Inference for Cure Fractions and Latency Effects

Why Cure-Model Inference Needs More Than a Standard Error

A cure model separates the probability of being susceptible to the event from the event-time distribution among susceptible individuals. Inference must therefore quantify uncertainty in two related but distinct quantities: the cure probability and the parameters describing latency among those who are susceptible. These quantities can have very different likelihood shapes. In particular, a cure-probability likelihood may be skewed or nearly flat, and a fitted cure probability may lie at the boundary of its allowable range.

A symmetric estimate-plus-or-minus-standard-error interval can be inadequate in those settings. This tutorial develops likelihood-based intervals, explains how to interpret them on useful scales, and shows when boundary estimates call for special care. The likelihood contributions below are those for independent right-censored observations under a mixture cure model; the central inferential ideas also apply when the latency model is extended.

Definition: Let $Y_i=\min(T_i,C_i)$ be the observed time for individual $i$, and let $\delta_i=1$ indicate an observed event and $\delta_i=0$ indicate right censoring. Write $\pi_i$ for the probability of cure and $S_u(t;\theta)$, $f_u(t;\theta)$ for the susceptible individuals’ latency survival and density, with latency parameters $\theta$. The individual likelihood contribution is $$ L_i= \begin{cases} (1-\pi_i)f_u(Y_i;\theta),&\delta_i=1,\\ \pi_i+(1-\pi_i)S_u(Y_i;\theta),&\delta_i=0. \end{cases} $$ The full likelihood is the product of these contributions, and the log likelihood is $\ell(\psi,\lambda)$, where $\psi$ is the parameter of interest and $\lambda$ denotes all remaining parameters.

Likelihood-based uncertainty relies on a correctly specified likelihood, independent observations (or an appropriate likelihood that accounts for dependence), and a fitting procedure that finds the relevant maximum. For ordinary likelihood-ratio approximations, the parameter of interest should be in the interior of its parameter space, the maximum should be identifiable, and the log likelihood should be sufficiently smooth and locally quadratic. Cure parameters near zero or one, and weak separation of long-term survivors from late events, can undermine these regularity conditions.

Wald Intervals and Profile-Likelihood Intervals

A Wald interval uses the estimate and its estimated standard error. For a scalar parameter $\psi$, its usual approximate 95% interval is $\hat{\psi}\pm1.96\,SE(\hat{\psi})$. This is most defensible when the likelihood is approximately quadratic around an interior maximum and the interval stays within the parameter space. A cure probability is constrained to $[0,1]$, so a Wald interval can extend below zero or above one. Even when it remains in range, symmetry may poorly describe an asymmetric likelihood.

A profile-likelihood interval directly uses the likelihood shape for the parameter of interest. For each fixed value $\psi$, maximize the log likelihood over nuisance parameters $\lambda$, defining $\ell_p(\psi)=\max_\lambda\ell(\psi,\lambda)$. Let $\hat{\psi}$ maximize this profile. Under regularity conditions, the likelihood-ratio statistic $2\{\ell_p(\hat{\psi})-\ell_p(\psi)\}$ is approximately distributed as chi-square with one degree of freedom when the fixed value is the true parameter. Thus an approximate 95% interval contains values satisfying $$ 2\{\ell_p(\hat{\psi})-\ell_p(\psi)\}\leq \chi^2_{1,0.95}=3.841, \qquad \ell_p(\psi)\geq \ell_p(\hat{\psi})-1.9205. $$

The approximation has a local quadratic justification. Near an interior maximum, let $I_p$ denote the profile information, that is, the negative second derivative of the profile log likelihood at the maximum. A second-order Taylor expansion gives $$ \ell_p(\psi)\approx \ell_p(\hat{\psi})-\frac{1}{2}I_p(\psi-\hat{\psi})^2. $$ Substitution into the likelihood-ratio statistic yields $2\{\ell_p(\hat{\psi})-\ell_p(\psi)\}\approx I_p(\psi-\hat{\psi})^2$. The approximate interval therefore satisfies $I_p(\psi-\hat{\psi})^2\leq3.841$, which is the familiar symmetric interval in the quadratic case. When the profile is skewed, its two likelihood-ratio endpoints need not be equally distant from the estimate.

Worked Example: A Profile-Likelihood Interval for the Cure Probability

Suppose a hypothetical fitted cure model has a profile log likelihood maximized at $\hat{\pi}=0.55$, with $\ell_p(0.55)=-104.000$. A grid search gives $\ell_p(0.45)=-107.000$, $\ell_p(0.50)=-104.500$, $\ell_p(0.60)=-104.3205$, and $\ell_p(0.65)=-106.000$. Assume the fitted likelihood is regular enough for the usual one-parameter likelihood-ratio cutoff. The 95% cutoff on the log-likelihood scale is $-104.000-1.9205=-105.9205$.

