The Annals of Statistics

Marginal asymptotics for the “large $p$, small $n$” paradigm: With applications to microarray data

Michael R. Kosorok and Shuangge Ma

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The “large $p$, small $n$” paradigm arises in microarray studies, image analysis, high throughput molecular screening, astronomy, and in many other high dimensional applications. False discovery rate (FDR) methods are useful for resolving the accompanying multiple testing problems. In cDNA microarray studies, for example, $p$-values may be computed for each of $p$ genes using data from $n$ arrays, where typically $p$ is in the thousands and $n$ is less than 30. For FDR methods to be valid in identifying differentially expressed genes, the $p$-values for the nondifferentially expressed genes must simultaneously have uniform distributions marginally. While feasible for permutation $p$-values, this uniformity is problematic for asymptotic based $p$-values since the number of $p$-values involved goes to infinity and intuition suggests that at least some of the $p$-values should behave erratically. We examine this neglected issue when $n$ is moderately large but $p$ is almost exponentially large relative to $n$. We show the somewhat surprising result that, under very general dependence structures and for both mean and median tests, the $p$-values are simultaneously valid. A small simulation study and data analysis are used for illustration.

Article information

Ann. Statist., Volume 35, Number 4 (2007), 1456-1486.

First available in Project Euclid: 29 August 2007

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Zentralblatt MATH identifier

Primary: 62A01: Foundations and philosophical topics 62H15: Hypothesis testing
Secondary: 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions

Brownian bridge Brownian motion empirical process false discovery rate Hungarian construction marginal asymptotics maximal inequalities median tests microarrays t-tests


Kosorok, Michael R.; Ma, Shuangge. Marginal asymptotics for the “large $p$, small $n$” paradigm: With applications to microarray data. Ann. Statist. 35 (2007), no. 4, 1456--1486. doi:10.1214/009053606000001433.

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