Semiparametric Proportional Hazards Mixture Cure Models

Model Structure and Notation

A mixture cure model separates two questions: whether an individual is susceptible to the event during the study’s event process, and, if susceptible, how their event time is distributed. The semiparametric proportional hazards version leaves the susceptible group’s baseline hazard unspecified while imposing proportional hazards for its covariate effects. This avoids choosing a particular latency distribution, but it does not remove the need to model who is susceptible.

Definition: Let $U_i=1$ indicate that individual $i$ is susceptible and $U_i=0$ indicate that the individual is cured with respect to the event under study. Write $p_i=P(U_i=1\mid z_i)$ for the susceptibility probability, modelled by $$ \operatorname{logit}(p_i)=\gamma^\top z_i. $$ The cure probability is $1-p_i$. Conditional on $U_i=1$, the latency time $T_i^*$ follows a Cox proportional hazards model: $$ \lambda_u(t\mid x_i,U_i=1)=\lambda_0(t)\exp(\beta^\top x_i), \qquad S_u(t\mid x_i)=\exp\{-\Lambda_0(t)\exp(\beta^\top x_i)\}. $$ Here $\lambda_0(t)$ is an unspecified baseline hazard and $\Lambda_0(t)=\int_0^t\lambda_0(s)\,ds$ is its cumulative hazard. For a cured individual, set $T_i^*=\infty$.

The incidence covariates $z_i$ and latency covariates $x_i$ may overlap, but they play different roles. A positive coefficient in $\gamma$ increases the odds of susceptibility, whereas a positive coefficient in $\beta$ increases the instantaneous event hazard among susceptible individuals. Neither coefficient is automatically a causal effect.

With right censoring, observe $Y_i=\min(T_i^*,C_i)$ and $\delta_i=I(T_i^*\leq C_i)$, where $C_i$ is the censoring time. An observed event establishes $U_i=1$. A censored observation does not: the person may be cured or may be susceptible and event-free up to $Y_i$. That unobserved membership is the central reason estimation is more involved than an ordinary Cox fit.

Observed-Data Likelihood

Assume individuals are independent, the incidence and latency models are correctly specified, and censoring is independent of the event time conditional on the model covariates. For clarity, the likelihood below omits the censoring distribution, which is treated as noninformative under that assumption. Write $S_{u,i}(t)=S_u(t\mid x_i)$ and $\lambda_{u,i}(t)=\lambda_0(t)\exp(\beta^\top x_i)$.

Likelihood contribution: An observed event at time $Y_i$ contributes $p_i\lambda_{u,i}(Y_i)S_{u,i}(Y_i)$. A right-censored observation at $Y_i$ contributes $(1-p_i)+p_iS_{u,i}(Y_i)$. Thus, apart from the censoring mechanism, $$ L_i= \{p_i\lambda_{u,i}(Y_i)S_{u,i}(Y_i)\}^{\delta_i} \{(1-p_i)+p_iS_{u,i}(Y_i)\}^{1-\delta_i}. $$

The event contribution requires susceptibility and a latency event at $Y_i$. The censored contribution adds two mutually exclusive possibilities: cured, with probability $1-p_i$, or susceptible and event-free through $Y_i$, with probability $p_iS_{u,i}(Y_i)$. In the semiparametric model, $\Lambda_0$ is represented through its jumps at observed event times and estimated rather than assigned a parametric distribution.

Worked Example: Event and Censoring Contributions

Suppose a hypothetical individual has susceptibility probability $p_i=0.70$. At an observed time of two years, suppose $\Lambda_0(2)=0.20$, $\exp(\beta^\top x_i)=1.5$, and the baseline hazard at that time is $0.08$ per year. Then $$ S_{u,i}(2)=\exp\{-0.20(1.5)\}=\exp(-0.30)\approx0.7408, \qquad \lambda_{u,i}(2)=0.08(1.5)=0.12\text{ per year}. $$

If an event is observed at two years, its likelihood contribution is $$ p_i\lambda_{u,i}(2)S_{u,i}(2) =0.70(0.12)(0.7408)\approx0.06223\text{ per year}. $$ The density contribution has units of inverse time, as an event-time density should.

