Joint Likelihood for Recurrent and Terminal Events

Why the Subject-Level Likelihood Matters

A subject may contribute several recurrent events, a terminal event or a censoring time, and repeated longitudinal measurements. These observations are linked: the terminal event stops the opportunity to experience further recurrences, while longitudinal measurements and event risks may both depend on subject-specific random effects. A subject-level likelihood puts these pieces together without treating the event times as unrelated observations.

The construction below is conditional on a subject's random effects. This is a useful first step: it makes explicit what the observed event history contributes for a fixed value of those effects. A marginal likelihood is then obtained by integrating that conditional contribution over the random-effects distribution.

Definition: For subject \(i\), let \(Y_i\) denote the observed longitudinal measurements, \(b_i\) the subject-specific random effects, \(D_i\) the terminal-event time, and \(C_i\) the censoring time. The observed end of follow-up is \(U_i=\min(D_i,C_i)\), and \(\Delta_i=I(D_i\leq C_i)\) indicates whether the terminal event was observed. Let \(N_i(t)\) count recurrent events by time \(t\), with observed event times \(0<t_{i1}<\cdots<t_{in_i}<U_i\). The likelihood contribution is conditional on the observed covariates and, initially, on \(b_i\).

Assume follow-up starts at time zero, event times are observed exactly, and the recurrent-event intensity \(\lambda_{Ri}(t\mid b_i,\mathcal H_{Ri}(t))\) is defined while the subject is alive. Here \(\mathcal H_{Ri}(t)\) is the observed recurrent-event history just before \(t\); in a simpler model, the intensity may depend on \(b_i\) and covariates but not on past recurrences. Let \(\lambda_{Di}(t\mid b_i,\mathcal H_{Ri}(t))\) be the terminal-event hazard while the subject is alive. The basic likelihood assumes that the longitudinal observations are conditionally independent of the event processes given the random effects and covariates, apart from any modeled history dependence in the hazards. It also assumes that censoring is ignorable for the parameters of interest, conditional on the modeled information. If censoring is informative, its mechanism may need its own likelihood contribution.

Conditional Likelihood Contributions

For a recurrent-event process with conditional intensity \(\lambda_{Ri}\), the likelihood of the observed event history up to \(U_i\) is the product of the intensity at each observed event time and the probability of no additional event in the intervals between events. That second part is the exponential of minus the integrated intensity over the whole at-risk interval. Thus,

L_{Ri}(b_i) = \left\{\prod_{j=1}^{n_i} \lambda_{Ri}(t_{ij}\mid b_i,\mathcal H_{Ri}(t_{ij})) \right\} \exp\left\{-\int_0^{U_i} \lambda_{Ri}(s\mid b_i,\mathcal H_{Ri}(s))\,ds\right\}.

The integral ends at \(U_i\), not at a later study horizon: after a terminal event or censoring, no further recurrent-event times are observed as part of this subject's risk history. If \(n_i=0\), the product is defined as 1, and only the no-event exponential remains.

The terminal-event contribution has a parallel form. Survival without a terminal event through \(U_i\) contributes the exponential of minus the integrated terminal hazard. If a terminal event is observed, its hazard at \(U_i\) is also included. Define \(\lambda_{Di}(t)\) as shorthand for the hazard evaluated at the subject's effects and observed recurrent history. Then

L_{Di}(b_i) = \{\lambda_{Di}(U_i)\}^{\Delta_i} \exp\left\{-\int_0^{U_i}\lambda_{Di}(s)\,ds\right\}.

When \(\Delta_i=0\), the hazard factor is 1: censoring contributes survival up to \(U_i\), not a terminal-event density. When \(\Delta_i=1\), the hazard at the observed terminal time is included. The event-process contribution, conditional on \(b_i\), is \(L_{Ri}(b_i)L_{Di}(b_i)\) under the stated sequential hazard specification. The longitudinal contribution is its conditional density \(f(Y_i\mid b_i)\). Therefore the subject's full conditional contribution is

L_i(b_i) = f(Y_i\mid b_i)\, L_{Ri}(b_i)\, L_{Di}(b_i).

This product is a conditional likelihood, not yet the likelihood contribution for an observed subject with unobserved random effects. If \(g(b_i)\) is the random-effects density, the marginal contribution is

L_i = \int L_i(b_i)g(b_i)\,db_i.

