Likelihood Contributions Under Right Censoring in Joint Models

Why censoring changes the likelihood contribution

In a survival study, an individual may experience the event during observation, or follow-up may end before the event is seen. A right-censored observation does not say that the event occurred at the recorded time. It says that the event time is later than that time. The likelihood must reflect this distinction: an observed event contributes a density, whereas a right-censored time contributes the probability of surviving beyond the censoring time.

This distinction is central when constructing the event-time part of a joint model. It also matters in a survival model on its own: treating censored follow-up as though it were an event changes the likelihood and can distort estimates. Here, the target is the survival likelihood for observed event and right-censored times under a parametric model.

Definition: For subject $i$, let $T_i$ be the event time and $C_i$ the censoring time. The observed follow-up time is $Y_i=\min(T_i,C_i)$, and the event indicator is $\delta_i=I(T_i\leq C_i)$. Thus $\delta_i=1$ denotes an observed event at $Y_i$, while $\delta_i=0$ denotes right censoring at $Y_i$. If $T_i$ has density $f_i(t;\theta)$ and survival function $S_i(t;\theta)=P(T_i>t)$, the survival-parameter likelihood contribution is $$ L_i(\theta)=f_i(Y_i;\theta)^{\delta_i}S_i(Y_i;\theta)^{1-\delta_i}. $$

When $\delta_i=1$, the contribution is $f_i(Y_i;\theta)$: the observation gives the event time, represented by a density because time is continuous. When $\delta_i=0$, the contribution is $S_i(Y_i;\theta)$: all that is known about $T_i$ is that $T_i>Y_i$. In particular, a censored observation is not a missing event time that can simply be discarded.

Assumptions behind the contribution

The usual contribution applies when event times are continuous and censoring is independent of the event time, at least conditional on the covariates and other modeled information. The censoring process must not carry unmodeled information about a subject’s event time after conditioning on that information. The event-time distribution must also be specified well enough to evaluate its density and survival function at the observed times.

For the product likelihood across subjects, subjects are assumed independent conditional on the model’s covariates and parameters. The observation scheme is taken to start from the stated time origin for each subject. If censoring is informative, or if entry into observation depends on survival up to entry, the likelihood needs additional treatment; the formula here is not by itself sufficient for those settings.

The censoring distribution is not usually part of the survival-parameter likelihood when it is independent and its parameters are distinct from $\theta$. More fully, the joint density of observed time and event indicator includes censoring-process factors as well as event-process factors. Under independent censoring, those censoring factors do not involve the survival parameters of interest and can be omitted when estimating them. If censoring and event processes share parameters or are dependent, that separation is not justified.

From density and survival to hazard

The earlier material on hazard and cumulative hazard gives $h_i(t;\theta)=f_i(t;\theta)/S_i(t;\theta)$ and $S_i(t;\theta)=\exp\{-H_i(t;\theta)\}$, where $H_i(t;\theta)=\int_0^t h_i(u;\theta)\,du$. Substituting these identities into the individual contribution gives an equivalent form:

L_i(\theta) = \left\{h_i(Y_i;\theta)\right\}^{\delta_i} \exp\{-H_i(Y_i;\theta)\}.

This representation separates the two kinds of information. An observed event contributes the hazard at the event time, as well as survival up to that time through the cumulative-hazard term. A censored observation contributes no event-time hazard factor, but it still contributes survival up to its censoring time.

Justification: Since $f_i(t;\theta)=h_i(t;\theta)S_i(t;\theta)$, the density-survival contribution is $$ [f_i(Y_i;\theta)]^{\delta_i}[S_i(Y_i;\theta)]^{1-\delta_i} = [h_i(Y_i;\theta)]^{\delta_i}[S_i(Y_i;\theta)]^{\delta_i+1-\delta_i} = [h_i(Y_i;\theta)]^{\delta_i}S_i(Y_i;\theta). $$ Using $S_i(Y_i;\theta)=\exp\{-H_i(Y_i;\theta)\}$ yields the hazard form. The survival term remains for both event and censored observations.

Worked contributions and likelihood assembly

Worked Example: One Event and One Censored Time

Suppose, hypothetically, event times follow an exponential model with rate $\lambda=0.20$ per month. In this example, $f(t;\lambda)=\lambda e^{-\lambda t}$ and $S(t;\lambda)=e^{-\lambda t}$. One subject experiences the event at 3 months; another is censored at 4 months. Calculate their contributions at the stated rate.

For the event at 3 months, $\delta=1$, so the contribution is the density: $$ L_{\mathrm{event}}=f(3;0.20)=0.20e^{-(0.20)(3)} =0.20e^{-0.6}\approx 0.1098\ \text{per month}. $$

For the subject censored at 4 months, $\delta=0$, so the contribution is the survival probability: $$ L_{\mathrm{censored}}=S(4;0.20)=e^{-(0.20)(4)} =e^{-0.8}\approx 0.4493. $$

The numerical values have different units: a continuous event-time density is measured per month, while a survival probability is dimensionless. Their roles in the likelihood are nevertheless clear: one records an event at a particular time, and the other records survival through a stated duration.

Worked Example: Product Likelihood for Independent Subjects

Consider four hypothetical independent subjects under the same exponential model, with $\lambda=0.25$ per month. Their observed times in months are 2 (event), 3 (censored), 4 (event), and 1 (censored). Calculate the likelihood at this rate.