If instead the individual is censored at two years, the contribution is $$ (1-p_i)+p_iS_{u,i}(2) =0.30+0.70(0.7408) =0.8186. $$ This is a probability of remaining without an observed event through two years under the mixture model; it includes both cured individuals and susceptible individuals whose event has not yet occurred.

Why the Partial-Likelihood Structure Is Different

In an ordinary Cox model, event status identifies the event-time risk sets, and the standard partial likelihood conditions on the observed event times. In a cure model, the susceptibility status of censored individuals is latent. Their contribution to a latency risk set is therefore uncertain. Applying an ordinary Cox partial likelihood to all observed individuals as if they were susceptible incorrectly treats cured people as members of the latency population.

A common semiparametric strategy uses the full observed-data likelihood and an expectation-maximization (EM) algorithm. In the E-step, calculate each censored individual’s current probability of being susceptible given their covariates and censoring history. If the current incidence and latency estimates are $p_i$ and $S_{u,i}(Y_i)$, the posterior susceptibility weight is $$ w_i=P(U_i=1\mid Y_i,\delta_i=0,x_i,z_i) =\frac{p_iS_{u,i}(Y_i)}{(1-p_i)+p_iS_{u,i}(Y_i)}. $$ For an observed event, $w_i=1$, because an event cannot occur for a cured individual. For a censored individual, $w_i$ lies between zero and one.

Given these weights, a Cox-type latency update uses weighted risk sets. To see the structure, let $t_j$ denote a distinct event time, $d_j$ the number of observed events at that time, and $R(t_j)=\{i:Y_i\geq t_j\}$ the observed risk set. In an EM iteration, treat the current $w_i$ as fixed. The expected complete-data latency log-likelihood contains event terms $\sum_i\delta_i\beta^\top x_i$, baseline jump terms $d_j\log(\Delta\Lambda_0(t_j))$, and cumulative-hazard terms $-\Delta\Lambda_0(t_j)\sum_{i\in R(t_j)}w_i\exp(\beta^\top x_i)$. Differentiating with respect to the baseline jump and setting the derivative to zero gives the weighted Breslow update:

\widehat{\Delta\Lambda}_0(t_j) =\frac{d_j}{\displaystyle\sum_{i\in R(t_j)}w_i\exp(\beta^\top x_i)}.

Substituting this profiled jump into the expected complete-data criterion produces a Cox-type weighted risk-set term. The event individuals have weight one; censored individuals enter the denominator according to their current expected susceptibility. This is an EM update, not a claim that the ordinary Cox partial likelihood is the observed-data likelihood. The incidence component is updated using the same latent-membership weights in its expected logistic log-likelihood.

Worked Example: A Weighted Risk Set in an EM Update

At a hypothetical event time, suppose two individuals remain at risk. Individual A has the observed event, so $w_A=1$ and $\exp(\beta^\top x_A)=1$. Individual B is censored later and currently has $p_B=0.50$ and $S_{u,B}(Y_B)=0.60$. Its posterior susceptibility weight is $$ w_B=\frac{0.50(0.60)}{0.50+0.50(0.60)} =\frac{0.30}{0.80}=0.375. $$ Suppose $\exp(\beta^\top x_B)=2$. For one event at this time, the weighted risk-set denominator is $$ 1(1)+0.375(2)=1.75, $$ so the weighted Breslow baseline-hazard increment is $$ \widehat{\Delta\Lambda}_0(t_j)=\frac{1}{1.75}=0.5714. $$ The corresponding Cox-type event term for A has numerator $\exp(\beta^\top x_A)=1$ and denominator $1.75$, or $1/1.75$. B contributes less than a fully susceptible individual with the same latency covariate because its current posterior probability of susceptibility is only 0.375. These are intermediate EM quantities; subsequent iterations update the weights as the model estimates change.

Interpreting the Two Components

The incidence regression is a logistic model for susceptibility, not for the event hazard. For a one-unit increase in $z_k$, holding the other incidence covariates fixed, $\exp(\gamma_k)$ is the multiplicative change in the odds of susceptibility. In the latency model, $\exp(\beta_k)$ is the hazard ratio among susceptible individuals for a one-unit increase in $x_k$, holding other latency covariates fixed. The hazard ratio is assumed constant over time; it does not compare the marginal hazards of the full population, which also depend on the cure fraction.

