Optimizing effective numbers of tests by vine copula modeling

Nico Steffen,Thorsten Dickhaus

doi:10.1515/demo-2020-0010

Abstract

Abstract In the multiple testing context, we utilize vine copulae for optimizing the effective number of tests. It is well known that for the calibration of multiple tests for control of the family-wise error rate the dependencies between the marginal tests are of utmost importance. It has been shown in previous work, that positive dependencies between the marginal tests can be exploited in order to derive a relaxed Šidák-type multiplicity correction. This correction can conveniently be expressed by calculating the corresponding „effective number of tests“ for a given (global) significance level. This methodology can also be applied to blocks of test statistics so that the effective number of tests can be calculated by the sum of the effective numbers of tests for each block. In the present work, we demonstrate how the power of the multiple test can be optimized by taking blocks with high inner-block dependencies. The determination of those blocks will be performed by means of an estimated vine copula model. An algorithm is presented which uses the information of the estimated vine copula to make a data-driven choice of appropriate blocks in terms of (estimated) dependencies. Numerical experiments demonstrate the usefulness of the proposed approach.

Highlights

Dependence modeling by means of copula functions has recently received a lot of attention in multiple testing, see [6], [2], [14], [13], [15], [3], [12], and Sections 2.2.4 and 4.4 of [5]
This correction can conveniently be expressed by calculating the corresponding „e ective number of tests“ for a given signi cance level. This methodology can be applied to blocks of test statistics so that the e ective number of tests can be calculated by the sum of the e ective numbers of tests for each block
Assuming that the dependency structure among the test statistics is a nuisance parameter, we propose to t a regular vine copula model to the observed data

Summary

Introduction

Dependence modeling by means of copula functions has recently received a lot of attention in multiple testing, see [6], [2], [14], [13], [15], [3], [12], and Sections 2.2.4 and 4.4 of [5]. In [6] it has explicitly been shown that the copula approach leads to the most general construction method for single-step multiple tests under known univariate marginal null distributions of test statistics or p-values, respectively In this context, concepts of positive dependency are of particular importance. We say that the RM-valued random vector (X , ..., XM) possesses the regular vine distribution (F, V, C ), if Fi is the marginal cdf of Xi for all ≤ i ≤ M, and if Ce is the (conditional) bivariate copula of (XBe,a , XBe,b ) given XDe for each edge e = {a, b} ∈ E(V). Under the aforementioned assumptions there exists an RM-valued random vector (X , ..., XM) possessing the regular vine distribution (F, V, C ), and Feγ|De is the conditional cdf of Xeγ given XDe , where γ ∈ { , } and e ∈ E(V)

Proposed methodology

Conclusion