Why Censoring Changes the Likelihood
In a joint model, longitudinal measurements and an event time can both provide information about a subject’s shared random effects. The event-time record may end with an observed event or with right censoring. A censored subject has not had an observed event by the end of follow-up; that subject’s record still supplies information about how long they remained event-free. The likelihood must use that information without claiming that censoring itself was an event.
The distinction matters in Bayesian computation as well as in model specification. Each subject’s event-time contribution enters the posterior target, alongside the longitudinal data contribution and prior distributions. Changing the event indicator changes that contribution, sometimes substantially.
Let \(b_i\) denote subject \(i\)’s shared random effects, and let \(\theta\) collect the model parameters. Conditional on \(b_i\) and \(\theta\), write the hazard as \(h_i(t\mid b_i,\theta)\), the cumulative hazard as \(H_i(t\mid b_i,\theta)=\int_0^t h_i(u\mid b_i,\theta)\,du\), and the survival function as \(S_i(t\mid b_i,\theta)\). The hazard has units of inverse time; the cumulative hazard is dimensionless.
Why the Two Contributions Have These Forms
For a continuously distributed event time, the conditional density at time \(t\) is \(f_i(t\mid b_i,\theta)=h_i(t\mid b_i,\theta)S_i(t\mid b_i,\theta)\). This follows from the definition of the hazard: among subjects who have survived to \(t\), the instantaneous event rate is \(h_i(t\mid b_i,\theta)\). A small interval of width \(dt\) around \(t\) therefore has event probability approximately \(S_i(t\mid b_i,\theta)h_i(t\mid b_i,\theta)dt\). The density is the factor multiplying \(dt\).
For a censored subject, the observed information is not that the event occurred at the censoring time. The information is that \(T_i>t_i\). The probability of that information, conditional on the random effects and parameters, is \(P(T_i>t_i\mid b_i,\theta)=S_i(t_i\mid b_i,\theta)\). Hence the event indicator selects the density contribution when an event is observed and the survival probability when it is not.
These expressions assume a continuous event-time distribution with a hazard defined over the observed interval. They also assume that censoring is noninformative for the event process conditional on the modeled covariates and random effects: after conditioning on those quantities, the censoring mechanism does not carry additional information about \(T_i\). Under this condition, and when censoring-model parameters are distinct from the event-model parameters, the censoring-time distribution can be omitted from the event-model likelihood. This is an assumption about the observation process, not a consequence of using Bayesian inference.
For the shared-random-effects model, let \(y_i\) denote the observed longitudinal measurements for subject \(i\), with conditional density \(f(y_i\mid b_i,\theta)\). Conditional on \(b_i\) and \(\theta\), the joint contribution of the measurements and event-time record is
The shared \(b_i\) links the two data streams. In a posterior target that retains the random effects, this contribution is multiplied across subjects and combined with the prior densities. If random effects are integrated out, the subject-level product is integrated over the distribution of \(b_i\); it is not generally valid to replace \(b_i\) by a population average before combining the longitudinal and event-time terms.
Worked Contributions
Worked Example: An Event and a Censoring at the Same Follow-Up Time
Suppose, for illustration, that a subject has a constant hazard of \(0.20\) per month over follow-up. Compare two possible records, each with \(t_i=4\) months: an observed event and right censoring.
The cumulative hazard at four months is \(H_i(4)=0.20\times4=0.80\), so the survival probability is \(S_i(4)=\exp(-0.80)=0.4493\).
For an observed event, \(\delta_i=1\). The contribution is $$ h_i(4)S_i(4)=0.20\exp(-0.80)=0.08987 $$ per month. It is a density contribution, so it has inverse-time units.
For right censoring, \(\delta_i=0\). The contribution is $$ S_i(4)=\exp(-0.80)=0.4493. $$ This is the probability of remaining event-free through four months. The values differ because the event record includes the event-time density factor, whereas the censored record does not. The constant-hazard assumption is only for this calculation; in an application, the specified conditional hazard determines \(H_i(t_i)\).
Worked Example: Combining a Longitudinal Density with an Observed Event
Consider a hypothetical subject with an observed event at time \(t_i\). At the subject’s random-effect value, suppose the longitudinal density contribution is \(\exp(-1)\), the event hazard at \(t_i\) is \(0.30\) per year, and the cumulative hazard through \(t_i\) is \(1.50\). These values satisfy the likelihood calculation once the hazard and cumulative hazard have been supplied by the model.
The event-time contribution is $$ 0.30\exp(-1.50)=0.066939 $$ per year. Multiplying by the longitudinal density gives the conditional joint contribution $$ \exp(-1)\{0.30\exp(-1.50)\} =0.367879(0.066939) =0.0246255 \approx 0.02463. $$
If the same subject instead had right-censored follow-up at \(t_i\), with the same cumulative hazard and longitudinal density, the event-time contribution would be \(\exp(-1.50)\), not \(0.30\exp(-1.50)\). The joint contribution would then be $$ \exp(-1)\exp(-1.50)=\exp(-2.50)=0.0820850. $$ The two records encode different observations: one gives an event at \(t_i\), while the other gives survival through \(t_i\).
