Likelihoods for Time-to-Event NLME Data

From Event Times to Likelihood Contributions

Time-to-event data record when an event occurs, or how long an individual is followed without an observed event. The likelihood must distinguish an observed event time from a follow-up time at which observation ended before the event. These records contribute different information: an exact event contributes a density, while a right-censored record contributes the probability of remaining event-free up to the censoring time.

In a nonlinear mixed-effects setting, an individual’s event-time distribution may depend on an individual random effect. The likelihood for that individual is then obtained by averaging the conditional contribution over the random-effects distribution. Whether the censoring process can be left out of that averaging depends on how censoring relates to the random effect, not merely on whether event time and censoring time are independent after conditioning on it.

Definition: Let \(T_i\) be the event time for individual \(i\), \(C_i\) the censoring time, \(Y_i=\min(T_i,C_i)\) the observed time, and \(\delta_i=I(T_i\leq C_i)\) the event indicator. Thus \(\delta_i=1\) denotes an observed event and \(\delta_i=0\) denotes right censoring. Conditional on observed covariates \(x_i\) and individual random effect \(b_i\), write the event-time survival function as \(S_i(t\mid b_i)\), hazard as \(h_i(t\mid b_i)\), cumulative hazard as \(H_i(t\mid b_i)\), and density as \(f_i(t\mid b_i)\).

For a continuous event-time distribution, the hazard is the instantaneous event rate among individuals still event-free just before time \(t\). The hazard has units of inverse time; the survival function is a probability and is dimensionless. Given the hazard, the cumulative hazard and survival function are

H_i(t\mid b_i)=\int_0^t h_i(u\mid b_i)\,du, \qquad S_i(t\mid b_i)=\exp\{-H_i(t\mid b_i)\}.

The event-time density is related to the hazard and survival by \(f_i(t\mid b_i)=h_i(t\mid b_i)S_i(t\mid b_i)\). An exact event therefore contributes a density, not a hazard alone. A censored observation contributes the survival probability through its observed follow-up time.

Conditional event-time contribution: For an exact event or a right-censored observation at time \(Y_i\), the event-time contribution conditional on \(b_i\) is

L_{T,i}(b_i) = \left[f_i(Y_i\mid b_i)\right]^{\delta_i} \left[S_i(Y_i\mid b_i)\right]^{1-\delta_i} = \left[h_i(Y_i\mid b_i)\right]^{\delta_i} \exp\{-H_i(Y_i\mid b_i)\}.

When \(\delta_i=1\), this is \(f_i(Y_i\mid b_i)=h_i(Y_i\mid b_i)\exp\{-H_i(Y_i\mid b_i)\}\). When \(\delta_i=0\), it is \(S_i(Y_i\mid b_i)=\exp\{-H_i(Y_i\mid b_i)\}\). A density has units of inverse time, whereas a survival contribution is dimensionless. This difference is appropriate: the likelihood combines densities for exact observations with probabilities for censored observations.

Why the Survival Formula Holds

For a continuous event time, \(S_i(t\mid b_i)=P(T_i>t\mid b_i)\), and its rate of decrease is the negative event density: \(\partial S_i(t\mid b_i)/\partial t=-f_i(t\mid b_i)\). By the hazard definition, \(h_i(t\mid b_i)=f_i(t\mid b_i)/S_i(t\mid b_i)\). Substitution gives the differential equation

\frac{\partial S_i(t\mid b_i)}{\partial t} =-h_i(t\mid b_i)S_i(t\mid b_i).

Since \(S_i(0\mid b_i)=1\), divide by \(S_i(t\mid b_i)\) and integrate from 0 to \(t\):

\int_1^{S_i(t\mid b_i)}\frac{dS}{S} = -\int_0^t h_i(u\mid b_i)\,du, \qquad \log S_i(t\mid b_i)=-H_i(t\mid b_i).

Exponentiating yields \(S_i(t\mid b_i)=\exp\{-H_i(t\mid b_i)\}\); multiplying by the hazard gives the density. This derivation assumes a continuous event-time distribution with a well-defined hazard and no point mass at the times under consideration.

Worked Example: Exact Event and Right Censoring Under a Constant Hazard

Suppose a hypothetical model has constant event hazard \(h(t)=0.20\) per day. One individual has an observed event on day 4; another is event-free when follow-up ends on day 4. Assume no random effects for this calculation.

