Logistic Regression for Cure Probability

Why Model the Susceptible Probability?

A mixture cure model distinguishes individuals who are susceptible to the event of interest from those who will never experience that event. Susceptible individuals have an event-time distribution; individuals in the cured, or non-susceptible, group have no event time within the model. Here, “cure” is a statistical label for this latent non-susceptibility, not necessarily a claim of biological cure.

The incidence part of the model describes how covariates relate to membership in the susceptible group. Logistic regression is a convenient choice because it models a probability while allowing a linear predictor on the log-odds scale. This is a model for who is susceptible, not for when an event occurs. The latency distribution addresses event timing and is a distinct component.

Definition: Let \(S=1\) indicate that an individual is susceptible to the event and \(S=0\) indicate that the individual is non-susceptible (cured) under the mixture model. For covariate vector \(x\), define \(\pi(x)=P(S=1\mid x)\), the susceptible probability. The cure probability is \(q(x)=P(S=0\mid x)=1-\pi(x)\).

The Logistic Incidence Model

For \(p\) covariates, let \(x=(x_1,\ldots,x_p)\) and let \(\beta_0,\beta_1,\ldots,\beta_p\) be incidence-model coefficients. The logistic incidence specification is

\operatorname{logit}\{\pi(x)\} =\log\left\{\frac{\pi(x)}{1-\pi(x)}\right\} =\beta_0+\sum_{j=1}^{p}\beta_jx_j =\eta(x).

Solving for the probability gives

\pi(x)=\frac{\exp\{\eta(x)\}}{1+\exp\{\eta(x)\}}, \qquad q(x)=1-\pi(x)=\frac{1}{1+\exp\{\eta(x)\}}.

The model assumes that the conditional log odds of susceptibility are linear in the specified covariates and terms (including any transformations or interactions explicitly included). It also assumes the resulting probabilities are an adequate description of the population being modeled. These are modeling assumptions, not consequences of using the logistic function. The function guarantees probabilities between zero and one; it does not guarantee that the covariates, functional forms, or coefficient values are appropriate.

For a one-unit increase in \(x_j\), holding the other modeled covariates fixed, the susceptible odds are multiplied by \(\exp(\beta_j)\) when the model has no interaction involving that change. Since cure odds are the reciprocal of susceptible odds, the corresponding multiplier for cure odds is \(\exp(-\beta_j)\). Neither multiplier is itself a probability difference: the probability change depends on the starting profile.

Key interpretation: A positive coefficient means higher susceptible odds and lower cure odds, conditional on the other modeled covariates. A negative coefficient means lower susceptible odds and higher cure odds. To report an effect on probability, calculate the probabilities for the covariate profiles of interest.

Why the Odds Interpretations Have Opposite Signs

The odds identity follows directly from the definitions. The susceptible odds are \(\pi(x)/\{1-\pi(x)\}\), while the cure odds are \(q(x)/\{1-q(x)\}\). Because \(q(x)=1-\pi(x)\), the numerator and denominator of the cure odds exchange places.

\frac{q(x)}{1-q(x)} =\frac{1-\pi(x)}{\pi(x)} =\left\{\frac{\pi(x)}{1-\pi(x)}\right\}^{-1} =\exp\{-\eta(x)\}.

If a covariate increases by one unit and changes the linear predictor by \(\beta_j\), the susceptible odds ratio is \(\exp(\beta_j)\). The cure odds ratio is therefore \(\exp(-\beta_j)\). This reciprocal relationship does not make the probability effects symmetric: for example, an odds ratio and its reciprocal do not imply equal-sized increases and decreases in probability.

Worked Example: A Binary Covariate and the Two Odds Ratios

Suppose a hypothetical incidence model uses a binary covariate \(x\), with \(x=1\) for the exposed profile and \(x=0\) for the reference profile:

\operatorname{logit}\{\pi(x)\}=-0.8+1.1x.

For \(x=0\), the linear predictor is \(\eta(0)=-0.8\). The susceptible probability is

\pi(0)=\frac{\exp(-0.8)}{1+\exp(-0.8)} =\frac{0.4493}{1+0.4493} =0.3100.

