Likelihood Contributions for Competing Events and Censoring

Why the Event Type Changes the Likelihood Contribution

In competing-risks data, each individual can experience at most one first event, and that event can belong to one of several causes. The time and cause observed together determine the event-process likelihood contribution. An individual who experiences the target cause contributes a target-cause hazard at the event time; an individual who experiences a competing cause contributes that cause's hazard instead. Someone who is right-censored contributes survival up to the censoring time, but no event hazard there.

In a joint model, these contributions are conditional on the individual's longitudinal random effects and biomarker history. Conditioning is important: it describes the event process for a specified latent trajectory, rather than averaging over differences in trajectories across individuals. The earlier material on cause-specific hazards supplies the hazard interpretation; here the focus is how that hazard and the observed outcome form an individual likelihood contribution.

Definition: Let \(T_i\) be individual \(i\)'s observed time, the minimum of the first event time and the right-censoring time. Let \(D_i\in\{0,1,\ldots,K\}\) denote the observed status: \(D_i=0\) means right-censored, and \(D_i=k\) means the first event was from cause \(k\). Conditional on random effects \(b_i\) and longitudinal history \(\mathcal H_i^L\), let \(\lambda_{ik}(t\mid b_i,\mathcal H_i^L)\) be the cause-specific hazard for cause \(k\). It is the instantaneous rate of cause \(k\) at time \(t\), among individuals who have experienced no event before \(t\), given the stated conditioning information.

The history available to a hazard at time \(t\) must be temporally appropriate: future measurements cannot determine a past hazard. In a standard joint model, the hazard often depends on the latent biomarker trajectory specified by \(b_i\), while the observed biomarker measurements inform that trajectory through the longitudinal model. Writing \(\mathcal H_i^L\) in the conditional contribution identifies the observed longitudinal information being held fixed; it does not mean that a noisy measurement is automatically the true biomarker value.

Assume the cause-specific hazards are nonnegative and integrable over the observed follow-up, the event causes are mutually exclusive first-event types, and event times have a continuous distribution. Define the total hazard and cumulative cause-specific hazards, conditional on \(b_i,\mathcal H_i^L\), by

\lambda_{i+}(t\mid b_i,\mathcal H_i^L) =\sum_{k=1}^K\lambda_{ik}(t\mid b_i,\mathcal H_i^L), \qquad \Lambda_{ik}(t\mid b_i,\mathcal H_i^L) =\int_0^t\lambda_{ik}(u\mid b_i,\mathcal H_i^L)\,du.

The conditional probability of remaining free of every event through time \(t\) is

S_i(t\mid b_i,\mathcal H_i^L) =\exp\left\{-\sum_{k=1}^K\Lambda_{ik}(t\mid b_i,\mathcal H_i^L)\right\}.

This is survival from all event causes, not survival from the target cause alone. It is the same factor whether the observed outcome is a target event, a competing event, or censoring, because every individual must remain event-free until the observed time.

Conditional Contributions for the Three Outcomes

Let cause 1 be the target cause. For an event of cause \(k\) at time \(T_i\), the conditional density is the probability of surviving all causes up to that time multiplied by the instantaneous cause-\(k\) hazard at that time. For right censoring at \(T_i\), the event-process contribution is just the probability of no event through that time.

Result: The conditional event-process contribution for individual \(i\) is

L_i^{E}(b_i\mid\mathcal H_i^L) = \begin{cases} S_i(T_i\mid b_i,\mathcal H_i^L)\, \lambda_{i1}(T_i\mid b_i,\mathcal H_i^L), & D_i=1,\\[4pt] S_i(T_i\mid b_i,\mathcal H_i^L)\, \lambda_{ik}(T_i\mid b_i,\mathcal H_i^L), & D_i=k,\quad k\ne 1,\\[4pt] S_i(T_i\mid b_i,\mathcal H_i^L), & D_i=0. \end{cases}

For any event, the contribution is a density in time and has units of inverse time when hazards are measured per unit time. For censoring, it is a dimensionless survival probability.

