Right-Censored Likelihood Contributions for Cure Models

Why censoring requires a mixture contribution

In a mixture cure model, some individuals are assumed never to experience the event of interest, while others are susceptible and have a finite event time governed by a latency distribution. In practice, cure status is usually not observed. A person who remains event-free until the end of follow-up may be cured, or may simply be susceptible with an event time later than the observed follow-up. The likelihood must represent both possibilities rather than assign the person to either group.

The introductory material on mixture cure models defines the cure fraction and the susceptible-group latency distribution. Here the focus is narrower: how to form the likelihood for each observed time when event status is observed only up to right censoring. The key distinction is that an observed event rules out cure under the model, whereas a censored observation is compatible with both latent groups.

Definition: Let $Z=0$ indicate that an individual is cured and $Z=1$ indicate that the individual is susceptible. Write $P(Z=0)=\pi$, so $P(Z=1)=1-\pi$, where $0\leq\pi\leq1$. A cured individual has event time $T=\infty$. Conditional on susceptibility, $T$ has survival function $S_u(t)=P(T>t\mid Z=1)$ and density $f_u(t)$. The marginal survival function is $S(t)=\pi+(1-\pi)S_u(t)$, and the marginal event-time density at finite $t$ is $f(t)=(1-\pi)f_u(t)$.

For an individual, let $C$ be the censoring time, $Y=\min(T,C)$ the observed time, and $\delta=I(T\leq C)$ the event indicator. Thus $\delta=1$ means an event was observed at time $Y$, while $\delta=0$ means the individual was still event-free at censoring time $Y$. We assume censoring is independent of the event time, conditional on any covariates included in the model, and that the event and censoring mechanisms have distinct parameters. These assumptions allow the censoring-process factors to be omitted when making likelihood-based inferences about the cure and latency parameters.

Deriving the individual contributions

If an event is observed at $Y=y$, the individual cannot be cured under the model. The event must come from the susceptible group. Its probability density is therefore the probability of susceptibility multiplied by the susceptible-group event-time density. If the observation is censored at $y$, all that is known about the event time is $T>y$. A cured individual satisfies this condition, and a susceptible individual satisfies it with probability $S_u(y)$. The latent possibilities must be added.

Result: Omitting factors that depend only on the censoring distribution, the observed-data likelihood contribution for one individual is $$ L_i(\pi,S_u,f_u) = \begin{cases} (1-\pi)f_u(y_i), & \delta_i=1,\\ \pi+(1-\pi)S_u(y_i), & \delta_i=0. \end{cases} $$ Equivalently, $L_i=\{(1-\pi)f_u(y_i)\}^{\delta_i}\{\pi+(1-\pi)S_u(y_i)\}^{1-\delta_i}$.

The derivation can be made explicit by summing over the unobserved $Z$. For an event, the contribution from $Z=0$ is zero, because a cured individual has no finite event time; the contribution from $Z=1$ is $(1-\pi)f_u(y)$. For censoring, the conditional probability of remaining event-free through $y$ is one for $Z=0$ and $S_u(y)$ for $Z=1$. Weighting each conditional contribution by its group probability gives the censoring expression.

L(y,\delta) =\sum_{z\in\{0,1\}}P(Z=z)\,L(y,\delta\mid Z=z)

In particular, for $\delta=0$,

L(y,0)=\pi\cdot 1+(1-\pi)S_u(y).

For $\delta=1$, the corresponding sum is $\pi\cdot0+(1-\pi)f_u(y)$. This is the observed-data likelihood: it sums over the unknown cure status instead of pretending that status was measured. For independent individuals, the likelihood for the cure and latency parameters is the product of their individual contributions. Event contributions are densities and therefore have units inverse to time; censoring contributions are probabilities and are dimensionless. The product can consequently carry units when there are observed events, as is usual for a likelihood involving continuous event times.

Worked Example: Contribution From an Observed Event

Suppose, hypothetically, that the cure fraction is $\pi=0.30$, and susceptible event times follow an exponential distribution with rate $\lambda=0.50$ per year. Thus $S_u(t)=e^{-\lambda t}$ and $f_u(t)=\lambda e^{-\lambda t}$. An individual has an event at $y=2$ years. The exponential model is assumed to describe the susceptible group, and censoring is assumed independent of event time.

