Why Model Recurrence and Terminal Causes Together?
A person may experience several types of recurrent events before one of several terminal events ends observation. For example, repeated episodes of two conditions might be observed until death from one of two causes. A recurrence model describes the rates of the repeated events; a competing-risks model describes which terminal cause occurs and when. A joint likelihood must account for both processes over the same observed time, stopping recurrence follow-up at the terminal event or censoring time.
The key is to distinguish recurrence event types from terminal causes. A recurrent event does not end follow-up, while a terminal event does. Moreover, a terminal event of one cause precludes observing a later terminal event of another cause. The likelihood therefore includes survival against all terminal causes up to the observed endpoint, and, if a terminal event is observed, the hazard for its particular cause at that time.
The intensity and hazard have units of events per unit time. They are conditional rates, not probabilities of an event at an exact time. Over a short interval of length \(dt\), the chance of a type-\(k\) recurrent event is approximately \(Y_i(t)\lambda_{ik}(t\mid\cdot)dt\), and the chance of terminal cause \(r\) is approximately \(h_{ir}(t\mid\cdot)dt\). The recurrence and terminal models may use different covariates or functions of \(b_i\), but their contributions must use compatible time scales and the same observed endpoint.
Observed Histories and the Conditional Likelihood
Let \(T_i\) be the terminal-event time and \(C_i\) the right-censoring time. The observed endpoint is \(\tilde T_i=\min(T_i,C_i)\). Set \(\Delta_i=1\) if a terminal event is observed and \(\Delta_i=0\) if follow-up is censored; when \(\Delta_i=1\), \(J_i\) identifies the terminal cause. Write \(t_{ikm}\) for the time of the \(m\)th observed recurrent event of type \(k\), and \(n_{ik}\) for the number of such events before \(\tilde T_i\).
Assume the following when using the factorized conditional likelihood below: the intensities are correctly specified conditional on the history and \(b_i\); the terminal causes are mutually exclusive, with one terminal event ending follow-up; and, conditional on observed covariates and \(b_i\), the recurrence and terminal processes are independent. Assume also that censoring is noninformative under the model, and that terminal causes are observed when a terminal event occurs. For continuous event times, simultaneous recurrence and terminal events have probability zero. If recurrence intensities depend on terminal-process information beyond the fact that follow-up has not ended, or terminal hazards depend on the recurrent history, a joint model can still be specified, but the independence factorization needs to be reconsidered.
The recurrent-event exponential contains the sum of the type-specific recurrence intensities because it accounts for having no recurrent event of any type in each portion of follow-up between observed events. Each observed recurrence contributes its own type-specific intensity. Similarly, the terminal survival term uses the sum of all cause-specific hazards: until \(\tilde T_i\), the subject must avoid every terminal cause. A terminal event contributes the hazard of its observed cause, not the sum of the hazards at the event time. If follow-up ends by censoring, there is no terminal-event hazard factor.
Why the Likelihood Has This Form
For a small interval \(dt\) with no recurrent event, the conditional probability of no event of any type is approximately \(1-Y_i(t)\sum_k\lambda_{ik}(t\mid\cdot)dt\). Multiplying these probabilities over intervals and taking the limit gives the exponential of the negative integrated total recurrence intensity. At an observed type-\(k\) event, the event-time density contributes \(\lambda_{ik}(t\mid\cdot)\). Repeating this for every event gives \(L_{i,\mathrm{rec}}\).
The same argument applies to terminal events. While the subject is terminal-event-free, the probability of avoiding every cause over \(dt\) is approximately \(1-\sum_r h_{ir}(t\mid\cdot)dt\), yielding the exponential term in \(L_{i,\mathrm{term}}\). If a terminal event of cause \(J_i\) occurs, the event-time density has the additional factor \(h_{iJ_i}(\tilde T_i\mid\cdot)\). If the subject is censored, observation ends without an observed terminal event, so that factor is absent.
These expressions are conditional on \(b_i\). If \(b_i\) is unobserved with density \(f(b_i)\), the subject-specific observed-data likelihood is obtained by integrating the product over its distribution:
In general, this integral is not the product of separately integrated recurrence and terminal likelihoods. Both contributions involve the same \(b_i\), which is why the conditional likelihood is formed first.
