Online False Discovery Rate Control for LORD & SAFFRON Under Positive, Local Dependence

TL;DR

This paper extends the theoretical guarantees of online FDR control methods, specifically LORD and SAFFRON, to hold under local dependence conditions, broadening their applicability beyond independence assumptions.

Contribution

It proves that LORD and SAFFRON control the FDR under local dependence, not just independence, and characterizes the limitations of the superuniformity assumption.

Findings

FDR control under local dependence for LORD and SAFFRON

Applicable to adaptive stopping rules such as rejection count thresholds

Implications for alpha investing and other online testing procedures

Abstract

Online testing procedures assume that hypotheses are observed in sequence, and allow the significance thresholds for upcoming tests to depend on the test statistics observed so far. Some of the most popular online methods include alpha investing, LORD++ (hereafter, LORD), and SAFFRON. These three methods have been shown to provide online control of the "modified" false discovery rate (mFDR) under a condition known as conditional superuniformity. However, to our knowledge, LORD & SAFFRON have only been shown to control the traditional false discovery rate (FDR) under an independence condition on the test statistics. Our work bolsters these results by showing that SAFFRON and LORD additionally ensure online control of the FDR under a "local" form of nonnegative dependence. Further, FDR control is maintained under certain types of adaptive stopping rules, such as stopping after a certain number of rejections have been observed. Because alpha investing can be recovered as a special case of the SAFFRON framework, our results immediately apply to alpha investing as well. In the process of deriving these results, we also formally characterize how the conditional superuniformity assumption implicitly limits the allowed p-value dependencies. This implicit limitation is important not only to our proposed FDR result, but also to many existing mFDR results.