Why Construct the Full Likelihood?
A multitype event history contains more information than a total event count. It records when recurrent events occurred, which type each event had, whether a terminal event occurred, and—if so—its type. A likelihood must account for both the observed events and the time during which each possible event could have occurred. In particular, the absence of a competing event during follow-up contributes information through its cumulative intensity, even though it contributes no event-time factor.
This tutorial constructs that likelihood from the intensity model. The main bookkeeping rule is simple: multiply by the intensity of the type that actually jumps, and subtract the integrated intensities of all types that could have jumped while the subject remained at risk.
Model and Likelihood Formula
Consider one subject observed from time 0 until \(\tau\), the earlier of the end of observation and the first terminal event. Let \(\mathcal R\) be the set of recurrent event types and \(\mathcal D\) the set of mutually exclusive terminal-event types. For each type \(j\), \(N_j(t)\) counts type-\(j\) events by time \(t\). A recurrent-event count can increase more than once; a terminal-event count can increase at most once, and no event channel remains active after a terminal event.
Let \(Y(t)\) indicate that the subject is under observation and has not yet had a terminal event. Let \(Z(t)\) be a covariate process whose value used at time \(t\) is predictable: it is determined by information available just before \(t\), rather than by the event at \(t\). Conditional on the modeled history, suppose type \(j\) has intensity \(\lambda_j(t\mid\mathcal F_{t-})=Y(t)\lambda_{0j}(t)\exp\{\beta_j^\mathsf{T}Z(t)\}\). Here \(\lambda_{0j}(t)\) is a nonnegative baseline intensity and \(\beta_j\) is a type-specific coefficient vector. More generally, the likelihood formula below applies to any specified predictable intensities \(\lambda_j(t\mid\mathcal F_{t-})\).
The formula is a path likelihood with respect to the usual reference measure for ordered event times and their type labels. For a terminal event, the observed type contributes its intensity at the terminal time; the indicator \(Y(t)\) then becomes zero, so there is no exposure after that time. If follow-up ends without a terminal event, there is no terminal-event factor at the censoring time. The likelihood includes exposure only up to the censoring time, conditional on the observation process; this treatment requires censoring to be independent or noninformative given the modeled history. Informative censoring requires an additional model.
The displayed intensity model assumes no simultaneous jumps among event types, as is appropriate for ordinary continuous-time intensities with probability-zero simultaneous events. If simultaneous jumps can occur with positive probability, the joint jump mechanism must be modeled rather than using a product of separate single-event intensity factors. The construction also presumes that the observed path is in the model's likelihood support: an event assigned zero intensity cannot have positive likelihood.
Why the Formula Has These Terms
Over a short interval of length \(dt\), conditional on the history just before the interval, the probability of no event of any type is \(1-\sum_j\lambda_j(t)dt+o(dt)\). The probability of one type-\(j\) event is \(\lambda_j(t)dt+o(dt)\). Multiplying these contributions over a fine partition of follow-up gives a factor \(\lambda_j(t)\) for each observed type-\(j\) jump and an exponential factor for the accumulated no-event probabilities.
The jump factors are densities in time, so this limit yields the stated path likelihood. Equivalently, its log-likelihood, for positive intensities at all observed jumps, is \(\ell=\sum_j\int_0^\tau\log\{\lambda_j(t)\}\,dN_j(t)-\sum_j\int_0^\tau\lambda_j(t)\,dt\). The first term is evaluated only at jumps of the corresponding type; the second accumulates over all time at risk. This is the event-process version of the compensator-and-jump decomposition.
Worked Example: Two Recurrent Types and a Terminal Cause
Suppose a hypothetical subject is followed from time 0 until a terminal event at time 4 years. Types A and B are recurrent; D1 and D2 are competing terminal types. The covariate \(Z(t)\) is 0 on \([0,2)\) and 1 on \([2,4]\), with the value at a jump taken from just before the jump. Assume the type-specific intensities while at risk, in events per year, are constant within each covariate period:
| Covariate value | A | B | D1 | D2 | Total intensity |
|---|---|---|---|---|---|
| 0 | 0.20 | 0.10 | 0.05 | 0.05 | 0.40 |
| 1 | 0.30 | 0.15 | 0.10 | 0.05 | 0.60 |
The observed history is an A event at year 1, a B event at year 3, and a D1 terminal event at year 4. These event labels are compatible with the model: A and B can recur, and D1 is one of the allowed terminal types. The subject remains at risk after each recurrent event and is no longer at risk after D1.
The event factors use the pre-event covariate values. They are \(0.20\) for A at year 1, \(0.15\) for B at year 3, and \(0.10\) for D1 at year 4. The integrated total intensity is \((2)(0.40)+(2)(0.60)=0.80+1.20=2.00\). Thus the likelihood contribution is \(L=(0.20)(0.15)(0.10)\exp(-2.00)=0.003\exp(-2.00)\approx0.0004060\). The units are per year cubed, as expected for a density in three event times. The D2 intensity supplies exposure in both periods even though no D2 event occurred.
Predictable Covariates and Cumulative Exposure
For the log-linear model, the integrated intensity for type \(j\) is \(\int_0^\tau Y(t)\lambda_{0j}(t)\exp\{\beta_j^\mathsf{T}Z(t)\}\,dt\). This integral is sometimes called cumulative exposure: it combines the time at risk, the baseline intensity, and the covariate-dependent multiplier. If \(Z(t)\) changes over time, its value must be integrated over each period in which it applies. At an event time, the jump factor uses \(Z(t-)\), the covariate value just before the jump. Using a value defined by the event itself would make the intensity nonpredictable and can create a circular model.