The lower endpoint lies between 0.45 and 0.50. Linear interpolation between the two grid values gives $$ \pi_L\approx0.45+0.05\left\{\frac{-105.9205-(-107.000)}{-104.500-(-107.000)}\right\} =0.45+0.05\left(\frac{1.0795}{2.500}\right) =0.4716. $$

The upper endpoint lies between 0.60 and 0.65. Interpolating on that segment gives $$ \pi_U\approx0.60+0.05\left\{\frac{-105.9205-(-104.3205)}{-106.000-(-104.3205)}\right\} =0.60+0.05\left(\frac{-1.6000}{-1.6795}\right) =0.6476. $$

The approximate profile-likelihood interval is therefore $(0.4716,0.6476)$. Its endpoints are not equally far from 0.55, reflecting the asymmetric profile. Under the model and the likelihood-ratio approximation, the procedure has approximately 95% repeated-sampling coverage; this does not mean there is a 95% probability that this fixed interval contains the parameter. The interpolation is only an approximation to the crossings, so a finer profile grid or numerical root-finding is preferable for final reporting.

Transforming Cure-Probability and Latency Effects

Cure probabilities are often modeled with logistic regression, $\operatorname{logit}(\pi_i)=x_i^\mathsf{T}\beta$. A coefficient interval is naturally on the log-odds scale, while an interval for a particular covariate pattern is obtained by transforming the linear predictor through the inverse-logit function. Because that function is monotone, transforming both endpoints preserves their order and gives an interval within $[0,1]$. For a contrast in a binary covariate, exponentiating its logistic coefficient and interval gives an odds ratio and corresponding interval.

Latency coefficients require equally careful interpretation. For example, in an accelerated failure-time model with a log-time coefficient $\gamma$, the time ratio associated with a one-unit covariate increase is $\exp(\gamma)$, among susceptible individuals. Exponentiate the estimate and both confidence limits to obtain the interval on the time-ratio scale. A latency effect describes event timing within the susceptible component; it is not an effect on the probability of cure.

Worked Example: A Cure-Probability Contrast From Logistic Regression

In a hypothetical model, treatment is coded $z=1$ versus $z=0$, and the cure probability is $\operatorname{logit}(\pi)=\beta_0+\beta_1z$. Suppose the estimate is $\hat{\beta}_1=-0.60$, with a profile-likelihood 95% interval $(-1.10,-0.12)$. The point estimate of the odds ratio is $$ \exp(-0.60)=0.5488. $$ The interval on the odds-ratio scale is $$ \bigl(\exp(-1.10),\exp(-0.12)\bigr)=(0.3329,0.8869). $$

This interval is entirely below 1, indicating that, under the fitted model and its assumptions, the treatment group has lower odds of cure than the reference group at the 5% two-sided likelihood-ratio level. This is an odds comparison, not a difference in cure probabilities. To obtain probabilities, suppose the fitted reference-group cure probability is 0.60. Its log odds are $\operatorname{logit}(0.60)=\log(1.5)=0.4055$. The treatment-group estimate is therefore $$ \hat{\pi}_1=\frac{\exp(0.4055-0.60)}{1+\exp(0.4055-0.60)} =\frac{0.8232}{1.8232}=0.4515. $$

The reference probability is treated here as fixed solely to illustrate the transformation. In actual inference, uncertainty in the intercept and coefficient must both be included; one can profile the treatment-group probability directly or use the joint fitted likelihood. A coefficient interval alone is not enough to construct a probability interval when the baseline probability is estimated.

Worked Example: A Latency Effect on the Time-Ratio Scale

Suppose a hypothetical accelerated failure-time cure model gives a latency coefficient $\hat{\gamma}=0.25$ for a one-unit increase in a covariate among susceptible individuals. Its profile-likelihood 95% interval is $(-0.05,0.55)$. The estimated time ratio and its interval are $$ \exp(0.25)=1.284,\qquad \bigl(\exp(-0.05),\exp(0.55)\bigr)=(0.9512,1.733). $$

The point estimate corresponds to a 1.284-fold event-time scale, or about 28.4% longer latency, for the covariate increase among susceptible individuals. The interval includes 1, so it does not rule out no latency effect at the 5% two-sided level using this interval. This conclusion concerns latency only: it does not establish whether the covariate changes the cure probability. The profile interval is asymmetric around the coefficient estimate, and exponentiating it preserves that asymmetry on the time-ratio scale.

Boundary Estimates and Failure Modes

The cure probability is constrained to $[0,1]$. If the likelihood is maximized at $\hat{\pi}=0$, the estimate is on a boundary; a symmetric Wald interval may include impossible negative values, and the ordinary chi-square likelihood-ratio approximation need not apply. A similar concern arises at $\hat{\pi}=1$. The exact behavior depends on the model and which parameters are identified at the boundary. Do not assume the usual cutoff is valid merely because software reports a standard error.

