Likelihood Contributions Under Right Censoring

Why Censoring Changes the Likelihood

In duration data, a recorded time does not always identify when the event occurred. A person may leave observation before experiencing the event, or a study may end while the person remains event-free. In that case the data give a lower bound on the event time, not its exact value. The likelihood must reflect what is known: an observed event contributes information about an event time at a particular value, while a right-censored record contributes information that the event time exceeds the censoring time.

The distinction is consequential. Treating a censored time as if it were an event time changes the statistical model and can distort estimates. The correct likelihood follows directly from the observed information, together with an assumption about how censoring relates to event times.

Definition: Let \(T\) be the event time and \(C\) the censoring time. For each subject, the observed time is \(Y=\min(T,C)\), and the event indicator is \(\delta=I(T\leq C)\). Thus \(\delta=1\) records an observed event at \(Y=T\), while \(\delta=0\) records right censoring at \(Y=C\) and tells us that \(T>C\). Write \(f_T(t\mid x;\theta)\) for the event-time density and \(S_T(t\mid x;\theta)=P(T>t\mid x)\) for its survival function, conditional on covariates \(x\) and parameters \(\theta\).

For a continuous event time, an observed event at \(y\) contributes the density \(f_T(y\mid x;\theta)\). A censored record at \(y\) contributes the probability \(P(T>y\mid x)=S_T(y\mid x;\theta)\). The density is not itself the probability of an event at one exact instant; it is the likelihood contribution associated with the observed event time.

Key result: Under independent, non-informative censoring, the event-time part of the individual likelihood contribution is

L_T(\theta; y,\delta,x) = f_T(y\mid x;\theta)^\delta S_T(y\mid x;\theta)^{1-\delta}.

When \(\delta=1\), this equals the event-time density at \(y\). When \(\delta=0\), it equals the probability of remaining event-free beyond \(y\).

Deriving the Two Contributions

The formula can be derived by distinguishing the two observable outcomes. Suppose, conditional on covariates \(x\), that \(T\) and \(C\) are independent. Let \(g_C(c\mid x;\psi)\) be the censoring-time density and \(G_C(c\mid x;\psi)=P(C>c\mid x)\) its survival function. Assume for this derivation that both times are continuous, so an exact tie has probability zero.

If \(\delta=1\), the event is observed at \(y\). For this to occur, the event time lies in a small interval around \(y\), and censoring has not occurred by \(y\). Independence makes the corresponding joint density contribution \(f_T(y\mid x;\theta)G_C(y\mid x;\psi)\). If \(\delta=0\), censoring occurs at \(y\) while the event has not yet occurred. Its joint contribution is \(g_C(y\mid x;\psi)S_T(y\mid x;\theta)\).

Combining the cases gives the full observed-data likelihood contribution:

L_{\mathrm{obs}}(\theta,\psi;y,\delta,x) = \left[f_T(y\mid x;\theta)G_C(y\mid x;\psi)\right]^\delta \left[g_C(y\mid x;\psi)S_T(y\mid x;\theta)\right]^{1-\delta}.

The censoring factors are part of the joint likelihood. They can be omitted when making inference about \(\theta\) if the censoring mechanism is independent of the event time conditional on included covariates, and its parameters \(\psi\) are distinct from and variation-independent of \(\theta\). The remaining factors then give the standard event-time likelihood. If censoring is fixed by design, such as a common administrative study end, the censoring mechanism is not ordinarily modeled with a density; the event-time contribution is still \(f_T(y)\) for an event and \(S_T(y)\) for a subject censored at the study end.

Worked Example: One Event and One Censored Record

Suppose a hypothetical event time follows an exponential distribution with rate \(\lambda=0.20\) per month. Its density and survival function are \(f_T(t)=\lambda e^{-\lambda t}\) and \(S_T(t)=e^{-\lambda t}\). Find the event-time contributions for an event at 3 months and for a record censored at 3 months.