These are event-process contributions, not the entire joint-model contribution for all observed data. A longitudinal measurement model contributes its own factor. If the censoring mechanism is modeled, it may also contribute a censoring factor. Under independent censoring conditional on the model's conditioning information, the censoring factor can be separated from the event-process contribution; it is not a cause-specific event hazard.

Why These Factors Have This Form

A short derivation makes clear why censoring and competing events differ. Conditional on \(b_i,\mathcal H_i^L\), the probability of no event in a short interval of length \(\Delta t\), given survival to its start \(t\), is approximately \(1-\lambda_{i+}(t)\Delta t\). Multiplying these conditional probabilities across successive short intervals and taking the limit gives survival as the exponential of minus the integrated total hazard.

S_i(t\mid b_i,\mathcal H_i^L) =\lim_{\max\Delta t\to0} \prod_j\{1-\lambda_{i+}(t_j\mid b_i,\mathcal H_i^L)\Delta t_j\} =\exp\left\{-\int_0^t\lambda_{i+}(u\mid b_i,\mathcal H_i^L)\,du\right\}.

For a small interval \([t,t+\Delta t)\), the conditional probability of a cause-\(k\) event in that interval, given survival to \(t\), is approximately \(\lambda_{ik}(t)\Delta t\). The probability of surviving to \(t\) and then having that event in the interval is therefore approximately \(S_i(t)\lambda_{ik}(t)\Delta t\). Dividing by \(\Delta t\) and taking the limit yields the cause-\(k\) event-time density \(S_i(t)\lambda_{ik}(t)\). If the observation ends with censoring at \(t\), the event process instead supplies only \(S_i(t)\): no event is observed at that time.

Worked Example: A Target-Cause Event

Suppose an individual's random effects and biomarker history are fixed at specified values. Over the relevant follow-up, the resulting conditional hazards are constant: \(0.12\) per month for the target cause and \(0.08\) per month for one competing cause. This hypothetical individual experiences the target event at month 2. Assume the hazards apply through that time and the event is correctly classified.

The total cumulative hazard through month 2 is

\Lambda_{i+}(2)=(0.12+0.08)\times 2=0.40.

Thus the conditional probability of remaining event-free through month 2 is \(S_i(2)=\exp(-0.40)=0.6703\). The target-event contribution uses the target hazard at the event time:

L_i^E =\exp(-0.40)\times 0.12 =0.6703\times 0.12 =0.08044\ \text{per month}.

This is a conditional event-time density contribution, not a probability that the event occurred at exactly month 2. For a continuous event time, the probability of an event at one exact time is zero; the density describes relative likelihood per unit time.

Worked Example: A Competing Event Is Observed

For a second hypothetical individual, suppose the conditional target-cause hazard is \(0.04\) per day and the competing-cause hazard is \(0.06\) per day throughout follow-up. A competing event occurs on day 5. The model assumptions are the same as above, and the observed event type is known.

The total cumulative hazard to day 5 is \((0.04+0.06)\times5=0.50\), so survival from both causes until then is \(\exp(-0.50)=0.6065\). Since the event was from the competing cause, its hazard—not the target-cause hazard—multiplies survival:

L_i^E =\exp(-0.50)\times0.06 =0.6065\times0.06 =0.03639\ \text{per day}.

Using \(0.04\) per day in this calculation would assign the observed event to the wrong cause. The target cause still contributes to the survival factor: its hazard helps determine the chance of having remained free of every event until day 5.

Worked Example: Right Censoring

A third hypothetical individual is followed for 4 months and then lost to follow-up without an observed event. Conditional on the specified random effects and biomarker history, suppose the target and competing hazards are constant at \(0.07\) and \(0.03\) per month. Assume censoring is independent of the event process given the conditioning information, so the calculation below is the event-process contribution.

The total cumulative hazard through censoring is \((0.07+0.03)\times4=0.40\). The event-process contribution is the probability of surviving all causes until month 4:

L_i^E=S_i(4)=\exp(-0.40)=0.6703.

There is no factor of \(0.07\) or \(0.03\) at month 4 because no event was observed then. If the censoring-time distribution is explicitly modeled, it has a separate contribution; it does not turn the censoring time into an event time.