Worked Example: Marginal Survival With Corrected Arithmetic

Suppose a hypothetical person has incidence linear predictor $0.40$, so the susceptibility probability is $$ p=\frac{\exp(0.40)}{1+\exp(0.40)}\approx0.5987. $$ For this person, suppose the latency linear predictor is $0.50$ and the baseline cumulative hazard at two years is $0.30$. The susceptible survival probability is $$ S_u(2\mid x)=\exp\{-0.30\exp(0.50)\} =\exp(-0.4946)\approx0.6098. $$

Marginal survival is the probability of being cured plus the probability of being susceptible and still event-free: $$ S(2\mid x,z)=1-p+pS_u(2\mid x) \approx1-0.5987+0.5987(0.6098) \approx0.7664. $$ Thus the model estimates a 0.7664 probability of remaining event-free through two years for this covariate profile, or a 0.2336 probability of an event by two years. These are model-based probabilities, not a statement that the individual’s cure status is known. The arithmetic uses the displayed rounded values; the unrounded model values give approximately the same result.

Assumptions, Diagnostics, and Limits

The model’s interpretation depends on assumptions that concern both components and their connection. In particular:

Proportional hazards among susceptible individuals: the latency hazard ratio for a covariate is constant over time. A covariate that changes its latency effect over time can make the Cox component inadequate.
Correct incidence specification: the logistic model must describe susceptibility adequately, including its covariate form and relevant interactions. A poor incidence model can be mistaken for a poor latency fit.
Noninformative censoring conditional on covariates: censoring must not provide unmodelled information about event time or cure status. If this fails, the likelihood contributions above are not justified without additional modelling.
Enough follow-up to distinguish a cure fraction from long latency: a long period without events can arise from either cure or slow event times. The data need sufficient follow-up and event information to separate these explanations.
Appropriate covariate and sampling structure: the displayed likelihood assumes independent individuals. Dependence, such as clustering, requires an appropriate extension rather than treating all observations as independent.

Diagnostics should examine whether the latency proportional-hazards assumption is plausible among the susceptible population and whether the fitted model reproduces observed event and censoring patterns over time. These checks are complicated by latent membership: a raw residual analysis that treats every censored person as susceptible does not directly diagnose the fitted cure model. Sparse events, few late follow-up observations, or risk sets dominated by low posterior weights can also make the latency estimates unstable.

The semiparametric model occupies a middle ground. A parametric cure model specifies a latency distribution and can be more efficient if that distribution is credible, but it risks misspecification. The Cox latency component avoids that particular distributional choice while retaining proportional hazards and requiring adequate follow-up to identify the mixture. A standard Cox model is simpler, but it has no cure component and should not be substituted when a nonzero cured fraction is substantively plausible. The model also cannot determine cure status with certainty for a censored individual merely because the fitted posterior weight is small.

Common mistake: “Censored individuals have not had the event, so they are all susceptible members of the Cox risk set.” This is wrong because a censored individual may be cured. The correction is to account for latent susceptibility, for example through posterior weights in an EM procedure, and to distinguish that procedure’s weighted risk-set update from an ordinary Cox partial likelihood.

Check Your Understanding

Answer each question using the distinction between susceptibility and latency.

If $\gamma_k>0$, what does this imply about susceptibility odds? Does it imply a larger latency hazard?
A censored person has $p_i=0.40$ and $S_{u,i}(Y_i)=0.50$. Calculate their posterior susceptibility weight.
Why is an ordinary Cox partial likelihood that treats everyone as susceptible inappropriate for a mixture cure model?
State one assumption needed to interpret $\exp(\beta_k)$ as a constant latency hazard ratio.

Solutions. (1) The susceptibility odds increase by the factor $\exp(\gamma_k)$ per unit of $z_k$, holding other incidence covariates fixed; this does not by itself imply a larger latency hazard. (2) $w_i=0.40(0.50)/\{0.60+0.40(0.50)\}=0.20/0.80=0.250$. (3) Some censored individuals may be cured and therefore do not belong to the susceptible latency population; their risk-set contribution is uncertain and must account for latent membership. (4) The latency proportional-hazards model must hold among susceptible individuals, so the covariate hazard ratio is constant over time.