Robust risk prediction with biomarkers under two‐phase stratified cohort design

Identification of novel biomarkers for risk prediction is important for disease prevention and optimal treatment selection. However, studies aiming to discover which biomarkers are useful for risk prediction often require the use of stored biological samples from large assembled cohorts, and thus the depletion of a finite and precious resource. To make efficient use of such stored samples, two‐phase sampling designs are often adopted as resource‐efficient sampling strategies, especially when the outcome of interest is rare. Existing methods for analyzing data from two‐phase studies focus primarily on single marker analysis or fitting the Cox regression model to combine information from multiple markers. However, the Cox model may not fit the data well. Under model misspecification, the composite score derived from the Cox model may not perform well in predicting the outcome. Under a general two‐phase stratified cohort sampling design, we present a novel approach to combining multiple markers to optimize prediction by fitting a flexible nonparametric transformation model. Using inverse probability weighting to account for the outcome‐dependent sampling, we propose to estimate the model parameters by maximizing an objective function which can be interpreted as a weighted C‐statistic for survival outcomes. Regardless of model adequacy, the proposed procedure yields a sensible composite risk score for prediction. A major obstacle for making inference under two phase studies is due to the correlation induced by the finite population sampling, which prevents standard inference procedures such as the bootstrap from being used for variance estimation. We propose a resampling procedure to derive valid confidence intervals for the model parameters and the C‐statistic accuracy measure. We illustrate the new methods with simulation studies and an analysis of a two‐phase study of high‐density lipoprotein cholesterol (HDL‐C) subtypes for predicting the risk of coronary heart disease.