Right-Censored Subject Likelihood Contributions

From Follow-Up Records to a Likelihood Contribution

In a joint model, the event process is described conditional on a subject’s random effects. For a subject who experiences the event during follow-up, the record contains an event time. For a subject who leaves observation without experiencing the event, the record says only that the event time exceeds the final follow-up time. These records supply different information and therefore have different likelihood contributions.

This tutorial focuses on the event-process contribution for one subject, conditional on that subject’s random effects. It is not the complete subject likelihood, which also involves the longitudinal measurements. The conditional event-time density and survival function are the ingredients established in the earlier material on event-time density conditional on random effects.

Definition: Let $T_i$ be subject $i$'s event time and $C_i$ the time at which that subject would be censored. The observed follow-up time and event indicator are $$ Y_i=\min(T_i,C_i), \qquad \delta_i=I(T_i\leq C_i). $$ Thus, $\delta_i=1$ indicates an observed event at $Y_i=T_i$, while $\delta_i=0$ indicates right censoring at $Y_i=C_i<T_i$. Conditional on random effects $b_i$, write $f_i(t\mid b_i)$ for the event-time density and $S_i(t\mid b_i)=P(T_i>t\mid b_i)$ for the event-time survival function.

The standard event-process contribution assumes a continuous event-time distribution and a censoring mechanism that is independent of the event time conditional on the random effects and relevant observed covariates. In notation, the assumption is $T_i\perp C_i\mid b_i,\text{ covariates}$. The censoring time may be fixed by the study design; if it is random, its distribution is not part of the event-process contribution shown here. If the censoring mechanism is informative even after conditioning on the variables in the model, omitting it may not give a valid event-process likelihood.

The event indicator convention assigns an event at a tie to $\delta_i=1$. With continuous, independent event and censoring times, exact ties have probability zero. If a study has a discrete time scale or a specified tie mechanism, that mechanism must be handled consistently rather than silently assuming a continuous-time model.

Deriving the Two Cases

First suppose the event is observed, so $\delta_i=1$ and $Y_i=T_i$. The observed value is a realization of the event time, and its conditional likelihood contribution is the event-time density evaluated at that value:

L_i^{(T)}=f_i(Y_i\mid b_i), \qquad \delta_i=1.

If the subject is censored, then $\delta_i=0$ and $Y_i=C_i$. The event time is not observed as a particular value. The information is instead that the event had not occurred by $Y_i$. The probability of that information, conditional on $b_i$, is the survival probability:

L_i^{(T)}=P(T_i>Y_i\mid b_i)=S_i(Y_i\mid b_i), \qquad \delta_i=0.

Combining these cases gives the standard right-censored event-process contribution:

L_i^{(T)} = \bigl[f_i(Y_i\mid b_i)\bigr]^{\delta_i} \bigl[S_i(Y_i\mid b_i)\bigr]^{1-\delta_i}.

This expression selects the density when $\delta_i=1$ and the survival probability when $\delta_i=0$. It does not claim that a censored subject experienced an event at the censoring time. The contribution is conditional on $b_i$; it does not integrate over the random-effects distribution.

Writing the Contribution With a Hazard

For a continuous event time, the conditional hazard $h_i(t\mid b_i)$ and cumulative hazard $H_i(t\mid b_i)$ satisfy $f_i(t\mid b_i)=h_i(t\mid b_i)S_i(t\mid b_i)$ and $S_i(t\mid b_i)=\exp\{-H_i(t\mid b_i)\}$, where $H_i(t\mid b_i)=\int_0^t h_i(u\mid b_i)\,du$. Substituting the density identity into the two-case contribution yields:

L_i^{(T)} = \bigl[h_i(Y_i\mid b_i)\bigr]^{\delta_i} \exp\{-H_i(Y_i\mid b_i)\}.

The cumulative-hazard term is present for both observed events and censored records. It accounts for remaining event-free up to the observed follow-up time. The hazard factor appears only for an observed event, because only then does the record specify an event occurring at that time. When $\delta_i=1$, the contribution has units of inverse time because it contains a density; when $\delta_i=0$, the survival contribution is a probability and is dimensionless. Likelihood contributions need not have the same units across different observation patterns.

Justification: For $\delta_i=1$, $f_i(Y_i\mid b_i)=h_i(Y_i\mid b_i)\exp\{-H_i(Y_i\mid b_i)\}$. For $\delta_i=0$, $S_i(Y_i\mid b_i)=\exp\{-H_i(Y_i\mid b_i)\}$. The single expression with the event indicator therefore reproduces the correct contribution in both cases.

Worked Example: An Observed Event Under a Constant Hazard

Suppose, hypothetically, that a subject has a constant conditional event hazard of $0.08$ per year over follow-up. The subject’s event is observed at $Y_i=6$ years, so $\delta_i=1$. Assume the specified hazard is valid over this period and censoring is independent under the model.

For a constant hazard, $H_i(t\mid b_i)=0.08t$, $S_i(t\mid b_i)=\exp(-0.08t)$, and $f_i(t\mid b_i)=0.08\exp(-0.08t)$. The subject contribution is

L_i^{(T)} = h_i(6\mid b_i)\exp\{-H_i(6\mid b_i)\} =0.08\exp(-0.08\times 6) =0.08\exp(-0.48) \approx 0.04950\ \text{per year}.

This is a density contribution, not the probability that the event occurred at exactly year 6. Its units are inverse years. The computation uses the event density because the event indicator is one.

