Survival or time-to-event data with time-varying covariates are common in practice, and exploring the non-stationarity in covariates is essential to accurately analyzing the nonlinear dependence of time-to-event outcomes on covariates. Traditional survival analysis methods such as Cox proportional hazards model have been extended to address the time-varying covariates through a counting process formulation, although sophisticated machine learning methods that can accommodate time-varying covariates have been limited. In this paper, we propose a non-parametric Bayesian survival model to analyze the nonlinear dependence of time-to-event outcomes on time-varying covariates. We focus on a computationally feasible Cox process called permanental process, which assumes the square root of hazard function to be generated from a Gaussian process, and tailor it for survival data with time-varying covariates. We verify that the proposed model holds with the representer theorem, a beneficial property for functional analysis, which offers us a fast Bayesian estimation algorithm that scales linearly with the number of observed events without relying on Markov Chain Monte Carlo computation. We evaluate our algorithm on synthetic and real-world data, and show that it achieves comparable predictive accuracy while being tens to hundreds of times faster than state-of-the-art methods.

Explainability is valuable to ensure that the model's forecasts are credible

Survival or time-to-event data with time-varying covariates are common in practice, and exploring the non-stationarity in covariates is essential to accurately analyzing the nonlinear dependence of time-to-event outcomes on covariates. Traditional survival analysis methods such as Cox proportional hazards model have been extended to address the time-varying covariates through a counting process formulation, although sophisticated machine learning methods that can accommodate time-varying covariates have been limited. In this paper, we propose a non-parametric Bayesian survival model to analyze the nonlinear dependence of time-to-event outcomes on time-varying covariates. We focus on a computationally feasible Cox process called permanental process, which assumes the square root of hazard function to be generated from a Gaussian process, and tailor it for survival data with time-varying covariates. We verify that the proposed model holds with the representer theorem, a beneficial property for functional analysis, which offers us a fast Bayesian estimation algorithm that scales linearly with the number of observed events without relying on Markov Chain Monte Carlo computation.

The accurate prediction of survival times for patients with severe diseases remains a critical challenge despite recent advances in artificial intelligence. This study introduces "SurvTimeSurvival: Survival Analysis On Patients With Multiple Visits/Records", utilizing the Transformer model to not only handle the complexities of time-varying covariates but also covariates data. We also tackle the data sparsity issue common to survival analysis datasets by integrating synthetic data generation into the learning process of our model. We show that our method outperforms state-of-the-art deep learning approaches on both covariates and time-varying covariates datasets. Our approach aims not only to enhance the understanding of individual patient survival trajectories across various medical conditions, thereby improving prediction accuracy, but also to play a pivotal role in designing clinical trials and creating new treatments.

In many longitudinal settings, time-varying covariates may not be measured at the same time as responses and are often prone to measurement error. Naive last-observation-carried-forward methods incur estimation biases, and existing kernel-based methods suffer from slow convergence rates and large variations. To address these challenges, we propose a new functional calibration approach to efficiently learn longitudinal covariate processes based on sparse functional data with measurement error. Our approach, stemming from functional principal component analysis, calibrates the unobserved synchronized covariate values from the observed asynchronous and error-prone covariate values, and is broadly applicable to asynchronous longitudinal regression with time-invariant or time-varying coefficients. For regression with time-invariant coefficients, our estimator is asymptotically unbiased, root-n consistent, and asymptotically normal; for time-varying coefficient models, our estimator has the optimal varying coefficient model convergence rate with inflated asymptotic variance from the calibration. In both cases, our estimators present asymptotic properties superior to the existing methods. The feasibility and usability of the proposed methods are verified by simulations and an application to the Study of Women's Health Across the Nation, a large-scale multi-site longitudinal study on women's health during mid-life.

Most work in neural networks focuses on estimating the conditional mean of a continuous response variable given a set of covariates.In this article, we consider estimating the conditional distribution function using neural networks for both censored and uncensored data. The algorithm is built upon the data structure particularly constructed for the Cox regression with time-dependent covariates. Without imposing any model assumption, we consider a loss function that is based on the full likelihood where the conditional hazard function is the only unknown nonparametric parameter, for which unconstraint optimization methods can be applied. Through simulation studies, we show the proposed method possesses desirable performance, whereas the partial likelihood method and the traditional neural networks with $L_2$ loss yield biased estimates when model assumptions are violated. We further illustrate the proposed method with several real-world data sets. The implementation of the proposed methods is made available at https://github.com/bingqing0729/NNCDE.

Survival data with time-varying covariates are common in practice. However, the traditional survival forests - conditional inference forest, relative risk forest and random survival forest - have accommodated only time-invariant covariates. Similarly, the recently proposed transformation forest, which incorporates the split statistics suitable for non-proportional hazard settings, has employed only time-invariant covariates. We generalize the conditional inference and relative risk forests to allow time-varying covariates. We compare their performance with that of the Cox model and transformation forest, adapted to accommodate time-varying covariates, through a comprehensive simulation study in which the Kaplan-Meier estimate serves as a benchmark. In general, the performance of the two proposed forests substantially improves over the Kaplan-Meier estimate when the estimation conditions become more favorable. Taking into an account all other factors, under the PH setting, the best method is always one of the two proposed forests, while under the non-PH setting, it is the adapted transformation forest. The K-fold cross-validation can be an effective tool to choose between the methods in practice. Finally, the performance of the proposed forest methods for time-invariant covariate data is broadly similar to that found for time-varying covariate data. We also propose a general framework for estimation of a survival function in the presence of time-varying covariates, which can be applied to any method that uses the counting process (pseudo-subject) approach to handling time-varying covariates. This novel estimate of a single survival function takes multiple survival estimation outputs corresponding to each pseudo-subject, and combines them in a theoretically-justified way to form a proper monotone-decreasing survival function estimate.