The event indicators are $1,0,1,0$. The event contributions are $0.25e^{-(0.25)(2)}$ and $0.25e^{-(0.25)(4)}$. The censored contributions are $e^{-(0.25)(3)}$ and $e^{-(0.25)(1)}$. Independence means the sample likelihood is their product: $$ L(\lambda) =(0.25)^2 \exp\{-0.25(2+3+4+1)\} =0.0625e^{-2.5} \approx 0.005130. $$

There are two observed events, so the rate factor is $\lambda^2$; all four observed times contribute to the survival part of the likelihood. Equivalently, at a general rate $\lambda$, this dataset gives $L(\lambda)=\lambda^2e^{-10\lambda}$, with time measured in months. The likelihood value compares support for different values of $\lambda$; it is not itself a probability that the parameter equals $0.25$.

Worked Example: A Covariate Changes the Hazard

Suppose a hypothetical parametric model has hazard $h_i(t)=\lambda_0\exp(\beta x_i)$, where $x_i$ is a fixed binary covariate. Let $\lambda_0=0.10$ per day and $\beta=\log(2)$. One subject with $x_i=1$ has an observed event at 5 days; another with $x_i=0$ is censored at 6 days. Use the hazard form to calculate each contribution.

For $x_i=1$, the hazard is constant at $0.10\exp\{\log(2)\}=0.20$ per day and the cumulative hazard by day 5 is $0.20(5)=1$. The event contribution is $$ L_1=h_1(5)e^{-H_1(5)} =0.20e^{-1}\approx 0.07358\ \text{per day}. $$

For $x_i=0$, the hazard is $0.10$ per day and the cumulative hazard by day 6 is $0.10(6)=0.6$. Since this subject is censored, the contribution is $$ L_2=e^{-H_2(6)}=e^{-0.6}\approx 0.5488. $$

The two subjects’ contributions are multiplied under independence. The covariate changes the modeled hazard, but it does not change the rule for whether an observation contributes a hazard-at-event factor: that depends on $\delta_i$.

Independent samples and the joint-model setting

For $n$ independent subjects, the survival likelihood is the product of the individual contributions:

L(\theta)=\prod_{i=1}^{n} \left\{f_i(Y_i;\theta)\right\}^{\delta_i} \left\{S_i(Y_i;\theta)\right\}^{1-\delta_i}.

Taking logarithms turns the product into a sum, which is usually more convenient for estimation:

\ell(\theta) = \sum_{i=1}^{n} \left[ \delta_i\log f_i(Y_i;\theta) +(1-\delta_i)\log S_i(Y_i;\theta) \right].

In a joint longitudinal and time-to-event model, a subject’s event hazard may depend on a latent subject-specific effect shared with the longitudinal process. Then the survival contribution is conditional on that latent effect, and it cannot generally be treated as an unrelated marginal factor after the effect is integrated out. A common subject-level likelihood has the form $\int L_{Y,i}(b_i)L_{T,i}(b_i)p(b_i)\,db_i$, where $L_{Y,i}$ and $L_{T,i}$ are the longitudinal and survival contributions conditional on $b_i$, and $p(b_i)$ is its distribution. This observation explains why constructing the censored survival contribution correctly is a necessary building block, but does not by itself specify the full joint model.

Choosing and checking the likelihood

The full parametric likelihood uses both event and censored observations and requires a specified event-time distribution. A common alternative is Cox partial likelihood, which avoids specifying a parametric baseline hazard for estimating regression effects, but it is not the same full parametric likelihood. The appropriate approach depends on the scientific aim and assumptions; neither makes the censoring indicator irrelevant.

Before fitting a model, verify that each record has a follow-up time and a correctly coded event indicator, with 1 meaning event and 0 meaning right censoring as defined here. Check that times use a consistent unit and refer to the intended time origin. Compare the recorded censoring patterns with the study’s follow-up process, and assess whether the assumed parametric survival shape is plausible using suitable diagnostics. A numerical optimizer converging does not establish that the distributional or censoring assumptions are correct.

Event coded as censored: the likelihood uses survival rather than event density, discarding the information that an event occurred.
Censoring coded as an event: the likelihood assigns a density at a time when the event was not observed.
Censored observations omitted: the analysis loses the information that those subjects remained event-free up to their censoring times.
Informative censoring ignored: if censoring carries information about event risk beyond modeled variables, the usual independent-censoring likelihood may be biased.
Wrong time scale or origin: even correctly coded indicators cannot repair a likelihood evaluated at incorrectly recorded follow-up times.

Common pitfall: “A subject censored at 4 months had no event, so their contribution should be zero or should be omitted.” This is incorrect. The observation establishes that the event time exceeds 4 months, so its survival contribution is $S(4;\theta)$, which is generally positive and depends on the survival parameters.

Check Your Understanding

Answer each question, then compare with the solutions below.

In words, what information does a subject censored at time $y$ contribute about the event time?
For an observed event at $y$, should the contribution be $f(y;\theta)$ or $S(y;\theta)$? Explain why.
In an exponential model with rate $\lambda=0.10$ per week, calculate the contribution for a subject censored at 5 weeks.
Two independent subjects have contributions $f_1(y_1;\theta)$ and $S_2(y_2;\theta)$. Write their joint survival likelihood contribution.

Solutions. (1) The subject is known to have remained event-free through time $y$, so the information is $T>y$. (2) Use the density $f(y;\theta)$, because the event was observed at that time. (3) The contribution is $S(5)=e^{-(0.10)(5)}=e^{-0.5}\approx0.6065$, a probability with no time unit. (4) Independence gives $f_1(y_1;\theta)S_2(y_2;\theta)$. Each term reflects whether that subject had an observed event or was censored.