The Annals of Statistics

Online rules for control of false discovery rate and false discovery exceedance

Adel Javanmard and Andrea Montanari

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Abstract

Multiple hypothesis testing is a core problem in statistical inference and arises in almost every scientific field. Given a set of null hypotheses $\mathcal{H}(n)=(H_{1},\ldots,H_{n})$, Benjamini and Hochberg [J. R. Stat. Soc. Ser. B. Stat. Methodol. 57 (1995) 289–300] introduced the false discovery rate ($\mathrm{FDR}$), which is the expected proportion of false positives among rejected null hypotheses, and proposed a testing procedure that controls $\mathrm{FDR}$ below a pre-assigned significance level. Nowadays $\mathrm{FDR}$ is the criterion of choice for large-scale multiple hypothesis testing.

In this paper we consider the problem of controlling $\mathrm{FDR}$ in an online manner. Concretely, we consider an ordered—possibly infinite—sequence of null hypotheses $\mathcal{H}=(H_{1},H_{2},H_{3},\ldots)$ where, at each step $i$, the statistician must decide whether to reject hypothesis $H_{i}$ having access only to the previous decisions. This model was introduced by Foster and Stine [J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 (2008) 429–444].

We study a class of generalized alpha investing procedures, first introduced by Aharoni and Rosset [J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 (2014) 771–794]. We prove that any rule in this class controls online $\mathrm{FDR}$, provided $p$-values corresponding to true nulls are independent from the other $p$-values. Earlier work only established $\mathrm{mFDR}$ control. Next, we obtain conditions under which generalized alpha investing controls $\mathrm{FDR}$ in the presence of general $p$-values dependencies. We also develop a modified set of procedures that allow to control the false discovery exceedance (the tail of the proportion of false discoveries). Finally, we evaluate the performance of online procedures on both synthetic and real data, comparing them with offline approaches, such as adaptive Benjamini–Hochberg.

Article information

Source
Ann. Statist., Volume 46, Number 2 (2018), 526-554.

Dates
Received: March 2016
Revised: January 2017
First available in Project Euclid: 3 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1522742428

Digital Object Identifier
doi:10.1214/17-AOS1559

Mathematical Reviews number (MathSciNet)
MR3782376

Zentralblatt MATH identifier
06870271

Subjects
Primary: 62F03: Hypothesis testing 62F05: Asymptotic properties of tests
Secondary: 62L99: None of the above, but in this section

Keywords
Hypothesis testing false discovery rate (FDR) false discovery exceedance (FDX) online decision making

Citation

Javanmard, Adel; Montanari, Andrea. Online rules for control of false discovery rate and false discovery exceedance. Ann. Statist. 46 (2018), no. 2, 526--554. doi:10.1214/17-AOS1559. https://projecteuclid.org/euclid.aos/1522742428


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Supplemental materials

  • Online rules for control of false discovery rate and false discovery exceedance. Due to space constraints, proof of theorems and some of the technical details as well as additional numerical studies are provided in the Supplementary Material [22].