For this hazard, \(H(4)=0.20(4)=0.80\) and \(S(4)=\exp(-0.80)=0.4493\). The exact-event contribution is the density \(f(4)=h(4)S(4)=0.20\exp(-0.80)=0.08987\) per day. The censored individual’s contribution is the survival probability \(S(4)=0.4493\), with no inverse-time units.

The two contributions are not interchangeable. Treating the censored record as an event would assign it the density \(0.08987\) per day and claim an event occurred at day 4. Treating the exact event as censored would instead discard the information that the event occurred at that time.

Introducing Individual Random Effects

Let \(b_i\) have population density \(g(b_i\mid\Omega)\), where \(\Omega\) denotes random-effects distribution parameters, and let \(\theta\) denote event-model parameters. Conditional on \(b_i\), the hazard and hence the event-time contribution may differ among individuals. Because \(b_i\) is unobserved, the individual’s event-time likelihood is the average of its conditional contribution over the random-effects distribution:

L_i(\theta,\Omega) = \int L_{T,i}(b_i;\theta)\,g(b_i\mid\Omega)\,db_i.

This integration is performed for each individual before multiplying contributions across independent individuals. Conditional independence of individuals given their random effects and population parameters leads to the marginal likelihood

L(\theta,\Omega)=\prod_{i=1}^{N}L_i(\theta,\Omega).

In general, the average of a product is not the product of averages. If several conditionally independent observations for one individual are part of a model, their conditional contributions must first be multiplied at the same \(b_i\), and that product is then integrated over \(g\). For a single terminal event record per individual, the formula above uses that individual’s event or censoring contribution.

Worked Example: Marginal Contributions With a Two-Point Random-Effects Distribution

Consider a hypothetical exponential event-time model with conditional hazard \(h_i(t\mid b_i)=0.20\exp(b_i)\) per day. Suppose \(b_i\) takes values \(-\log 2\) and \(\log 2\), each with probability \(1/2\). The corresponding hazards are \(0.10\) and \(0.40\) per day. One independent individual has an event at day 3; a second has follow-up censored at day 3.

For the event, the conditional density at day 3 is \(\lambda\exp(-3\lambda)\). At the low hazard it is \(0.10\exp(-0.30)=0.07408\) per day; at the high hazard it is \(0.40\exp(-1.20)=0.1205\) per day. Averaging with the two random-effects probabilities gives

L_{\text{event}} =\tfrac12(0.0740818)+\tfrac12(0.1204777) =0.09728\ \text{day}^{-1}.

For the censored individual, the conditional contributions are survival probabilities. They are \(\exp(-0.30)=0.7408\) and \(\exp(-1.20)=0.3012\). Thus

L_{\text{censored}} =\tfrac12(0.7408182)+\tfrac12(0.3011942) =0.5210.

The individuals have independent random effects, so the sample contribution is the product of the two marginal contributions: \(0.0972798(0.5210062)=0.05068\) per day. Averaging each individual’s contribution over the random-effects distribution is what accounts for the unobserved heterogeneity in this likelihood.

When Censoring Can Be Omitted

The event-time contribution alone is justified only under an appropriate censoring assumption. A clear sufficient condition is that, conditional on observed covariates, the censoring time is independent of both the event time and the individual random effect: \(C_i\mathrel{\perp}(T_i,b_i)\mid x_i\). In addition, the censoring mechanism must not share unknown parameters with the event model if its likelihood factor is to be omitted for inference about the event-model parameters. A censoring time fixed by the study design can also be conditioned upon when that design is noninformative for event time and random effects.

To see the issue, conditional independence of \(T_i\) and \(C_i\) given \(b_i\) is not sufficient by itself. For a continuous censoring time with density \(f_C\) and survival \(S_C\), the conditional joint contribution to the observed data is

\left[f_T(Y_i\mid b_i)S_C(Y_i\mid b_i)\right]^{\delta_i} \left[S_T(Y_i\mid b_i)f_C(Y_i\mid b_i)\right]^{1-\delta_i}.

Even if \(T_i\) and \(C_i\) are independent conditional on \(b_i\), the censoring factor may depend on \(b_i\). It then remains inside the integral over \(b_i\), and the observed censoring time can provide information about that random effect. If censoring depends on the random effect or is otherwise informative, use a joint likelihood for event and censoring processes, integrating their joint contribution over the random-effects distribution, rather than silently dropping the censoring factor.