For the observed event, \(\delta=1\), so the contribution is the density:

f_T(3)=0.20e^{-0.20(3)} =0.20e^{-0.60} \approx 0.1098\ \text{per month}.

For the censored record, \(\delta=0\), so the contribution is the probability that the event time exceeds 3 months:

S_T(3)=e^{-0.20(3)} =e^{-0.60} \approx 0.5488.

The values differ because the records convey different information. The first identifies an event time at 3 months and contributes a density; the second establishes only that the event had not occurred by 3 months and contributes a survival probability. The rate and units are specified, and the exponential formulas define a valid continuous event-time model.

From Individual Contributions to a Sample Likelihood

For \(n\) independent subjects, conditional on their covariates, the likelihood is the product of the individual event-time contributions. Writing the data as \((y_i,\delta_i,x_i)\), for \(i=1,\ldots,n\), gives

L_T(\theta) = \prod_{i=1}^{n} f_T(y_i\mid x_i;\theta)^{\delta_i} S_T(y_i\mid x_i;\theta)^{1-\delta_i}.

The corresponding log-likelihood is

\ell_T(\theta) = \sum_{i=1}^{n} \left[ \delta_i\log f_T(y_i\mid x_i;\theta) + (1-\delta_i)\log S_T(y_i\mid x_i;\theta) \right].

This expression makes it straightforward to include both types of records without replacing censored times by assumed event times. A term for an observed event uses the log density; a term for a censored observation uses the log survival probability.

Worked Example: Exponential Likelihood With Mixed Records

In a hypothetical sample, four subjects have observed times in months and event indicators \((2,1),(3,0),(5,1),(4,0)\). Assume independent subjects and an exponential event-time distribution with rate \(\lambda>0\) per month. The censoring mechanism is independent and its parameters are distinct from \(\lambda\). Construct the event-time likelihood and find the maximum-likelihood estimate of \(\lambda\).

There are two observed events, at 2 and 5 months, and two censored records, at 3 and 4 months. Each event contributes \(\lambda e^{-\lambda y_i}\); each censored record contributes \(e^{-\lambda y_i}\). Therefore,

L_T(\lambda) = (\lambda e^{-2\lambda}) (e^{-3\lambda}) (\lambda e^{-5\lambda}) (e^{-4\lambda}) = \lambda^2 e^{-14\lambda}.

The exponent uses the sum of all four observed times: \(2+3+5+4=14\) months. Taking logarithms gives \(\ell_T(\lambda)=2\log\lambda-14\lambda\). Differentiating and setting the score to zero,

\frac{d\ell_T}{d\lambda} = \frac{2}{\lambda}-14=0 \quad\Longrightarrow\quad \hat{\lambda}=\frac{2}{14} \approx 0.1429\ \text{per month}.

The second derivative is \(-2/\lambda^2<0\) for every \(\lambda>0\), so this stationary point maximizes the log-likelihood. The estimate uses the two events and the exposure information in all four observed times; the censored observations are not treated as events. This calculation assumes the exponential model is correctly specified and the censoring assumptions stated above hold.

Independent Censoring: What the Assumption Does

In practice, censoring may depend on measured covariates. The relevant assumption is then typically conditional independent censoring: \(T\) and \(C\) are independent given the covariates included in the analysis. This permits censoring and event times to be related through those covariates, while ruling out residual dependence after conditioning on them. If important predictors of both event and censoring are omitted, the assumption may fail.

Under this assumption and distinct censoring parameters, censoring determines which part of the event-time information is observed but does not add a parameter-dependent factor to the likelihood for \(\theta\). This is why the standard survival likelihood can use the density for events and the survival probability for censored observations. It does not mean censoring is harmless in every setting: the assumption must be credible for the target population and the event-time model.

Worked Example: Keeping the Censoring Factors in a Joint Likelihood

Consider three hypothetical independent subjects. Their records are an event at 2 months, censoring at 3 months, and an event at 5 months. Let event times be exponential with rate \(\lambda\) per month, and let independent censoring times be exponential with rate \(\gamma\) per month. Both rates are positive, and \(\lambda\) and \(\gamma\) are distinct parameters.