The integration is over the product of the longitudinal and event contributions. In general, it is not valid to integrate each factor separately and multiply the resulting marginal quantities: the same \(b_i\) appears in all components. Numerical integration is often used when this integral has no convenient closed form.

Why These Terms Have This Form

The survival factor follows directly from the definition of a hazard. If an event process has hazard \(\lambda(t)\), then over a short interval of length \(dt\), the conditional event probability is approximately \(\lambda(t)dt\). The probability of no event over an interval is therefore built from factors approximately equal to \(1-\lambda(t)dt\). Multiplying these factors over time and taking the continuous-time limit gives the survival probability \(\exp\{-\int\lambda(s)\,ds\}\). An observed event at time \(t\) adds its hazard factor \(\lambda(t)\), yielding an event-time density.

For recurrent events, apply this reasoning along the observed history. Every observed event contributes its intensity at its time, and the integrated intensity accounts for all opportunities for another event during follow-up. For the terminal event, the same survival argument applies, with an additional hazard factor only if the terminal event is observed. This explains why a censored subject still contributes information: the subject remained free of the terminal event through the censoring time.

Worked Example: Two Recurrences and Censored Follow-Up

Suppose a hypothetical subject has recurrent events at 0.5 and 2.0 years and is censored at \(U_i=3\) years without a terminal event. Conditional on a fixed \(b_i\), suppose the recurrent-event intensity is constant at \(0.4\) events per year and the terminal-event hazard is constant at \(0.2\) per year. For illustration, suppose one longitudinal measurement is \(y=1.4\), with conditional distribution \(Y_i\mid b_i\sim N(1.0,0.5^2)\).

The recurrent-event contribution is \(L_{Ri}(b_i)=(0.4)(0.4)\exp\{-(0.4)(3)\}=0.16e^{-1.2}\). Since \(e^{-1.2}\approx0.3011942\), this is \(0.16(0.3011942)=0.0481911\), or \(0.04819\). The time units are per year squared because there are two event-time density factors.

No terminal event was observed, so \(\Delta_i=0\), and the terminal contribution is \(L_{Di}(b_i)=\exp\{-(0.2)(3)\}=e^{-0.6}\approx0.5488116\). The event-process contribution is \(0.0481911(0.5488116)\approx0.02645\).

The longitudinal density is \(f(y\mid b_i)=\{0.5\sqrt{2\pi}\}^{-1}\exp\{-(1.4-1.0)^2/[2(0.5)^2]\}\). The standardized residual is \(0.4/0.5=0.8\), so the exponent is \(-0.8^2/2=-0.32\). Thus the density is \(0.7978846e^{-0.32}\approx0.5794\) per biomarker unit. The full conditional contribution is approximately \(0.5794(0.02645)=0.01532\). This is a likelihood density, not a probability of observing this exact continuous measurement and exact event times.

Worked Example: An Observed Terminal Event

Consider a different hypothetical subject followed for \(U_i=2\) years, with one recurrence at year 0.8 and a terminal event observed at year 2. Conditional on \(b_i\), suppose the recurrent intensity is constant at \(0.3\) events per year and the terminal hazard is constant at \(0.25\) per year. Assume the terminal hazard does not vary with the recurrent history in this example.

For the recurrence, the event-time intensity is \(0.3\) and the integrated intensity is \((0.3)(2)=0.6\). Therefore \(L_{Ri}(b_i)=0.3e^{-0.6}=0.3(0.5488116)=0.1646435\), or \(0.1646\).

Here \(\Delta_i=1\), so the terminal contribution includes the hazard at year 2: \(L_{Di}(b_i)=0.25e^{-(0.25)(2)}=0.25e^{-0.5}=0.25(0.6065307)=0.1516327\), or \(0.1516\). Their product is \(0.1646435(0.1516327)\approx0.02497\). The terminal hazard factor is essential: omitting it would treat an observed terminal event as though the subject had merely been censored at year 2.

From Conditional to Marginal: A Small Integration Example

The integral over \(b_i\) is not an optional correction for a technical detail. The observed data do not reveal a subject's random effects, so the likelihood for those data must average over their possible values. The following deliberately small discrete example illustrates the calculation; it is not an assumption that random effects in general have a two-point distribution.