Because an event was observed, this individual must be susceptible under the cure-model assumptions. The likelihood contribution is

L(y,1)=(1-\pi)f_u(y) =(1-0.30)(0.50)e^{-(0.50)(2)} =0.35e^{-1} \approx 0.1288\ \text{per year}.

The factor $1-\pi=0.70$ accounts for the probability of belonging to the susceptible group; $f_u(2)=0.50e^{-1}$ per year is the susceptible event-time density at two years. This contribution is a density, not the probability of an event occurring at exactly two years.

Worked Example: Contribution From a Censored Observation

Use the same hypothetical model, with $\pi=0.30$ and $\lambda=0.50$ per year. A second individual is censored event-free at $y=3$ years. Under independent censoring, the observation tells us only that the event time exceeds three years.

A cured individual has probability one of remaining event-free through three years. A susceptible individual has probability $S_u(3)=e^{-1.5}$. Therefore the contribution is

L(3,0)=\pi+(1-\pi)S_u(3) =0.30+0.70e^{-1.5} =0.30+0.70(0.2231) \approx 0.4562.

This is a probability and has no time units. The observed censoring is compatible with cure, but does not prove cure. For example, within this model the posterior probability of cure for an individual censored at three years is the cured-group part of the contribution divided by the total:

P(Z=0\mid T>3) =\frac{\pi}{\pi+(1-\pi)S_u(3)} =\frac{0.30}{0.4562} \approx 0.6576.

Thus the model assigns about a 65.8% probability of cure for this hypothetical censored individual, conditional on remaining event-free through three years. This posterior probability depends on the specified model and parameter values; it is not directly observed in the data.

From individual terms to a sample likelihood

Let $n$ independent individuals have observations $(y_i,\delta_i)$, for $i=1,\ldots,n$. Multiplication of the individual contributions gives

L(\pi,\text{latency parameters}) =\prod_{i=1}^{n} \left\{(1-\pi)f_u(y_i)\right\}^{\delta_i} \left\{\pi+(1-\pi)S_u(y_i)\right\}^{1-\delta_i}.

The log-likelihood is the sum of the logarithms of these contributions. An observed event contributes $\log(1-\pi)+\log f_u(y_i)$; a censored observation contributes $\log\{\pi+(1-\pi)S_u(y_i)\}$. The two terms have different forms because an event identifies susceptibility, while censoring leaves the latent group unresolved. These formulas specify the likelihood; they do not by themselves prescribe a particular estimation algorithm.

Worked Example: Likelihood for Three Hypothetical Individuals

Consider three independent hypothetical individuals under an exponential susceptible latency model with $\pi=0.25$ and $\lambda=0.40$ per year. Suppose the records are: an event at one year, censoring at two years, and an event at three years. Assume the censoring mechanism is independent and is not being modeled for inference on $\pi$ and $\lambda$.

For the event at one year,

L_1=(1-\pi)\lambda e^{-\lambda(1)} =0.75(0.40)e^{-0.40} \approx 0.2011\ \text{per year}.

For the individual censored at two years,

L_2=\pi+(1-\pi)e^{-\lambda(2)} =0.25+0.75e^{-0.80} \approx 0.5870.

For the event at three years,

L_3=(1-\pi)\lambda e^{-\lambda(3)} =0.75(0.40)e^{-1.20} \approx 0.09036\ \text{per year}.

The sample likelihood contribution, omitting censoring-process factors, is

L=L_1L_2L_3 \approx (0.2011)(0.5870)(0.09036) \approx 0.01067\ \text{per year}^2.

There are two observed event times, so the product has units inverse years squared. The censored observation contributes a dimensionless probability, not a density. The corresponding log-likelihood is approximately $\log(0.01067)=-4.541$, with the conventional understanding that the log is applied to a likelihood expressed using the chosen time unit.

