Semiparametric estimation of GLMs with interval-censored covariates via an augmented Turnbull estimator

TL;DR

This paper addresses regression with interval-censored covariates in generalized linear models by introducing GELc, a likelihood-based estimator that augments Turnbull's nonparametric estimator for the censored covariate distribution. GELc jointly estimates the GLM parameters and a nonparametric covariate distribution using augmented Turnbull intervals and self-consistent updating, with the authors proving consistency and asymptotic normality under mild regularity. The method is validated via extensive simulations showing favorable finite-sample performance and good coverage, and demonstrated on two real datasets (carotenoids and ACTG359) to illustrate practical applicability; an R package ICenCov provides implementation. Overall, GELc offers robust inference for GLMs with interval-censored covariates by avoiding parametric misspecification of the covariate distribution while providing standard errors and feasible computation.

Abstract

Interval-censored covariates are frequently encountered in biomedical studies, particularly in time-to-event data or when measurements are subject to detection or quantification limits. Yet, the estimation of regression models with interval-censored covariates remains methodologically underdeveloped. In this article, we address the estimation of generalized linear models when one covariate is subject to interval censoring. We propose a likelihood-based approach, GELc, that builds upon an augmented version of Turnbull's nonparametric estimator for interval-censored data. We prove that the GELc estimator is consistent and asymptotically normal under mild regularity conditions, with available standard errors. Simulation studies demonstrate favorable finite-sample performance of the estimator and satisfactory coverage of the confidence intervals. Finally, we illustrate the method using two real-world applications: the AIDS Clinical Trials Group Study 359 and an observational nutrition study on circulating carotenoids. The proposed methodology is available as an R package at github.com/atoloba/ICenCov.

Figures (9)

• Figure 1: Simulated observed intervals for $n=300$ under the four censoring scenarios: the true $Z_i$ values ($\mu=0$) and the three scenarios of censoring process with increasing mean widths. Observations are ordered by the true $Z_i$ values to facilitate comparison of how censoring interval widths vary across scenarios.
• Figure 2: Results for the gamma regression with true parameters $\alpha=10$, $\gamma = -0.05$ and $\phi = 1$, showing relative bias (with Monte Carlo error bars) and empirical standard error. From left to right: $\alpha$, $\gamma$, and $\phi$ parameters.
• Figure B.3: Relative bias of $\gamma$ for increasing mean width of observed intervals (x-axis) and sample size, in the Gamma regression model for varying $\gamma_0$ and $\phi_0$.
• Figure B.4: Relative bias of $\gamma$ for increasing mean width of observed intervals (x-axis) and sample size, in the Logistic regression model for varying $\gamma_0$.
• Figure B.5: Relative bias of $\alpha$ for increasing mean width of observed intervals (x-axis) and sample size, in the Gamma regression model for $\alpha_0=10$, and varying $\gamma_0$ and $\phi_0$.

Theorems & Definitions (15)

• proof
• Lemma A.1
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• Corollary A.2
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• Lemma A.3
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• Lemma A.4
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• Lemma A.5