For the event at 2 months, the joint contribution is \(f_T(2)G_C(2)=\lambda e^{-2\lambda}e^{-2\gamma}\). For censoring at 3 months, it is \(g_C(3)S_T(3)=\gamma e^{-3\gamma}e^{-3\lambda}\). For the event at 5 months, it is \(f_T(5)G_C(5)=\lambda e^{-5\lambda}e^{-5\gamma}\). Multiplication gives

L_{\mathrm{obs}}(\lambda,\gamma) = \lambda^2\gamma e^{-10\lambda}e^{-10\gamma}.

The total observed time is \(2+3+5=10\) months, so it appears in the exponent for each rate. For inference about \(\lambda\), the censoring factor \(\gamma e^{-10\gamma}\) contains no \(\lambda\) and can be omitted under the stated independence and parameter-distinctness assumptions. The event-time likelihood is therefore \(L_T(\lambda)=\lambda^2e^{-10\lambda}\). The censoring factor is not absent from the full joint likelihood; it is irrelevant to the likelihood for \(\lambda\) under these conditions.

When the Standard Likelihood Is Not Justified

If subjects with a high underlying event risk are more likely to be lost to follow-up, censoring may be informative. For a censored subject, the relevant event-time information is then not necessarily the marginal probability \(P(T>y\mid x)\). It may instead depend on the subject’s event-time risk given the censoring process. Using the standard contribution without addressing this dependence can produce misleading inference.

Possible responses depend on the study design and the scientific question. Relevant predictors of censoring may be measured and conditioned on; the censoring process may be modeled jointly with event time; or a justified weighting or sensitivity analysis may be used. These approaches require their own assumptions. Merely observing that censoring times are recorded separately does not establish independent censoring.

Common pitfall: Treating every observed time \(y_i\) as an event and using \(f_T(y_i)\) for both event and censored records. This asserts that the event occurred at the censoring time, which the data do not show. For \(\delta_i=0\), the observed fact is \(T_i>y_i\), so the event-time contribution is \(S_T(y_i)\), not \(f_T(y_i)\).

A second caution is to say that independent censoring alone always allows censoring factors to be dropped. The standard argument also requires that the censoring mechanism contributes no parameter-dependent information about the event-time parameter—for example, that its parameters are distinct and variation-independent, or that censoring is fixed by design. When these conditions are not credible, the joint observed-data likelihood should be examined rather than automatically discarding the censoring mechanism.

Record what was observed.
For each subject, identify \(Y=\min(T,C)\) and whether \(\delta=1\) (event) or \(\delta=0\) (right censoring).

Choose the event-time factor.
Use \(f_T(Y\mid x;\theta)\) for an observed event and \(S_T(Y\mid x;\theta)\) for a censored time.

Assess censoring assumptions.
State the conditioning covariates and justify conditional independence and parameter distinctness, or retain and model the relevant censoring factors.

Combine contributions.
Under independent subjects, multiply the individual contributions and optimize the resulting likelihood for the event-time parameters.

The central accounting rule is concise: an observed event supplies a density contribution, while a right-censored record supplies a survival contribution. Correctly applying that rule preserves the distinction between an exact event time and a lower bound, and makes the censoring assumptions visible in the likelihood.

Check Your Understanding

Answer each question using the event-time likelihood and the assumptions stated in the tutorial.

A subject is censored at 8 months. What event-time information does this record provide, and which likelihood factor represents it?
For a continuous event time, explain why an observed event at \(y\) contributes a density rather than a survival probability.
Write the event-time likelihood for three independent records \((1,1),(4,0),(6,1)\), using \(f_T\) and \(S_T\).
State conditional independent censoring in words and explain why conditioning on covariates matters.
Why can the censoring factors be omitted for inference about \(\theta\) in the joint likelihood example, and what assumption would make that omission questionable?