Single-index Semiparametric Transformation Cure Models with Interval-censored Data

TL;DR

The paper tackles interval-censored survival data with a cure fraction by introducing a flexible single-index semiparametric transformation cure model (SMCI). It jointly models the incidence with a single-index link $p({\bm X})=g(\boldsymbol{\gamma}^{\top}{\bm X})$ and the latency with a semiparametric transformation $S_u(t|{\bm Z})=\exp\{-G(\exp(\boldsymbol{\beta}^{\top}{\bm Z})\Lambda(t))\}$, where $G$ is induced by gamma frailty and includes PH and PO as special cases. An EM algorithm with four-layer data augmentation handles interval censoring and latent components, with kernel (SMCI-K) or spline (SMCI-S) approaches for the incidence regression and monotone I-splines for the baseline transform. Simulations show that SMCI-K and SMCI-S often outperform the spline-based and logistic-based mixtures, particularly under non-logistic incidence, and the Alzheimer's Disease Neuroimaging Initiative analysis demonstrates improved fit and reveals nonmonotonic incidence patterns and APOE4 effects on both incidence and latency. The framework offers scalable, interpretable analysis for high-dimensional covariates in interval-censored contexts and suggests avenues for theoretical development and extensions to more flexible latency structures.

Abstract

Interval censored data commonly arise in medical studies when the event time of interest is only known to lie within an interval. In the presence of a cure subgroup, conventional mixture cure models typically assume a logistic model for the uncure probability and a proportional hazards model for the susceptible subjects. However, in practice, the assumptions of parametric form for the uncure probability and the proportional hazards model for the susceptible may not always be satisfied. In this paper, we propose a class of flexible single-index semiparametric transformation cure models for interval-censored data, where a single-index model and a semiparametric transformation model are utilized for the uncured and conditional survival probability, respectively, encompassing both the proportional hazards cure and proportional odds cure models as specific cases. We approximate the single-index function and cumulative baseline hazard functions via the kernel technique and splines, respectively, and develop a computationally feasible expectation-maximisation (EM) algorithm, facilitated by a four-layer gamma-frailty Poisson data augmentation. Simulation studies demonstrate the satisfactory performance of our proposed method, compared to the spline-based approach and the classical logistic-based mixture cure models. The application of the proposed methodology is illustrated using the Alzheimers dataset.

Figures (17)

• Figure 1: Boxplots of ASE for the SMCI-K (white boxplots), SMCI-S (grey boxplots) and GORMC (dark boxplots) models with $n=200$ and 500.
• Figure 2: Estimated baseline survival functions $S_u(\cdot|{\bm Z}=0)$ for Scenario 1. The solid curves show the true values, the dashed curves pertain to averaged estimates, and the dotted curves represent the upper and lower 2.5% quantiles of the estimates.
• Figure 3: Estimated baseline survival functions $S_u(\cdot|{\bm Z}=0)$ for Scenario 2. The solid curves show the true values, the dashed curves pertain to averaged estimates, and the dotted curves represent the upper and lower 2.5% quantiles of the estimates.
• Figure 4: Estimated baseline survival functions $S_u(\cdot|{\bm Z}=0)$ for Scenario 3. The solid curves show the true values, the dashed curves pertain to averaged estimates, and the dotted curves represent the upper and lower 2.5% quantiles of the estimates.
• Figure 5: Analysis of the ADNI study: (a) The Kaplan-Meier survival curve; (b) the loglikelihood at the nonparametric maximum likelihood estimates plotted as a function of $r$ in the logarithmic transformations.