Assumptions, alternatives, and common errors

The contribution formulas rely on a particular model structure. Cure means that the event never occurs, not merely that it occurs after the end of study. Among susceptible individuals, $S_u(t)$ and $f_u(t)$ must describe a valid event-time distribution, with $f_u(t)$ the density corresponding to $S_u(t)$. The cure probability and latency distribution may vary with covariates in richer models, but the same individual contribution applies after evaluating the appropriate individual-specific quantities. The censoring mechanism must be independent of event time conditional on modeled covariates for the event-time likelihood to be used without jointly modeling informative censoring.

To see what is omitted under independent censoring, let $S_C$ and $f_C$ denote the censoring-time survival function and density. The joint observed-data contribution factors into an event-time part and a censoring part:

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

Here $f_T(y)=(1-\pi)f_u(y)$ and $S_T(y)=\pi+(1-\pi)S_u(y)$. If the censoring parameters are distinct from the cure-model parameters, the second bracket does not affect likelihood inference on the latter and can be omitted. If censoring is informative after conditioning on modeled covariates, that separation is not justified; a censoring model or another appropriate analysis is needed.

The ordinary right-censored survival likelihood has the familiar structure $f_T(y)^{\delta}S_T(y)^{1-\delta}$. A mixture cure model does not change that observation scheme. Instead, it supplies the marginal cure-model density and survival function, which produce the event and censoring terms above. A latent-variable or complete-data formulation may instead write separate contributions conditional on $Z$, but if $Z$ is unobserved, the observed likelihood must sum over its possible values. Treating the unknown $Z$ as if it were observed gives a different likelihood.

A common mistake is to say, “This person was censored, so the person must be cured,” and to use $\pi$ as the censored contribution. That discards susceptible individuals whose event times exceed their censoring times. The correct contribution is $\pi+(1-\pi)S_u(y)$: the cured probability plus the susceptible probability of surviving event-free to $y$. Another mistake is to use $S_u(y)$ alone, which assumes the censored person is known to be susceptible and omits the cured group. Neither shortcut represents the observed data under a mixture cure model.

Useful checks include verifying that the event indicator is coded consistently, that event densities are evaluated at event times rather than censoring times, and that each censored term is a mixture survival probability. A censoring contribution should lie between zero and one. An event contribution is a density and need not be less than one, but must be nonnegative and have units inverse to the time unit. Numerical underflow in very small density or survival terms is a computational concern; summing log contributions is generally more stable than multiplying many small terms. If the fitted model assigns essentially zero density to observed event times, or gives implausible survival probabilities for censored times, the assumed latency distribution, covariate specification, data coding, or censoring assumptions warrant review.

Key takeaway: An event contributes $(1-\pi)f_u(y)$, because it establishes susceptibility. A right-censored time contributes $\pi+(1-\pi)S_u(y)$, because both cure and sufficiently long susceptible survival remain possible.

Check Your Understanding

Answer each question using the mixture cure likelihood contributions, then compare with the solutions.

In a model with cure fraction $\pi$, what is the contribution for an event at time $y$, and why is there no cured-group term?
For a censored individual, explain why the contribution is not simply $\pi$.
Suppose $\pi=0.20$, $S_u(y)=0.50$, and an individual is censored at $y$. Calculate the contribution and the posterior probability of cure conditional on being event-free through $y$.
For an exponential susceptible latency distribution with rate $\lambda$, write the censored contribution at time $y$.
State the role of independent censoring in using the event-time likelihood without a censoring model.

Solutions. (1) The contribution is $(1-\pi)f_u(y)$. A cured individual cannot have a finite event, so its event contribution is zero. (2) Censoring is also possible for a susceptible person whose event time exceeds $y$, so the full term is $\pi+(1-\pi)S_u(y)$. (3) The contribution is $0.20+0.80(0.50)=0.60$. The posterior cure probability is $0.20/0.60=1/3$, or about 0.333. (4) Since $S_u(y)=e^{-\lambda y}$, the censored contribution is $\pi+(1-\pi)e^{-\lambda y}$. (5) Conditional independence permits the joint likelihood to factor into event-time and censoring-process terms; when censoring parameters are distinct, the censoring-only factor can be omitted for inference on cure and latency parameters.

Right-Censored Likelihood Contributions for Cure Models

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