The Net Reclassification Index (NRI) introduced by Pencina and colleagues [1, 2] is designed to quantify the prediction increment provided by a new biomarker. It has become popular for evaluating and selecting novel markers. The published variance formulae for NRI statistics do not account for the fact that risks are estimated based on risk models fit to data, and thus are not valid in practice when estimated risks are used [3]. Kerr and colleagues [4] showed that the confidence intervals constructed based on a bootstrap estimate of the variance and Normal approximation had the best performance among various methods they examined, including the one based on bootstrap quantiles. This paper establishes asymptotic Normality of NRI statistics when true risks are unknown and are estimated. Our results provide theoretical support for constructing confidence intervals for NRI statistics based on a Normal approximation. We also derive explicit variance formulae for NRI statistics that are calculated based on estimated risks. In addition, we examine finite sample distributional behavior of NRI statistics in a simulation study. These results provide some guidance on the sample size required for adopting a Normal approximation for NRI inference in practice.



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