Worked Example: Censoring That Depends on the Random Effect

Suppose a hypothetical individual has an event hazard \(\lambda=0.10\) per day when \(b\) is low and \(\lambda=0.40\) per day when \(b\) is high, with prior probabilities \(1/2\) each. Now suppose the censoring hazard is \(\mu=0.05\) per day for the low-\(b\) group and \(\mu=0.20\) per day for the high-\(b\) group. The observation is right-censored on day 3. Assume event and censoring times are independent conditional on \(b\), and both hazards are constant.

For a censored observation, the joint density of censoring at day 3 while the event has not yet occurred is \(S_T(3\mid b)f_C(3\mid b)=\exp\{-3(\lambda+\mu)\}\mu\). At low \(b\), this is \(0.05\exp(-0.45)=0.03188\) per day. At high \(b\), it is \(0.20\exp(-1.80)=0.03306\) per day. Integrating the joint contribution gives

L_{\text{joint,censored}} =\tfrac12(0.0318814)+\tfrac12(0.0330598) =0.03247\ \text{day}^{-1}.

The event-only marginal survival at day 3 would be \(\tfrac12\exp(-0.30)+\tfrac12\exp(-1.20)=0.5210\), but that is not the full joint likelihood for this observation. Censoring is more likely under the high-\(b\) state, so observing its time changes the information about \(b\). Integrating only the event survival would omit that information.

Practical Checks and Limitations

Before constructing an event-time likelihood, verify that each record’s time and event indicator have the intended meanings, and that the conditional hazard is nonnegative and yields a valid survival function. Confirm that the density for an event includes both the hazard at the observed time and survival up to that time. For censored records, confirm that the contribution is survival through the censoring time. Check time units: if time is changed from days to hours, hazards and densities must be expressed in corresponding units.

Numerical integration over random effects can be challenging when the conditional contribution varies sharply with \(b_i\), when the random-effects distribution is poorly represented, or when a small data set provides little information about its parameters. Compare integration accuracy across suitable numerical settings and inspect whether estimates or likelihood values change materially. A finite quadrature approximation is a numerical method, not a different likelihood definition.

Common alternatives answer different questions or make different assumptions. A conditional likelihood evaluates contributions at specified random effects, but those effects are generally unobserved; the population likelihood integrates them out. A full joint event-and-censoring likelihood models both processes and is needed when censoring is informative or depends on random effects. Simply deleting censored individuals or imputing their censoring times as event times does not represent the observed-data likelihood. The event-time contribution also depends on the chosen hazard model and random-effects distribution; a correctly written likelihood does not guarantee those choices are suitable.

Common pitfall: “Event time and censoring time are independent given the random effect, so I can always integrate only the event contribution.” This is not generally valid: the censoring contribution can still depend on the random effect and therefore affect its posterior weighting. Either justify an ignorability condition that lets the censoring factor be omitted, or include the censoring process in the joint likelihood and integrate the joint contribution.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

At time \(t\), what is the conditional likelihood contribution for an exact event, and what is it for a right-censored record?
For a constant hazard of \(0.05\) per hour, calculate the event-time contribution for an exact event at 2 hours and the contribution for censoring at 2 hours.
If \(b_i\) is unobserved, in what order should you combine the conditional contribution and the random-effects density?
Why does conditional independence of event and censoring times given \(b_i\) not always justify omitting the censoring process?
Name one assumption under which the censoring mechanism can be omitted from the event-time likelihood.

Solutions. (1) The exact-event contribution is \(f_i(t\mid b_i)=h_i(t\mid b_i)S_i(t\mid b_i)\); the censored contribution is \(S_i(t\mid b_i)\). (2) \(H(2)=0.05(2)=0.10\), so the event density is \(0.05\exp(-0.10)=0.04524\) per hour, while the censoring contribution is \(S(2)=\exp(-0.10)=0.9048\). (3) Multiply the conditional contribution by \(g(b_i\mid\Omega)\), integrate over \(b_i\), then multiply the resulting individual marginal contributions across independent individuals. (4) The censoring factor may depend on \(b_i\), so the censoring observation can inform that effect and must then remain in the joint integral. (5) For example, censoring may be independent of both event time and random effect conditional on observed covariates, with no shared unknown parameters that require modeling the censoring factor.