Thus the cure probability is \(q(0)=1-0.3100=0.6900\). For \(x=1\), \(\eta(1)=-0.8+1.1=0.3\), so

\pi(1)=\frac{\exp(0.3)}{1+\exp(0.3)} =\frac{1.350}{1+1.350} =0.5745, \qquad q(1)=1-0.5745=0.4255.

The susceptible odds ratio comparing \(x=1\) with \(x=0\) is \(\exp(1.1)=3.004\). In contrast, the cure odds ratio is \(\exp(-1.1)=0.3329\). The probabilities show the context of those odds effects: the modeled susceptible probability is 0.3100 in the reference profile and 0.5745 in the exposed profile. These are model-based associations, not evidence that exposure causes a change in susceptibility.

Continuous Covariates and Profile Predictions

For a continuous covariate, the coefficient describes the change in log susceptible odds per unit of measurement, assuming a linear effect on that scale and holding other covariates fixed. Units matter. If age is measured in years, a coefficient is a per-year effect; if age is centered or rescaled, interpretation follows that coding. When a change of several units is of interest, multiply the coefficient by that change before exponentiating.

Worked Example: A Ten-Year Age Contrast

Consider the hypothetical model \(\operatorname{logit}\{\pi(x)\}=-2+0.04(x-50)\), where \(x\) is age in years. At age 50, \(\eta=-2\), giving

\pi(50)=\frac{\exp(-2)}{1+\exp(-2)} =\frac{0.1353}{1.1353} =0.1192, \qquad q(50)=0.8808.

At age 60, \(\eta=-2+0.04(60-50)=-1.6\). Therefore,

\pi(60)=\frac{\exp(-1.6)}{1+\exp(-1.6)} =\frac{0.2019}{1.2019} =0.1680, \qquad q(60)=0.8320.

A ten-year increase corresponds to a susceptible odds ratio of \(\exp(0.04\times10)=\exp(0.4)=1.492\), and a cure odds ratio of \(\exp(-0.4)=0.6703\). Under this model, the predicted susceptible probability is 0.1192 at age 50 and 0.1680 at age 60. The positive association is conditional on the model’s linear log-odds assumption over this age range.

The logistic transformation also explains why an odds ratio does not determine a fixed probability difference. Differentiating with respect to a continuous covariate \(x_j\), when it enters linearly without interactions, gives

\frac{\partial \pi(x)}{\partial x_j} =\beta_j\pi(x)\{1-\pi(x)\}.

The factor \(\pi(x)\{1-\pi(x)\}\) varies with the profile. Thus even when the log-odds change is constant per unit, the probability change is not constant across profiles.

Worked Example: Predictions With an Interaction

Suppose a hypothetical model includes a binary treatment indicator \(T\), a binary marker \(M\), and their interaction:

\operatorname{logit}\{\pi(T,M)\} =-1.2+0.9T-0.5M+0.4TM.

Calculate the linear predictor for each profile, then apply the logistic transformation. For \(T=0,M=0\), \(\eta=-1.2\), so \(\pi=0.2315\) and \(q=0.7685\). For \(T=1,M=0\), \(\eta=-1.2+0.9=-0.3\), giving \(\pi=0.4256\) and \(q=0.5744\). For \(T=0,M=1\), \(\eta=-1.2-0.5=-1.7\), giving \(\pi=0.1545\) and \(q=0.8455\). For \(T=1,M=1\), \(\eta=-1.2+0.9-0.5+0.4=-0.4\), giving \(\pi=0.4013\) and \(q=0.5987\).

The treatment susceptible-odds ratio is \(\exp(0.9)=2.460\) when \(M=0\), but \(\exp(0.9+0.4)=\exp(1.3)=3.669\) when \(M=1\). The interaction changes the treatment contrast on the log-odds scale. It would be incorrect to interpret \(0.9\) as the treatment coefficient applying to both marker groups or to describe a single treatment odds ratio without specifying the marker profile.

