A novel approach to estimate the Cox model with temporal covariates and its application to medical cost data

TL;DR

This paper introduces a new method to estimate the Cox model with time-dependent covariates, modeling the covariates as a bounded variational process linked to an ODE, useful for medical datasets with limited longitudinal data.

Contribution

The paper presents a novel approach that models temporal covariates via a bounded variational process using ODEs, requiring minimal longitudinal observations for Cox model estimation.

Findings

The estimators are consistent and asymptotically normal.

Variable selection with Adaptive LASSO is feasible and has the oracle property.

The method performs well in simulations and real data applications.

Abstract

We propose a novel approach to estimate the Cox model with temporal covariates. Our new approach treats the temporal covariates as arising from a longitudinal process which is modeled jointly with the event time. Different from the literature, the longitudinal process in our model is specified as a bounded variational process and determined by a family of Initial Value Problems associated with an Ordinary Differential Equation. Our specification has the advantage that only the observation of the temporal covariates at the time to event and the time to event itself are required to fit the model, while it is fine but not necessary to have more longitudinal observations. This fact makes our approach very useful for many medical outcome datasets, like the New York State Statewide Planning and Research Cooperative System and the National Inpatient Sample, where it is important to find the hazard rate of being discharged given the accumulative cost but only the total cost at the discharge time is available due to the protection of patient information. Our estimation procedure is based on maximizing the full information likelihood function. The resulting estimators are shown to be consistent and asymptotically normally distributed. Variable selection techniques, like Adaptive LASSO, can be easily modified and incorporated into our estimation procedure. The oracle property is verified for the resulting estimator of the regression coefficients. Simulations and a real example illustrate the practical utility of the proposed model. Finally, a couple of potential extensions of our approach are discussed.