A defensible approach is to inspect the profile over the allowable parameter range and determine whether it closes on each side. If an endpoint is at the boundary, report that fact. When standard asymptotic calibration is doubtful, a parametric bootstrap can calibrate the relevant likelihood-ratio statistic under the fitted model, provided the model and simulation procedure are appropriate. A one-sided upper limit at a boundary may be useful, but its calibration must match the inferential question; it should not be presented as an ordinary two-sided interval without explanation.

Worked Example: A Cure Estimate at the Boundary

Consider hypothetical data for which the maximum likelihood estimate is $\hat{\pi}=0$. The estimated log likelihood is $-52.000$, and the log likelihood at $\pi=0.08$, after re-optimizing nuisance parameters, is $-53.355$. Suppose a parametric bootstrap under the fitted model has been used to calibrate a one-sided 95% upper limit, and its simulated likelihood-ratio cutoff is 2.71.

At $\pi=0.08$, the likelihood-ratio statistic is $$ 2\{-52.000-(-53.355)\}=2(1.355)=2.710. $$ This reaches the stated bootstrap cutoff, so the illustrative one-sided upper limit is approximately 0.08, giving the boundary-aware interval $[0,0.08]$. The number is conditional on the hypothetical bootstrap calibration; it is not a universal cutoff or a general recipe for boundary problems. The interval says that values up to about 0.08 remain compatible with this calibrated procedure, not that the cure probability is known to be zero.

Other warning signs include a profile that is nearly flat over a wide range, abrupt jumps caused by failed optimization, or estimates that change substantially with starting values. Flatness may signal weak identifiability, particularly when follow-up is too short to distinguish a cured fraction from very long susceptible survival. Check that the nuisance parameters were successfully re-optimized at each fixed value of the parameter of interest; otherwise the plotted curve is not the profile likelihood. A fine grid, multiple starting values, and direct checks of the fitted log likelihood help distinguish genuine shape from numerical artifacts.

Common mistake: Treating $\hat{\pi}\pm1.96\,SE(\hat{\pi})$ as automatically valid because the software reports it. This can yield an interval outside $[0,1]$, hide an asymmetric likelihood, or rely on a normal approximation at a boundary. Inspect the profile, use a likelihood-based interval when its approximation is justified, and use an appropriately calibrated method when boundary behavior makes the standard approximation unreliable.

Profile likelihood is not automatically superior in every respect. It can be computationally demanding, depends on successful nuisance-parameter optimization, and its usual cutoff still relies on regularity conditions. Wald intervals are quick and can be adequate for a well-identified interior estimate with a nearly quadratic likelihood. Bootstrap intervals can address nonstandard sampling behavior but require a credible data-generating model and enough successful refits. None of these methods repairs weak identification, an incorrect latency distribution, or insufficient follow-up; an interval may honestly be wide because the data do not separate cure from long latency.

Choose the estimand.
Specify whether the target is a cure probability for a covariate pattern, a cure-model coefficient, or a latency parameter, and state the scale for interpretation.

Check the fitted likelihood.
Confirm convergence, inspect constraints and starting-value sensitivity, and ensure nuisance parameters are re-optimized when profiling.

Construct and interpret uncertainty.
Use a Wald interval only when its local approximation is credible; otherwise consider a profile interval or calibrated bootstrap, then transform endpoints to the reporting scale.

Check Your Understanding

Answer each question, then compare with the solutions below.

For a one-parameter 95% profile-likelihood interval, what log-likelihood drop from the maximum corresponds to the usual chi-square cutoff?
A cure probability estimate is 0.04 and its Wald standard error is 0.05. What is the approximate Wald 95% interval, and what is the immediate concern?
A log-time latency coefficient is $-0.20$, with interval $(-0.50,0.10)$. Convert the estimate and limits to the time-ratio scale and interpret the interval.
Why might a flat profile for the cure probability occur even when the optimizer converges successfully?

Solutions. (1) The cutoff is $\chi^2_{1,0.95}=3.841$ for twice the log-likelihood drop, so the log-likelihood drop is $3.841/2=1.9205$. (2) The interval is $0.04\pm1.96(0.05)=0.04\pm0.098$, or approximately $(-0.058,0.138)$. It includes an impossible negative cure probability and suggests that the symmetric normal approximation is unsuitable; inspect the constrained likelihood and consider boundary-aware inference. (3) The estimate is $\exp(-0.20)=0.8187$, and the interval is $(\exp(-0.50),\exp(0.10))=(0.6065,1.105)$. The point estimate corresponds to shorter latency among susceptible individuals, but the interval includes a time ratio of 1, so it does not rule out no latency effect at the stated level. (4) If follow-up is insufficient to distinguish a cured fraction from susceptible individuals with very long event times, many cure probabilities can fit nearly equally well. Convergence alone does not establish that the parameter is well identified.