Assumptions, Diagnostics, and Limits

The incidence model concerns a latent status. In ordinary binary logistic regression, the response is observed for each individual. In a cure model, an individual’s susceptible status is generally not observed directly: someone without an event by the end of follow-up may be cured or may simply have a long latency time. Therefore, the model-based probability \(\pi(x)\) cannot generally be estimated by labeling every event-free individual “cured” and applying ordinary logistic regression. The event-time information and censoring must be handled through the cure-model likelihood.

Interpretation depends on a sufficiently informative follow-up period and a credible latency specification as well as on the incidence model. The introductory material on identifiability and follow-up in mixture cure models explains why finite follow-up can make a long-latency group difficult to distinguish from a cured fraction. A well-fitting logistic curve cannot resolve that fundamental ambiguity by itself.

Useful checks include verifying the event and covariate coding, inspecting the fitted probability across meaningful covariate profiles, and checking whether the chosen linear predictor is plausible. Extremely large estimated coefficients or fitted probabilities near zero or one can signal sparse covariate patterns or separation. Curvature in a continuous covariate’s relationship with the log odds suggests that a straight-line term may be inadequate. Interactions should be considered when scientifically justified, since a main-effect coefficient in an interaction model is conditional on the interacting covariate being zero.

Observed event proportions by covariate group can be informative descriptively, but they are not direct estimates of susceptible proportions when follow-up is incomplete or censoring occurs. Likewise, apparent long-term survival plateaus should be interpreted with caution if follow-up is short or the tail of the latency distribution is uncertain. Predictions should be reported for covariate profiles within the support of the data; extrapolated probabilities rely heavily on the specified linear predictor.

Approach	What it models	Important distinction
Logistic incidence model	Conditional probability of latent susceptibility	Coefficients act on susceptible log odds; event times and censoring still matter for fitting the cure model.
Intercept-only incidence model	A common susceptible probability across modeled profiles	It does not represent covariate differences in susceptibility.
Alternative binary-response link	The susceptible probability using a different link function	Coefficient scales and interpretations differ; predictions must still respect the latent-status setting.
Ordinary time-to-event model without a cured fraction	Event-time risk under its specified survival model	It does not explicitly represent a non-susceptible group with no event time.

Finally, a fitted coefficient is an association conditional on the included covariates and the model specification. A causal interpretation requires additional design and assumptions beyond this regression model. Logistic incidence modeling also does not replace the latency component: two profiles can have the same susceptible probability but different event-time distributions among susceptible individuals.

Common mistake: “A positive coefficient means the cure odds increase by \(\exp(\beta_j)\).” This reverses the direction. A positive coefficient multiplies susceptible odds by \(\exp(\beta_j)\), while cure odds are multiplied by \(\exp(-\beta_j)\). For probability statements, calculate \(\pi(x)\) and \(q(x)\) at the stated profiles instead of treating an odds ratio as a probability ratio.

Check Your Understanding

Answer each question, then compare with the solutions below.

A profile has linear predictor \(\eta=0.4\). Calculate its susceptible probability and cure probability.
A continuous covariate has coefficient \(-0.2\) per unit and enters without interactions. State the susceptible and cure odds ratios for a three-unit increase.
A binary covariate has coefficient \(0.7\). Which odds increase, and by what factor, when the covariate changes from zero to one, holding other covariates fixed?
Why is it generally incorrect to label every person without an observed event as cured and use those labels as the response in ordinary logistic regression?

Solutions. (1) \(\pi=\exp(0.4)/\{1+\exp(0.4)\}=1.492/2.492=0.5987\), so \(q=1-0.5987=0.4013\). (2) The change in the linear predictor is \(-0.2(3)=-0.6\). The susceptible odds ratio is \(\exp(-0.6)=0.5488\), and the cure odds ratio is \(\exp(0.6)=1.822\). (3) Susceptible odds increase by \(\exp(0.7)=2.014\); cure odds change by the reciprocal factor, \(\exp(-0.7)=0.4966\). (4) Event-free status at the end of follow-up does not reveal latent susceptibility: a susceptible person may have a latency time longer than the observed follow-up. Censoring and event-time information must therefore be handled in the cure-model analysis.