Assumptions, Diagnostics, and Alternatives

The formulas require a correctly specified conditional hazard for each cause, including the way the hazard depends on the random effects and longitudinal information. They also require correct event-time and cause recording, and a risk history that ends at the first event or censoring time. When censoring is treated as ignorable for the event-process contribution, censoring must be independent of the event process conditional on the variables used in that assumption. If this condition is implausible, the censoring mechanism may need explicit modeling or a sensitivity analysis.

Useful checks include confirming that hazards are nonnegative, that cumulative hazards increase rather than decrease with time, and that all hazards and times use compatible units. Examine event counts by cause and follow-up time; a very sparse cause can make its hazard difficult to estimate. Compare fitted event-time patterns by cause with observed patterns, and check whether predicted survival systematically departs from observed event-free proportions. For joint models, also assess whether the longitudinal model adequately represents the biomarker trajectories: a poor trajectory model can distort the conditional hazard and its contribution even when the formula is applied correctly.

A cause-specific hazard analysis may code a competing event as leaving the risk set when estimating the hazard for a particular cause. That risk-set convention does not make the observed competing event right censoring in the full competing-risks event-process likelihood: the full contribution includes the hazard for the cause that actually occurred. Likewise, a subdistribution-hazard analysis targets a different quantity. Neither should be substituted for the cause-specific event contribution without changing the statistical model and estimand.

Observed outcome	Event-process factor at \(T_i\)	What distinguishes it
Target event, \(D_i=1\)	\(S_i(T_i)\lambda_{i1}(T_i)\)	Survival from all causes, then the target-cause hazard
Competing event, \(D_i=k\ne1\)	\(S_i(T_i)\lambda_{ik}(T_i)\)	Survival from all causes, then the observed competing-cause hazard
Right censoring, \(D_i=0\)	\(S_i(T_i)\)	Survival through censoring, with no event hazard at that time

One limitation is that these are conditional contributions: the random effects are held fixed. In applications they are generally latent, not known constants. A conditional expression alone therefore does not average over random-effects uncertainty or constitute the complete likelihood for all observed data. That distinction is especially important when interpreting a fitted hazard for an individual trajectory versus a population-level event pattern.

A Common Mistake

A common error is to treat a competing event as though the individual were simply right-censored at that time and then use only the target-cause hazard, or to omit the competing-cause hazard from the survival factor. The reasoning is that the target event did not occur, so the competing event supposedly contributes no event information. The correction is that the competing event is an observed first event: its own hazard multiplies survival, and survival up to that time accounts for every cause. Only genuine right censoring has no event-hazard factor at the observed time.

Key takeaway: Conditional on random effects and longitudinal history, every outcome contributes survival from all causes up to the observed time. An observed event adds the hazard for its actual cause; right censoring adds no event hazard.

Check Your Understanding

Use the conditional event-process contribution only. Assume constant hazards over the stated follow-up and independent censoring where censoring is observed.

With target hazard \(0.05\) per month and competing hazard \(0.15\) per month, an individual has a target event at month 3. Write and calculate the contribution.
With hazards \(0.02\) and \(0.08\) per week for the target and competing causes, respectively, an individual has the competing event at week 2. Which hazard multiplies survival, and what is the contribution?
With total hazard \(0.06\) per month, an individual is right-censored at month 5. What is the event-process contribution, and why is there no hazard factor at month 5?
Explain why a competing event's contribution is not the same as a censoring contribution at the same time.

Solutions. (1) The total cumulative hazard is \((0.05+0.15)\times3=0.60\), so \(L_i^E=\exp(-0.60)\times0.05=0.02744\) per month. (2) The competing-cause hazard multiplies survival: \(L_i^E=\exp\{-(0.02+0.08)\times2\}\times0.08=\exp(-0.20)\times0.08=0.06550\) per week. (3) \(L_i^E=\exp(-0.06\times5)=\exp(-0.30)=0.7408\). No event was observed at month 5, so the event process contributes survival only. (4) A competing event is an observed event and contributes its cause-specific hazard as well as survival from all causes up to that time; censoring contributes survival only.