Smoothed Nested Testing on Directed Acyclic Graphs

TL;DR

This paper introduces a framework for multiple hypothesis testing on nested structures modeled as directed acyclic graphs, employing smoothing techniques to improve power while controlling error rates.

Contribution

It proposes a general smoothing framework for nested hypothesis testing on DAGs, ensuring error control and demonstrating power gains in simulations and real data.

Findings

Smoothing improves testing power significantly.

Framework controls error rates under independence and positive dependence.

Application to biology data shows practical effectiveness.

Abstract

We consider the problem of multiple hypothesis testing when there is a logical nested structure to the hypotheses. When one hypothesis is nested inside another, the outer hypothesis must be false if the inner hypothesis is false. We model the nested structure as a directed acyclic graph, including chain and tree graphs as special cases. Each node in the graph is a hypothesis and rejecting a node requires also rejecting all of its ancestors. We propose a general framework for adjusting node-level test statistics using the known logical constraints. Within this framework, we study a smoothing procedure that combines each node with all of its descendants to form a more powerful statistic. We prove a broad class of smoothing strategies can be used with existing selection procedures to control the familywise error rate, false discovery exceedance rate, or false discovery rate, so long as the original test statistics are independent under the null. When the null statistics are not independent but are derived from positively-correlated normal observations, we prove control for all three error rates when the smoothing method is arithmetic averaging of the observations. Simulations and an application to a real biology dataset demonstrate that smoothing leads to substantial power gains.