Worked Example: Evaluating a Covariate-Dependent Likelihood
Consider one subject at risk from time 0 to a D terminal event at year 3.5. There is one recurrent type R and one terminal type D. The covariate is 0 for the first year and 1 from year 1 to 3.5. Assume constant baseline intensities \(\lambda_{0R}=0.20\) and \(\lambda_{0D}=0.10\) per year, with intensities \(\lambda_R=0.20\exp(\beta_R Z)\) and \(\lambda_D=0.10\exp(\beta_D Z)\). R events occur at years 0.5 and 2; D occurs at year 3.5. These event types and times are supported by the model, and follow-up stops at D.
The event factors are \(0.20\) at the first R event, \(0.20\exp(\beta_R)\) at the second R event, and \(0.10\exp(\beta_D)\) at the D event. The total cumulative intensity is \(1(0.20+0.10)+2.5\{0.20\exp(\beta_R)+0.10\exp(\beta_D)\}\). Therefore, \(L=0.004\exp(\beta_R+\beta_D)\exp[-0.30-0.50\exp(\beta_R)-0.25\exp(\beta_D)]\).
For illustration, set \(\beta_R=\log 2\) and \(\beta_D=\log 3\). Then the event factors are \(0.20\), \(0.40\), and \(0.30\), whose product is \(0.024\). The cumulative intensity is \(0.30+2.5(0.40+0.30)=2.05\), so \(L=0.024\exp(-2.05)\approx0.003090\). This calculation uses the same covariate value in the integrated intensity throughout the second period and at each event in that period.
Censoring and the Exposure from Unobserved Types
Censoring ends the observed interval; it is not itself a terminal event. If a subject is censored while still at risk, all channel intensities contribute through the censoring time, but no channel receives an event factor there. Likewise, a competing terminal type that did not occur has no jump factor, yet its intensity contributes to the integrated total while the subject remains at risk. Omitting that exposure changes the likelihood because it treats the absence of that event type as if it carried no information.
Worked Example: Recurrent Event Followed by Censoring
Suppose a hypothetical subject has constant intensities of \(0.30\), \(0.10\), and \(0.20\) events per year for recurrent type R and terminal types D1 and D2, respectively. The subject has one R event at year 1 and is censored at year 2 without a terminal event. Assume censoring is noninformative given the modeled history, the event is allowed, and the subject remains at risk throughout the two years.
The observed event factor is \(0.30\). All three intensities contribute to exposure, so the cumulative intensity is \(2(0.30+0.10+0.20)=1.20\). Hence \(L=0.30\exp(-1.20)\approx0.09036\) per year. There is no D1 or D2 event factor and no factor for a censoring event.
If one incorrectly omitted D2 from the exposure, the result would be \(0.30\exp[-2(0.30+0.10)]\approx0.1348\), substantially different from the correct likelihood contribution. The difference arises because two years without a D2 event are part of the observed path.
Practical Checks and Alternatives
A reliable construction starts with the path and its risk interval, not with algebra. List each observed event with its type and pre-event covariates; then integrate the total intensity over every at-risk interval. The following checks catch common failures:
- Check the stopping rule. After a terminal event, set \(Y(t)=0\). Do not add recurrent-event or terminal-event exposure afterward.
- Check the event factors. Each observed jump contributes only its own type-specific intensity, evaluated at the pre-jump history.
- Check all active channels. Every possible event type contributes to the integrated intensity while the subject is at risk, including terminal types that did not occur.
- Check the support. A path with an event at a time its type-specific intensity is zero has likelihood zero under that model.
- Check the observation endpoint. Censoring truncates exposure; absent a censoring model, it does not create an event factor.
A common mistake is to use the total intensity as the factor for an observed event, reasoning that some event occurred at that time. That factor loses the observed type information. If the observed event is type \(j\), its factor is \(\lambda_j(t)\), while the total \(\sum_k\lambda_k(t)\) belongs in the integrated no-event term. Another tempting shortcut is to fit separate type-specific event models and omit the other types' intensities from each exposure. That is not the full likelihood for the joint event history. The log-likelihood has a sum of type-specific jump terms, but its integrated term sums across all channels; shared parameters can therefore couple the type-specific contributions.
Conditioning on the times at which any event occurred and modeling only the event-type labels is a different likelihood: it can describe type probabilities given an event, but it discards information in the waiting times. Conversely, the full intensity likelihood uses both waiting-time and type information. The construction here also has limits: it depends on a correctly specified intensity and observation mechanism, on the assumed treatment of covariates, and on the no-simultaneous-jumps condition. Intensity misspecification, informative censoring, or structural simultaneous events require a different or expanded model.
Check Your Understanding
Answer each question using the event-factor and cumulative-exposure distinction.
- A subject has one recurrent event of type A, then a terminal event of type D1. Which type-specific intensities appear as event factors?
- During follow-up, a subject remains at risk for terminal types D1 and D2 but experiences D1. Does D2 contribute to the likelihood?
- A subject has recurrent events at years 1 and 3, then is censored at year 4 with no terminal event. Is there an event factor at year 4?
- Why must a time-varying covariate in an intensity be evaluated using information available just before an event?
- With constant intensities \(0.2\) for recurrent type R and \(0.1\) for terminal type D, a subject has one R event and is censored after 2 years. Write the likelihood contribution.
Solutions. (1) The event factors are \(\lambda_A\) at the A event and \(\lambda_{D1}\) at the D1 event; there is no post-D1 exposure. (2) Yes. D2 has no event factor, but its intensity is integrated over all time at risk. (3) No. Censoring ends observation but is not an event in the modeled channels. (4) The intensity must be predictable, based on the history immediately before the event, so it cannot depend on information revealed by that event. (5) The event factor is \(0.2\), and the cumulative intensity is \(2(0.2+0.1)=0.6\), giving \(L=0.2\exp(-0.6)\approx0.1098\) per year.