Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cure models: from mixture to matrix distributions

Martin Bladt, Jorge Yslas

TL;DR

This paper introduces a flexible class of cure-rate models based on phase-type distributions, modeling the cure mechanism as a latent Markov jump process with two absorbing states: immunity and the event. It unifies incidence (cure vs. susceptible) and latency (time-to-event for susceptibles) via closed-form expressions and extends to time-inhomogeneous PH forms to capture diverse tail behavior. A regression framework framed as a Mixture-of-Experts allows covariates to shape both the cure probability and the susceptible survival, and the authors prove denseness of this class, ensuring any well-behaved cure model can be approximated. Estimation is carried out through EM, with automatic dimension selection and residual-based goodness-of-fit diagnostics; simulations and a leukemia data application demonstrate superior fit and interpretability relative to classical cure models, highlighting the practical value of the approach.

Abstract

Cure rate models address survival data in which a proportion of individuals will never experience the event of interest. Existing parametric approaches are predominantly based on finite mixtures, which impose restrictive assumptions on both the cure mechanism and the distribution of susceptible event times. A cure model based on phase-type distributions is introduced, leveraging their latent Markov jump process representation to allow immunity to occur either at baseline or dynamically during follow-up. This structure yields a flexible and interpretable formulation of long-term survival while encompassing classical mixture cure models as special cases. A unified regression framework is developed for covariate effects on both the cure rate and the susceptible survival distribution, and the proposed model class is dense, reducing the impact of parametric misspecification. Estimation is performed via expectation-maximization algorithms, accompanied by an automatic model selection strategy. Simulation studies and a real-data example demonstrate the practical advantages of the approach.

Cure models: from mixture to matrix distributions

TL;DR

This paper introduces a flexible class of cure-rate models based on phase-type distributions, modeling the cure mechanism as a latent Markov jump process with two absorbing states: immunity and the event. It unifies incidence (cure vs. susceptible) and latency (time-to-event for susceptibles) via closed-form expressions and extends to time-inhomogeneous PH forms to capture diverse tail behavior. A regression framework framed as a Mixture-of-Experts allows covariates to shape both the cure probability and the susceptible survival, and the authors prove denseness of this class, ensuring any well-behaved cure model can be approximated. Estimation is carried out through EM, with automatic dimension selection and residual-based goodness-of-fit diagnostics; simulations and a leukemia data application demonstrate superior fit and interpretability relative to classical cure models, highlighting the practical value of the approach.

Abstract

Cure rate models address survival data in which a proportion of individuals will never experience the event of interest. Existing parametric approaches are predominantly based on finite mixtures, which impose restrictive assumptions on both the cure mechanism and the distribution of susceptible event times. A cure model based on phase-type distributions is introduced, leveraging their latent Markov jump process representation to allow immunity to occur either at baseline or dynamically during follow-up. This structure yields a flexible and interpretable formulation of long-term survival while encompassing classical mixture cure models as special cases. A unified regression framework is developed for covariate effects on both the cure rate and the susceptible survival distribution, and the proposed model class is dense, reducing the impact of parametric misspecification. Estimation is performed via expectation-maximization algorithms, accompanied by an automatic model selection strategy. Simulation studies and a real-data example demonstrate the practical advantages of the approach.
Paper Structure (17 sections, 2 theorems, 26 equations, 6 figures, 2 tables, 3 algorithms)

This paper contains 17 sections, 2 theorems, 26 equations, 6 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Absorption times of Markov jump-processes
  3. General specification
  4. The cure rate
  5. Censoring and estimation
  6. Inhomogeneity
  7. Regression on the cure rate and survival shape
  8. Phase-type MoE
  9. Denseness on cure models
  10. Model selection and goodness-of-fit
  11. Automatic selection of $r$
  12. Goodness-of-fit
  13. Cox-Snell residuals
  14. Modified Cox-Snell residuals
  15. Simulation studies
  16. ...and 2 more sections

Key Result

Proposition 2.2

The cure rate of $\tau \sim \hbox{PH}(\pmb{\pi}, \boldsymbol{\bm T})$ is given by Moreover, the survival function $S_\tau = 1 - F_\tau$ of $\tau$ admits the representation where $S_u$ denotes the survival function of a phase-type distribution with vector of initial probabilities and sub-intensity matrix $\boldsymbol{\bm T}_{1:(r-1),1:(r-1)}$.

Figures (6)

  • Figure 5.1: Density function of the latency: mixture of three Gamma distributions, $\Gamma(1, 4)$, $\Gamma(4, 2)$, and $\Gamma(8, 1)$, with weights $0.1$, $0.5$, and $0.4$.
  • Figure 5.2: KM estimates (black step functions with confidence bands) and fitted survival curves for the simulated dataset under each model specification. While standard models like the Exponential and Weibull show significant divergence, the phase-type model's survival curve (bottom panel) overlaps the KM curve almost perfectly, including the plateau representing the cured fraction.
  • Figure 5.3: Average models' performance across different censoring levels in 100 replications. Top left: average loglikelihoods, with the phase-type model (red) consistently highest. Top right: loglikelihood differences relative to the phase-type model, all negative. Bottom: average estimated susceptible fraction $\hat{p}$, with the true value $p=0.8$ shown as a dashed line.
  • Figure 5.4: Average Cramér-von Mises distances for CS (left) and modified CS residuals (right) across censoring levels. Lower values indicate a better fit. The phase-type model (red) achieves smaller distances in both cases, with a more notable difference in the modified residuals, demonstrating its superior fit to the latency distribution.
  • Figure 6.1: KM estimates and fitted survival curves from the phase-type MoE model for the leukemia data, stratified by transplant type (Allogeneic in yellow, Autologous in blue). The model captures both the initial steep decline and the plateau for both populations.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 3.1
  • Remark 3.1
  • Definition 3.2
  • Proposition 3.4
  • proof
  • Remark 4.1