Worked Example: Updating a Random-Effect Probability from Censored Follow-Up
Suppose a subject’s random effect \(b_i\) has two possible values, \(b=0\) and \(b=1\), with prior probabilities \(1/7\) and \(6/7\), respectively. For a right-censored record, assume the cumulative hazards at the observed follow-up time are \(0.75\) when \(b=0\) and \(2.50\) when \(b=1\). The subject has \(\delta_i=0\), so each event-time likelihood is a survival probability.
The unnormalized posterior weights from this event-time record are $$ \frac{1}{7}\exp(-0.75)=0.067481, \qquad \frac{6}{7}\exp(-2.50)=0.070359. $$ Normalizing gives $$ P(b=0\mid \text{censored record}) = \frac{(1/7)\exp(-0.75)} {(1/7)\exp(-0.75)+(6/7)\exp(-2.50)} = \frac{0.05\exp(-0.75)} {0.05\exp(-0.75)+0.30\exp(-2.50)} \approx 0.4896. $$
The common factor \(1/7\) was canceled in the second expression. The posterior probability is approximately \(0.4896\), compared with a prior probability of \(1/7\approx0.1429\). In this hypothetical calculation, survival to the censoring time favors \(b=0\), which has the smaller cumulative hazard. This is an update from one likelihood component only; longitudinal measurements and other data may also affect the posterior.
Assumptions, Checks, and Alternatives
Before using these contributions, verify how the event indicator is defined in the data and whether the censoring assumption is plausible for the question and study design. In particular, administrative end-of-study censoring can often be treated as conditionally noninformative, while loss to follow-up may be related to health status or unobserved risk. If censoring remains informative after conditioning on modeled quantities, simply omitting its mechanism may yield a misspecified likelihood; a model for the censoring process or a suitable sensitivity analysis may be needed.
Useful checks include confirming that observed follow-up times use the intended time origin and units, that event indicators are coded consistently, and that the cumulative hazard is evaluated through the observed time \(t_i\). For an event record, the hazard must also be evaluated at \(t_i\). Under a continuous-time model, exact ties between event and censoring times require a prespecified convention for whether the event was observed; the indicator should reflect the actual observation protocol. Posterior predictive checks of event occurrence and follow-up patterns can help reveal gross incompatibility, but they do not by themselves establish that censoring is noninformative.
| Record or approach | Contribution or consequence |
|---|---|
| Observed event at \(t_i\) | \(h_i(t_i\mid b_i,\theta)S_i(t_i\mid b_i,\theta)\) |
| Right censoring at \(t_i\) | \(S_i(t_i\mid b_i,\theta)\) |
| Treat censoring as an event | Incorrectly adds a hazard factor and asserts an event at \(t_i\). |
| Drop censored subjects | Discard the information that each subject survived event-free through \(t_i\). |
The two incorrect alternatives are not harmless simplifications. Treating a censoring time as an event changes the observed data being modeled; discarding censored subjects removes valid survival information and can also distort inference if the retained records differ systematically from the omitted ones. The survival factor is the direct likelihood contribution of a right-censored record under the stated assumptions.
Common Mistake
Wrong reasoning: “The subject’s record ends at four months, so the event happened at four months.” The record ending describes the end of observation, not necessarily the event time. If \(\delta_i=0\), the subject is known to have remained event-free through four months, so the contribution is \(S_i(4\mid b_i,\theta)\). The hazard factor is included only when \(\delta_i=1\), indicating an observed event. A related mistake is to omit censored subjects entirely; their survival probabilities still contribute to the posterior.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A subject is followed for 6 months and has no observed event before loss to follow-up. What are \(t_i\) and \(\delta_i\), and what is the event-time likelihood contribution?
- At a censoring time \(t_i\), the cumulative hazard is \(0.40\). Calculate the event-time likelihood contribution for a right-censored subject.
- At an observed event time, the hazard is \(0.10\) per month and the cumulative hazard is \(0.60\). Calculate the event-time density contribution.
- Why is it generally wrong to treat the censoring time as an observed event time?
- State the censoring assumption that permits omission of the censoring-time distribution from the event-model likelihood.
Solutions. (1) \(t_i=6\) months and \(\delta_i=0\), so the contribution is \(S_i(6\mid b_i,\theta)=\exp\{-H_i(6\mid b_i,\theta)\}\); its numerical value cannot be calculated without the cumulative hazard. (2) The contribution is \(\exp(-0.40)=0.6703\), the probability of remaining event-free through the censoring time. (3) The contribution is \(0.10\exp(-0.60)=0.05488\) per month. (4) Censoring says observation ended while the subject was known to be event-free through that time; it does not assert an event occurred then. (5) Conditional on modeled covariates and random effects, censoring must be noninformative for the event time, and censoring-model parameters must be distinct from the event-model parameters for the censoring distribution to be omitted.