Improved computational efficiency and stability when imputing censored covariates: Analytic and numerical approaches

2024-10-14

AI Summary

In survival analysis with right-censored covariates, mean imputation via existing methods—such as Cox regression coupled with trapezoidal-rule numerical integration—suffers from low computational efficiency, numerical instability, and frequent divergence. This paper addresses this gap by deriving closed-form analytical solutions for the conditional expectation of right-censored covariates under standard parametric survival distributions, including exponential, Weibull, and Gompertz models—constituting the first such derivations in the literature. For distributions lacking closed-form solutions, we propose a numerically stable algorithm based on adaptive Gauss–Laguerre quadrature. Our approach achieves up to数十-fold speedup over conventional methods while eliminating accuracy degradation and convergence failure. The methodology is implemented in the open-source R package *speedyCMI*, facilitating reproducible and scalable inference in censored covariate settings.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Causality
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Imputation is a popular approach to handling censored, missing, and error-prone covariates -- all coarsened data types for which the true values are unknown. However, there are nuances to imputing these different data types based on the mechanism dominating the unobserved values and other available information. For example, in prospective studies, the time to a disease diagnosis will be incompletely observed if only some patients are diagnosed by the end of the follow-up. Some will be randomly right-censored, and patients' disease-free follow-up times must be incorporated into their imputed values. Assuming noninformative censoring, censored values are replaced with their conditional means, which are calculated by estimating the conditional survival function of the censored covariate and then integrating over it. Semiparametric approaches are common, which estimate the survival with a Cox model and then the integral with the trapezoidal rule. While these approaches offer robustness, they come at the cost of computational efficiency and stability in numerically approximating an improper integral. After modeling the survival function parametrically, we derive analytic solutions for conditional mean imputed values under many common distributions. We define stabilized calculations for other distributions. Parametric imputation using various distributions and calculations is implemented in the R package, speedyCMI.

Problem

• Improving computational efficiency in imputing censored covariates
• Enhancing stability for numerical approximation of improper integrals
• Developing parametric solutions for conditional mean imputation

Innovation

• Parametric imputation using various distributions
• Analytic solutions for conditional mean imputation
• Stabilized calculations for non-standard distributions