hazards describe instantaneous rates conditional on remaining event-free. They answer different questions and should not be treated as interchangeable. A cause-specific hazard ratio is conditional and is not, by itself, a risk ratio for cumulative incidence. Results remain conditional on the joint model, its random-effects distribution, its censoring assumptions, and the chosen association structure; observational associations do not establish causation.
Common Mistake
Check Your Understanding
Answer each question, then check the brief solutions.
- For \(K=4\), what is the all-cause event hazard in terms of the four cause-specific hazards?
- A subject is censored at time \(t\). Which event-hazard factors appear in that subject's conditional likelihood contribution?
- Under constant hazards \(h_1=0.12\) and \(h_2=0.08\) per year, calculate the probability of remaining event-free through 1 year.
- Why can a cause-specific cumulative incidence not generally be calculated from that cause's hazard alone?
- What does a shared association constraint \(\alpha_1=\cdots=\alpha_K\) assert, and what does it not assert?
Solutions. (1) \(h_{i+}(t\mid b_i)=h_{i1}(t\mid b_i)+h_{i2}(t\mid b_i)+h_{i3}(t\mid b_i)+h_{i4}(t\mid b_i)\). (2) Only the survival factor \(\exp\{-\int_0^t\sum_k h_{ik}(u\mid b_i)\,du\}\) appears; there is no event-hazard factor. (3) The total hazard is \(0.12+0.08=0.20\), so \(S(1)=\exp(-0.20)=0.8187\). (4) Other causes remove subjects from the event-free state, so the incidence integrand uses survival based on the sum of all hazards. (5) It asserts an equal longitudinal-association coefficient across causes under the specified model; it does not assert equal baseline hazards, equal covariate effects, or equal cumulative incidences.