Worked Example: Two Recurrence Types and an Observed Terminal Cause
Hypothetical data. Suppose a subject is followed for 3 years and then has terminal cause 1. Two recurrent-event types have constant intensities \(\lambda_{i1}=0.20\) and \(\lambda_{i2}=0.10\) events per year. There are two type-1 events and one type-2 event. The constant terminal hazards are \(h_{i1}=0.04\) and \(h_{i2}=0.06\) per year. Assume the models meet the conditional independence and censoring assumptions above, and take these rates as fixed conditional on the subject’s covariates and \(b_i\).
The recurrence contribution includes three event-time intensity factors and the probability of no other recurrence over 3 years: $$ L_{i,\mathrm{rec}}=\exp\{-(0.20+0.10)(3)\}(0.20)^2(0.10) =\exp(-0.9)(0.004)\approx 0.001626. $$
The terminal contribution includes survival against both causes for 3 years and the hazard of the observed cause 1: $$ L_{i,\mathrm{term}}=\exp\{-(0.04+0.06)(3)\}(0.04) =\exp(-0.3)(0.04)\approx 0.02963. $$
Thus, using unrounded expressions for the product, $$ L_i=\exp(-0.9)(0.004)\exp(-0.3)(0.04) =\exp(-1.2)(0.00016)\approx 0.00004819. $$
This is a conditional likelihood contribution, not a probability that the model or the observed history is true. In particular, the units of the event-time hazard factors mean a continuous-time event density is not itself a dimensionless probability.
Worked Likelihoods for Censoring and Cause Labels
Worked Example: Right Censoring Without a Terminal Event
Hypothetical data. A second subject is followed for 2.5 years and then administratively censored, with no observed terminal event. The constant recurrence intensities are \(0.12\) and \(0.08\) events per year; one type-1 event and two type-2 events occur. The two terminal hazards are \(0.03\) and \(0.05\) per year. Assume censoring is independent under the model and the specified rates are appropriate conditional on covariates and \(b_i\).
For recurrence, the total intensity is \(0.20\) per year and the product of event-time factors is \(0.12(0.08)^2=0.000768\): $$ L_{i,\mathrm{rec}}=\exp\{-(0.20)(2.5)\}(0.000768) =\exp(-0.5)(0.000768)\approx 0.0004658. $$
The subject remained free of both terminal causes until censoring, so the terminal contribution is survival against their combined hazard, with no cause-specific event factor: $$ L_{i,\mathrm{term}}=\exp\{-(0.03+0.05)(2.5)\} =\exp(-0.2)\approx 0.8187. $$
The joint contribution is $$ L_i=(0.000768)\exp(-0.5)\exp(-0.2) =(0.000768)\exp(-0.7)\approx 0.0003814. $$
Censoring does not mean that the subject had a terminal event of an unknown cause. It records that observation ended before such an event was observed. Consequently, the likelihood has a terminal survival term through 2.5 years, but no terminal hazard factor at that time.
Worked Example: Using the Observed Terminal Cause
Hypothetical data. A third subject has one recurrence of each of two types and a terminal event of cause 2 at 2 years. The recurrence intensities are \(0.15\) and \(0.05\) events per year; the terminal hazards are \(0.03\) and \(0.07\) per year. Assume constant rates and the same conditional-independence and observation assumptions as above.
The recurrence contribution is $$ L_{i,\mathrm{rec}}=\exp\{-(0.15+0.05)(2)\}(0.15)(0.05) =\exp(-0.4)(0.0075)\approx 0.005027. $$
The terminal contribution uses both hazards in the survival term, but uses cause 2 alone as the event-time factor: $$ L_{i,\mathrm{term}}=\exp\{-(0.03+0.07)(2)\}(0.07) =\exp(-0.2)(0.07)\approx 0.05731. $$
Multiplying the unrounded expressions gives $$ L_i=\exp(-0.4)(0.0075)\exp(-0.2)(0.07) =\exp(-0.6)(0.000525)\approx 0.0002881. $$
Using \(0.03\) rather than \(0.07\) as the event-time factor would describe a cause-1 terminal event, not the observed cause-2 event. Using \(0.03+0.07\) as the event-time factor would omit the observed cause label from the likelihood.
Assumptions, Diagnostics, and Alternatives
A likelihood calculation can be arithmetically correct but statistically inappropriate if its assumptions do not fit the data collection or event process. Check that all recurrent events are counted only while the subject is under observation and before the terminal endpoint; that the recurrence types and terminal causes are defined consistently; and that terminal cause is known for observed terminal events. Also check that the intensity models describe the rates over the observed time scale and account for relevant history, covariates, and any intended random effects.
