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J Am Stat Assoc. 2018;113(522):882-892. doi: 10.1080/01621459.2017.1295866. Epub 2017 Feb 28.

Nonparametric Maximum Likelihood Estimators of Time-Dependent Accuracy Measures for Survival Outcome Under Two-Stage Sampling Designs.

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Department of Biostatistics, Vanderbilt University Medical Center, Nashville, TN 37232.
Department of Biostatistics, Harvard School of Public Health, Boston, Massachusetts 02115.
Division of Gastroenterology, University of Michigan, Ann Arbor, MI 48109.
Public Health Sciences Division, Fred Hutchinson Cancer Research Center, Seattle, WA 98109.


Large prospective cohort studies of rare chronic diseases require thoughtful planning of study designs, especially for biomarker studies when measurements are based on stored tissue or blood specimens. Two-phase designs, including nested case-control (Thomas, 1977) and case-cohort (Prentice, 1986) sampling designs, provide cost-effective strategies for conducting biomarker evaluation studies. Existing literature for biomarker assessment under two-phase designs largely focuses on simple inverse probability weighting (IPW) estimators (Cai and Zheng, 2011; Liu et al., 2012). Drawing on recent theoretical development on the maximum likelihood estimators for relative risk parameters in two-phase studies (Scheike and Martinussen, 2004; Zeng et al., 2006), we propose nonparametric maximum likelihood based estimators to evaluate the accuracy and predictiveness of a risk prediction biomarker under both types of two-phase designs. In addition, hybrid estimators that combine IPW estimators and maximum likelihood estimation procedure are proposed to improve efficiency and alleviate computational burden. We derive large sample properties of proposed estimators and evaluate their finite sample performance using numerical studies. We illustrate new procedures using a two-phase biomarker study aiming to evaluate the accuracy of a novel biomarker, des-γ-carboxy prothrombin, for early detection of hepatocellular carcinoma (Lok et al., 2010).


Case-cohort sampling; Negative predictive value; Nested case-control sampling; Nonparametric maximum likelihood estimator; Positive predictive value; Receiver Operating Characteristics Curve (ROC curve); Two-phase study

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