A Bayesian Approach for the Cox Proportional Hazards Model with Covariates Subject to Detection Limit

Abstract

The research on biomarkers has been limited in its effectiveness because biomarker levels can only be measured within the thresholds of assays and laboratory instruments, a challenge referred to as a detection limit (DL) problem. In this paper, we propose a Bayesian approach to the Cox proportional hazards model with explanatory variables subject to lower, upper, or interval DLs. We demonstrate that by formulating the time-to-event outcome using the Poisson density with counting process notation, implementing the proposed approach in the OpenBUGS and JAGS is straightforward. We have conducted extensive simulations to compare the proposed Bayesian approach to the other four commonly used methods and to evaluate its robustness with respect to the distribution assumption of the biomarkers. The proposed Bayesian approach and other methods were applied to an acute lung injury study, in which a panel of cytokine biomarkers was studied for the biomarkers' association with ventilation-free survival.

Keywords: Bayesian; Biomarker; Detection limit; Lung Injury; Proportional hazards models.

Figure 1

Power Analysis for Simulation Studies.

Figure 2

Histogram Plot and QQ-plot for a Randomly Selected Dataset from Simulation III.

Figure 3

Histogram Plot of log(ICAM.1) in ALI Study Completed with Imputed Values.

Figure 4

QQ-plots for Logarithmic Transformation of Observed log(IL8) and log(ICAM-1) in ALI Study.