Worked Example: Right Censoring Under a Constant Hazard

Now suppose, hypothetically, that a different subject has the same conditional constant hazard of $0.08$ per year but is administratively censored after 6 years without an observed event. Thus $Y_i=6$ years and $\delta_i=0$. Administrative censoring at the stated time is treated as independent of the event time.

The subject’s record establishes that the event time exceeds 6 years. Therefore the contribution is the survival probability, not the event density:

L_i^{(T)} =S_i(6\mid b_i) =\exp\{-H_i(6\mid b_i)\} =\exp(-0.08\times 6) =\exp(-0.48) \approx 0.6188.

The contribution is the conditional probability of remaining event-free beyond 6 years, approximately 0.6188. Although the same cumulative hazard appears in the previous example, this censored record has no hazard multiplier: no event was observed at year 6.

Worked Example: An Observed Event Under a Time-Varying Hazard

Consider a hypothetical subject whose conditional hazard over time is specified as $h_i(t\mid b_i)=0.05\exp(0.2t)$ per year, with time measured in years. Assume this hazard model and the independent-censoring condition apply. An event is observed at $Y_i=2$ years, so $\delta_i=1$.

Integrating the hazard gives the cumulative hazard at two years:

H_i(2\mid b_i) =\int_0^2 0.05\exp(0.2u)\,du =0.25\{\exp(0.4)-1\} \approx 0.12296.

The hazard at the event time is $h_i(2\mid b_i)=0.05\exp(0.4)\approx0.07459$ per year. The contribution is therefore

L_i^{(T)} =0.05\exp(0.4)\exp\bigl[-0.25\{\exp(0.4)-1\}\bigr] \approx 0.07459\times0.8843 \approx 0.06596\ \text{per year}.

The survival factor, about 0.8843, represents survival through two years; the hazard factor supplies the event density at the observed event time. The arithmetic checks the units as well: the survival factor is dimensionless and the hazard factor is per year.

Assumptions, Checks, and Common Errors

Before using the formula, verify that the recorded time is the time from the stated origin to either the event or censoring, and that $\delta_i$ uses the stated event convention. The hazard and cumulative hazard must be defined over the full interval from the time origin to $Y_i$, and the cumulative hazard must be the integral of that hazard. A negative cumulative hazard, a survival value outside the interval from zero to one, or a density that is nonpositive signals an error in the hazard specification or its calculation.

The independent-censoring assumption is substantive. A study may censor subjects because of administrative end-of-study, loss to follow-up, or withdrawal. If censoring is related to the event time after conditioning on the modeled variables, the simple event-process contribution may not suffice. The censoring process may need to be modeled or addressed through an appropriate design or analysis. Likewise, a misspecified event hazard can produce incorrect density and survival contributions even when censoring is independent.

Pitfall: treating censoring as an event. The wrong reasoning is “follow-up ended at 6 years, so use the event density at 6.” The correction is to use $S_i(6\mid b_i)$ when $\delta_i=0$, because the observed information is that the event time exceeded 6 years.
Pitfall: using only the hazard for an observed event. The hazard at $Y_i$ alone is not the event-time density. The contribution is $h_i(Y_i\mid b_i)\exp\{-H_i(Y_i\mid b_i)\}$, including survival through the preceding follow-up.
Pitfall: treating the censoring contribution as the whole joint-model contribution. This expression concerns only the event process conditional on random effects. It does not include the longitudinal measurement contribution or average over random effects.

A subject-level event-process likelihood is different from a survival curve summary or a Cox partial-likelihood contribution: it directly represents the observed event-time or censoring information conditional on the specified hazard. The expression here is useful when the conditional hazard or survival function is specified, but it does not by itself resolve how to model a baseline hazard, how to combine longitudinal records with event information, or how to integrate over random effects. Those are separate modeling steps.

Record the observed pair.
Write $Y_i=\min(T_i,C_i)$ and $\delta_i=I(T_i\leq C_i)$, checking the time origin and event coding.

Select the observation type.
For $\delta_i=1$, use the event density; for $\delta_i=0$, use the probability of surviving beyond $Y_i$.

Evaluate the conditional contribution.
Use $f_i(Y_i\mid b_i)^{\delta_i}S_i(Y_i\mid b_i)^{1-\delta_i}$, or its hazard form, and check the units and the censoring assumption.

Key takeaway: An observed event contributes its conditional event-time density; right censoring contributes the conditional probability of remaining event-free through the observed follow-up time. In hazard notation, both cases share the cumulative-hazard survival factor, while the hazard multiplier is included only for an observed event.

Check Your Understanding

For each item, identify or calculate the conditional event-process contribution. Solutions follow the questions.

If $Y_i=4$ years and $\delta_i=0$, should the contribution be $f_i(4\mid b_i)$ or $S_i(4\mid b_i)$? Explain what the record tells us.
For a constant hazard of $0.10$ per year, calculate the contribution for an event observed at 3 years.
Under the same constant hazard, calculate the contribution for a subject censored event-free at 3 years.
In $h_i(Y_i\mid b_i)\exp\{-H_i(Y_i\mid b_i)\}$, which factor represents survival through follow-up, and which represents the event at the observed time?

Solutions. (1) Use $S_i(4\mid b_i)$: the record says that the event time exceeds 4 years. (2) The contribution is $0.10\exp(-0.10\times3)=0.10\exp(-0.3)\approx0.07408$ per year. (3) It is $\exp(-0.10\times3)=\exp(-0.3)\approx0.7408$, a survival probability. (4) The exponential term is survival through follow-up; the hazard multiplier supplies the event-time density factor and appears only when an event is observed.

Right-Censored Subject Likelihood Contributions

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