Cox-Integrated Spline Sieve Estimation for Doubly Censored Data

Abstract

When considering doubly censored data, the failure time of interest refers to the duration between the initial and a subsequent event, where both events may be censored. Prior studies investigating the association between these events have yielded statistically inconclusive results due to their reliance exclusively on censored data (initial event). To address this limitation, a spline-based sieve maximum likelihood estimator (MLE) was developed by using integrated spline (I-splines) and M-splines (derivative of I-splines), integrated with Cox Proportional Hazards, to investigate the association between HIV infection and AIDS incubation period. Unlike prior studies, the proposed approach incorporates exactly observed event times and censored data to improve dependency estimates. The estimator treats the initial event as a covariate, allowing for a direct investigation of its relationship with the failure time. Spline functions simultaneously estimate parametric/non-parametric components, simplifying model implementation; multiple imputation ensures robustness to missing data. The asymptotic properties of the estimator are rigorously established. In simulation studies with sample sizes n = 50, 100, 200, 300, the proposed estimator achieved a reduction up to 93% in mean squared error and a 95% reduction in bias compared with existing methods, with coverage probabilities close to nominal levels. In the AIDS dataset, the proposed method revealed a statistically significant association between HIV infection timing and AIDS incubation (log HR = 0.580)  that earlier analyses failed to detect (log HR = -0.11), and produced more precise treatment effect estimates. These findings highlight the enhanced sensitivity and statistical power of the proposed Cox-integrated spline sieve approach.