Likelihood Contributions for Right-Censored Event Times

Why Censoring Changes the Likelihood Contribution

In a time-to-event PK/PD analysis, an individual may experience the event during observation, or observation may end first. If the event is observed at time \(t\), the data identify its event time. If the individual is event-free when follow-up ends at time \(c\), the event time is not known; the data tell us only that it exceeds \(c\). These two outcomes provide different information and therefore have different likelihood contributions.

The distinction matters in drug-response analyses because follow-up can end for reasons such as study completion or withdrawal before an event occurs. Treating a censored time as if it were an event, or discarding it, misrepresents the observed data and can distort estimates of the hazard and its relation to drug exposure.

Write the hazard as \(h_i(t;\theta)\), conditional on the model parameters \(\theta\) and the individual’s observed covariate or exposure history. Its cumulative hazard is \(H_i(t;\theta)=\int_0^t h_i(u;\theta)\,du\), and the survival function is \(S_i(t;\theta)=\exp[-H_i(t;\theta)]\). The event-time density is \(f_i(t;\theta)=h_i(t;\theta)S_i(t;\theta)\). These standard relationships connect the instantaneous event rate to the probability of remaining event-free and to the density of an event time.

Deriving the Individual Likelihood Contribution

An observed event at time \(y_i\) contributes the event-time density at that time. A right-censored observation at time \(y_i\) contributes the probability of surviving beyond that time. Thus, conditional on the covariate and exposure information,

The second equality follows by substituting \(f_i=h_iS_i\) for an event and \(S_i=\exp(-H_i)\) for either outcome. When \(\delta_i=1\), the contribution is \(h_i(y_i;\theta)\exp[-H_i(y_i;\theta)]\). When \(\delta_i=0\), the hazard factor is raised to the zero power, leaving only \(\exp[-H_i(y_i;\theta)]\).

This construction assumes that the event-time distribution is described by the stated hazard model and that censoring is independent of the event time, at least conditional on variables included in the model. Under this independent-censoring assumption, the censoring process does not supply additional information about \(\theta\) that must be included in the event-time likelihood. If censoring is informative after conditioning on the modeled variables, this likelihood alone is generally insufficient; the censoring mechanism may need to be modeled.

For independent individuals, the joint likelihood is the product of the individual contributions. Taking logarithms turns the product into a sum:

This is the usual event-time log-likelihood for exact events and right-censored observations. The cumulative-hazard term appears for every individual, because everyone is known to have remained event-free up to the observed time. The log-hazard term appears only for an observed event. A PK/PD model can enter through \(h_i\), including through an individual’s exposure trajectory; the likelihood form itself does not depend on the particular parameterization.

Log-Likelihoods for Parametric Hazards

The general log-likelihood is especially convenient when the cumulative hazard has a tractable expression or can be evaluated numerically. For a constant hazard \(h_i(t;\theta)=\lambda\), where \(\lambda>0\), the cumulative hazard is \(H_i(y_i)=\lambda y_i\). The individual contribution becomes \(\lambda^{\delta_i}\exp(-\lambda y_i)\), and the full log-likelihood is

The first sum counts observed events; the second includes the observed duration for every event and censored individual. This form illustrates why censored individuals still contribute: their duration adds evidence that no event occurred before follow-up ended.

With a time-varying hazard, the same likelihood applies, but the cumulative hazard must integrate the hazard over time. If the hazard is generated from a PK/PD model and changes as exposure changes, the integration must use the relevant exposure history for each individual. For an event, the hazard is evaluated at the event time, while the cumulative hazard accounts for the entire preceding interval.

Justification, Checks, and Common Failure Modes

The density relationship used above follows directly from the survival function. For a continuous event time, \(S(t)=P(T>t)\), and the event-time density is \(f(t)=-dS(t)/dt\). Since \(S(t)=\exp[-H(t)]\) and \(dH(t)/dt=h(t)\), differentiation gives

This derivation requires a continuous event-time distribution with a well-defined hazard and cumulative hazard over the interval. A model with point masses or a different observation scheme needs a likelihood appropriate to that structure; the continuous exact-event density should not be applied mechanically.

Several checks help identify errors in implementation. The hazard should be nonnegative, and its integral should be nondecreasing and nonnegative. For each individual, numerical integration should correspond to the observed exposure history and run from the time origin to \(y_i\). The log-likelihood should contain one \(-H_i(y_i;\theta)\) term per individual, and \(\log h_i(y_i;\theta)\) only for events. If an event contribution is reported as a probability rather than a density, check whether the hazard’s inverse-time units have been overlooked.

A common mistake is to regard a right-censored time as if an event occurred at the censoring time, thereby adding \(\log h_i(y_i;\theta)\). That reasoning confuses “known to be event-free through \(y_i\)” with “event observed at \(y_i\).” The correction is to use the survival contribution \(\exp[-H_i(y_i;\theta)]\) for censoring and reserve the hazard factor for observed events.

Another useful comparison is with a nonparametric survival estimate: it can describe survival without specifying the same parametric hazard form, but it does not provide this parametric likelihood for estimating a specified PK/PD hazard model. The parametric likelihood uses the chosen hazard model to combine event times, censoring times, and exposure histories. Its advantage is a direct route to parameter estimation and model-based prediction; its limitation is dependence on the adequacy of the assumed hazard and exposure-history specification. A well-calculated likelihood cannot compensate for an implausible or misspecified hazard.