Partially linear Cox model with neural networks for left-truncated data

In the past few decades, artificial neural networks (ANNs) have exhibited their superior capabilities for capturing nonlinear data patterns in supervised learning. Inspired by such desirable advantages, we herein explore the integration of ANNs with partially linear Cox model in the context of left truncation. This sampling scheme tends to enroll individuals with slower disease progression and thus leads to a biased sample of survival times in the target population. The proposed model comprises both parametric covariate effects and a nuisance function of uninterested covariates that is approximated with ANNs, offering a balance between interpretability and flexibility. We consider the conditional maximum likelihood estimation and derive a profile likelihood that is free of the baseline hazard function. An iterative algorithm embedded with stochastic gradient descent is proposed to minimize the negative profile log-likelihood, leading to the estimators of the regression parameters and ANNs simultaneously. Extensive simulation studies and an application demonstrate that the proposed method outperforms the traditional approaches regarding the estimation accuracy and predictive capability.