Maximum Likelihood for Parametric Survival Models

Why Censoring Changes the Likelihood

A parametric survival model describes the distribution of an event time \(T\), such as time from enrollment to disease progression. In many studies, the event is not observed for every participant: follow-up may end first, or a participant may withdraw. A censored observation still tells us something important. If a person is known to be event-free through month 8 and then leaves the study, the event time is not known to equal 8 months; it is known to exceed 8 months.

Maximum likelihood estimation uses that distinction directly. An observed event contributes a density at its observed time. A right-censored observation contributes the probability of surviving beyond its last observed time. Treating censored times as events instead changes the data being modeled and generally biases estimates.

The likelihood above is for the event-time parameters. It omits the censoring-process contribution under the usual independent-censoring assumption: the censoring mechanism does not depend on the event time, and its parameters are distinct from the event-time parameters of interest. If censoring is independent only after conditioning on measured covariates, that assumption and the event-time model should be stated conditionally on those covariates. If censoring is informative, the displayed likelihood alone may not give valid inference for the event-time distribution.

From Likelihood to Score Equations

Taking logarithms turns a product of contributions into a sum, which is easier to maximize. The log-likelihood and score vector are

An interior maximum-likelihood estimate \(\hat{\theta}\) ordinarily satisfies \(U(\hat{\theta})=0\). This is a necessary condition, not by itself proof of a maximum: one must also check the likelihood and parameter boundaries. For many parametric survival models, the score equations have no closed-form solution and are solved numerically. The observed information, \(J(\hat{\theta})=-\partial^2\ell(\theta)/\partial\theta\partial\theta^\mathsf{T}\) evaluated at \(\hat{\theta}\), can provide an approximate covariance matrix \(J(\hat{\theta})^{-1}\) under regularity conditions and adequate information in the data.

The different contributions are also visible in the score:

Thus a censored observation is not discarded and is not treated as an event. Its score contribution is the derivative of the log probability of remaining event-free through the censoring time.

A General Derivation and a Weibull Estimate

A useful special case illustrates how the score depends on the event indicators. For an exponential model, each event contributes \(\log\lambda-\lambda Y_i\), while each censored observation contributes \(-\lambda Y_i\). Summing gives

Equating the score to zero yields \(\hat{\lambda}=d/\sum_iY_i\). The numerator counts observed events; the denominator is total observed time at risk. This result follows from the censored-data likelihood, not from counting censored participants as event-free forever.

For a further example, parameterize a Weibull model by shape \(k>0\) and scale \(\lambda>0\): \(S(t)=\exp[-(t/\lambda)^k]\) and \(f(t)=\frac{k}{\lambda}(t/\lambda)^{k-1}\exp[-(t/\lambda)^k]\). If \(k\) is fixed, the log-likelihood terms involving \(\lambda\) are

Differentiating gives \(U(\lambda)=-dk/\lambda+k\sum_iY_i^k/\lambda^{k+1}\). An interior score root therefore satisfies \(\hat{\lambda}^k=\sum_iY_i^k/d\), or \(\hat{\lambda}=(\sum_iY_i^k/d)^{1/k}\). This closed form applies only when the shape \(k\) is fixed; estimating both shape and scale generally requires solving the joint score equations.

Reading Individual Contributions and Checking Assumptions

The density and survival contributions can differ substantially in size, as the next example shows. For the lognormal model, \(\log T\sim N(\mu,\sigma^2)\), where \(\sigma>0\). Its density is \(f(t)=\phi((\log t-\mu)/\sigma)/(t\sigma)\), and its survival function is \(S(t)=1-\Phi((\log t-\mu)/\sigma)\), with \(\phi\) and \(\Phi\) the standard normal density and distribution function.

The likelihood calculation is meaningful only if the assumed event-time model and censoring assumptions are defensible. In practice, examine whether the model’s implied survival pattern is plausible over the observed follow-up, whether the data contain enough events to estimate the parameters, and whether numerical optimization converges to a stable maximum. A nearly flat likelihood, estimates at parameter boundaries, large standard errors, or materially different estimates from different starting values are warnings to investigate. A model can converge successfully and still be a poor description of the event-time distribution.

The independent-censoring assumption also deserves substantive scrutiny. Administrative study closure is often plausibly independent of an individual’s event time, conditional on study entry and relevant design factors. Withdrawal because a participant’s health is worsening may be informative. If event risk and censoring remain associated after conditioning on observed covariates, the ordinary event-time likelihood may be inadequate. The likelihood also assumes the event-time distribution is correctly specified; precise estimation under a misspecified distribution does not make the model correct.

How This Likelihood Differs From Other Survival Approaches

Parametric maximum likelihood uses a specified distribution, such as exponential, Weibull, or lognormal, to describe event times. Its advantages include a compact set of parameters and the ability to estimate survival beyond observed event times, but that extrapolation depends on the model being suitable. The Kaplan–Meier estimator instead estimates survival nonparametrically over observed follow-up and does not impose one of these parametric distributions; it does not provide the same parametric extrapolation.

The Cox proportional hazards model uses a partial likelihood for regression coefficients rather than specifying a full parametric distribution for the baseline event-time distribution. It is a different modeling strategy, not a reason to change the event-versus-censoring contributions in a parametric likelihood. Choosing among parametric distributions requires attention to fit and scientific plausibility, not merely which model gives the largest likelihood when models have different structures or parameter counts.