Useful diagnostics include plotting recurrent-event rates by follow-up time and event type, comparing observed and model-predicted terminal events by cause, and inspecting whether residual patterns vary with time, prior event count, or covariates. A systematic rise in recurrence residuals after earlier events can indicate that an intensity model has omitted event-history dependence. A discrepancy in the proportion of terminal events attributed to each cause can signal poorly calibrated cause-specific hazards. Sparse events for one terminal cause can make its hazard estimates unstable. These checks do not prove the model is correct, but they can reveal departures that make the likelihood unreliable.
| Approach | What enters the likelihood | Main limitation |
|---|---|---|
| Separate cause-specific terminal hazards | Survival against the sum of terminal hazards, plus the observed cause’s hazard if an event occurs | Requires the causes to be recorded and the cause-specific models to be adequate |
| Collapse terminal causes | A single terminal hazard for any terminal event | Discards cause information and cannot describe differences between terminal causes |
| Treat other terminal causes as ordinary censoring in a single-cause analysis | Only the hazard for the selected cause is modeled | Does not give the full likelihood for recurrence and all competing terminal outcomes; interpretation depends on the target analysis |
The first approach is appropriate when the joint likelihood is meant to represent the observed terminal cause and timing. Collapsing causes can be reasonable if the scientific question concerns only time to any terminal event, but it changes the modeled outcome. Treating competing causes as censored can be useful for a cause-specific hazard analysis, but it does not make those subjects disappear from the full event-history likelihood: their observed terminal causes still contribute to terminal survival and event terms when modeling all causes together.
A further limitation is that conditional independence given \(b_i\) is a substantive modeling assumption, not a consequence of writing a product. If recurrence and terminal risk remain dependent after conditioning on the modeled covariates and random effects, the factorized conditional model may be inadequate. Likewise, informative censoring requires its own model or a justified adjustment. The introductory material on joint likelihoods gives the broader context for combining event-process contributions; here the essential extension is to sum over both recurrence types and terminal causes in their respective no-event terms.
Common Mistakes
- Using only the observed terminal cause in the survival term. Before a terminal event or censoring, the subject must remain free of every terminal cause, so the survival integrand is \(\sum_r h_{ir}(t)\).
- Using the sum of terminal hazards as the event-time factor. If a cause is observed, use its own hazard at the event time; the sum belongs in the survival term.
- Adding a terminal event factor at censoring. A censored subject contributes terminal survival up to censoring, not a terminal hazard at that time.
- Continuing recurrence follow-up after a terminal event. Recurrence exposure and its integrated intensity stop at \(\tilde T_i\).
- Multiplying marginal likelihoods when sharing an unobserved random effect. Form the conditional product first, then integrate it over the random-effect distribution.
Check Your Understanding
Answer each question using the conditional-likelihood rules above. Brief solutions follow the questions.
- With three terminal causes and censoring at time \(c\), what belongs in the terminal contribution?
- For two recurrent types with constant intensities \(0.10\) and \(0.05\) per year over 2 years, one event of each type, calculate the recurrence contribution.
- A terminal event of cause 1 occurs at time \(t\). Which hazards appear in the survival term, and which appears as the event-time factor?
- Why is a terminal event of another cause not just an ordinary censored observation in the full likelihood for all terminal causes?
- If recurrence and terminal contributions share an unobserved random effect, what must be integrated over that effect?
Solutions. (1) The terminal contribution is \(\exp\{-\int_0^c\sum_{r=1}^{3}h_{ir}(t)\,dt\}\), with no terminal event factor. (2) The total recurrence intensity is \(0.15\) per year and the event-factor product is \(0.10(0.05)=0.005\), so \(L_{\mathrm{rec}}=\exp\{-(0.15)(2)\}(0.005)=\exp(-0.3)(0.005)\approx 0.003704\). (3) All cause-specific hazards appear summed in the survival term; the event-time factor is \(h_{i1}(t)\). (4) Its observed cause and time contribute information to the terminal process, including survival against all causes before that event and the hazard for the observed cause. (5) Integrate the product of the conditional recurrence and terminal contributions against the random-effect distribution, not the two separately integrated contributions.