Worked Example: Integrating a Conditional Contribution

Suppose a hypothetical calculation uses two quadrature points for \(b_i\), each with weight 0.5. At the low point, the recurrent intensity is \(0.2\) per year and the terminal hazard is \(0.1\) per year; at the high point they are \(0.6\) and \(0.3\) per year. The subject has one recurrence at year 1, is censored at year 2, and has no terminal event. The conditional longitudinal densities at the two points are stipulated to be 0.3 and 0.8 per measurement unit, respectively.

At the low point, the event contribution is \(0.2e^{-(0.2)(2)}e^{-(0.1)(2)}=0.2e^{-0.6}=0.1097623\). Multiplying by the longitudinal density gives the full conditional contribution \(0.3(0.1097623)=0.0329287\).

At the high point, the event contribution is \(0.6e^{-(0.6)(2)}e^{-(0.3)(2)}=0.6e^{-1.8}=0.0991793\). The full conditional contribution is \(0.8(0.0991793)=0.0793434\).

The marginal contribution is the weighted average of these full conditional contributions: \(0.5(0.0329287)+0.5(0.0793434)=0.0561361\), or \(0.05614\). Averaging the event contribution first and the longitudinal density separately would not generally give the same answer, because each conditional contribution combines both components at the same random-effects value.

Diagnostics, Alternatives, and Limitations

A correct likelihood expression is only as useful as its assumptions. Check that event times are ordered and strictly before \(U_i\), that the recurrent-event integral uses the correct at-risk interval, and that the terminal indicator agrees with the observed event status. A zero-event subject should have no event-intensity factors but should retain the integrated-intensity term. A censored subject should retain terminal survival through censoring and should not receive a terminal hazard factor.

Model diagnostics should also examine whether the chosen recurrent and terminal hazards describe observed event patterns, whether fitted cumulative hazards are compatible with the data, and whether residual patterns suggest time dependence or history dependence omitted from the model. If longitudinal measurements are scheduled or missing in a way related to unobserved outcomes or event risk, the assumption that their observation process can be ignored may fail. Likewise, informative censoring can invalidate the simple contribution unless the censoring process is modeled or otherwise addressed.

A common alternative is to analyze recurrences and the terminal event separately. That can be simpler, but separate analyses do not by themselves provide this joint subject-level contribution or account for their shared random effects. Conversely, a joint likelihood is more demanding: it depends on the specified conditional distributions and hazards, and its marginal likelihood may require numerical integration. The earlier material on recurrent events with a terminal event motivates the event-process setting; the focus here is how its observed history enters the likelihood.

Pitfall: multiplying event-time densities without a survival term. This omits the probability of no additional event during the observed risk time. Include \(\exp\{-\int_0^{U_i}\lambda_{Ri}(s)\,ds\}\).
Pitfall: extending the recurrence integral beyond follow-up. A terminal event ends the recurrent-event risk interval. Integrate only to \(U_i\).
Pitfall: adding a terminal hazard for a censored subject. Censoring contributes survival through \(U_i\); the hazard factor appears only when \(\Delta_i=1\).
Pitfall: treating a conditional contribution as marginal. If \(b_i\) is unobserved, integrate the complete conditional contribution against \(g(b_i)\).

Key takeaway: For each subject, multiply the longitudinal density by the recurrent-event history contribution and the terminal survival/event contribution, all conditional on the same random effects. Then integrate this complete product over the random-effects distribution to obtain the marginal likelihood.

Check Your Understanding

Answer each question, then compare with the solutions below.

A subject has no recurrence before censoring at year 4. What is the recurrent-event contribution if the intensity is a constant \(0.25\) events per year?
A subject is censored at year 3, and the terminal hazard is a constant \(0.2\) per year. Should the terminal contribution include a hazard factor?
With a constant recurrent intensity of \(0.5\) events per year, one recurrence at year 1, and follow-up through year 2, write and evaluate the recurrent-event contribution.
Why must the longitudinal and event contributions be multiplied before integrating over the random effects?

Solutions. (1) There are no event-time factors, so \(L_{Ri}(b_i)=\exp\{-(0.25)(4)\}=e^{-1}\approx0.3679\). (2) No. The subject contributes terminal survival \(e^{-(0.2)(3)}=e^{-0.6}\approx0.5488\), with no hazard factor because no terminal event was observed. (3) The contribution is \(0.5\exp\{-(0.5)(2)\}=0.5e^{-1}\approx0.1839\). (4) The same unobserved random effects govern the conditional longitudinal and event contributions. The marginal likelihood averages their product over \(g(b_i)\); separately averaging the factors generally does not preserve their dependence through \(b_i\).