Etienne Roquain

Optimal weighting for false discovery rate control

How to weigh the Benjamini-Hochberg procedure? In the con- text of multiple hypothesis testing, we propose a new step-wise procedure that controls the false discovery rate (FDR) and we prove it to be more powerful than any weighted Benjamini-Hochberg procedure. Both finite- sample and asymptotic results are presented. Moreover, we illustrate good performance of our procedure in simulations and a genomics application. This work is particularly useful in the case of heterogeneous p-value distri- butions.

Some nonasymptotic results on resampling in high dimension, I: Confidence regions

We study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure. The random vector is supposed to be either Gaussian or to have a symmetric and bounded distribution. The dimensionality of the vector can possibly be much larger than the number of observations and we focus on a non-asymptotic control of the confidence level, following ideas inspired by recent results in learning theory. We consider two approaches, the first based on a concentration principle (valid for a large class of resampling weights) and the second on a direct resampled quantile, specifically using Rademacher weights. Several intermediate results established in the approach based on concentration principles are of self-interest. We also discuss the question of accuracy when using Monte-Carlo approximations of the resampled quantities. We present an application of these results to the one-sided and two-sided multiple testing problem, in which we derive several resampling-based step-down procedures providing a non-asymptotic FWER control. We compare our different procedures in a simulation study, and we show that they can outperform Bonferronis or Holms procedures as soon as the observed vector has sufficiently correlated coordinates.

SOME NONASYMPTOTIC RESULTS ON RESAMPLING IN HIGH DIMENSION, II: MULTIPLE TESTS

In the context of correlated multiple tests, we aim to nonasymptotically control the family-wise error rate (FWER) using resampling-type procedures. We observe repeated realizations of a Gaussian random vector in possibly high dimension and with an unknown covariance matrix, and consider the one- and two-sided multiple testing problem for the mean values of its coordinates. We address this problem by using the confidence regions developed in the companion paper [Ann. Statist. (2009), to appear], which lead directly to single-step procedures; these can then be improved using step-down algorithms, following an established general methodology laid down by Romano and Wolf [J. Amer. Statist. Assoc. 100 (2005) 94-108]. This gives rise to several different procedures, whose performances are compared using simulated data.

Spatial clustering of array CGH features in combination with hierarchical multiple testing.

We propose a new approach for clustering DNA features using array CGH data from multiple tumor samples. We distinguish data-collapsing (joining contiguous DNA clones or probes with extremely similar data into regions) from clustering (joining contiguous, correlated regions based on a maximum likelihood principle). The model-based clustering algorithm accounts for the apparent spatial patterns in the data. We evaluate the randomness of the clustering result by a cluster stability score in combination with cross-validation. Moreover, we argue that the clustering really captures spatial genomic dependency by showing that coincidental clustering of independent regions is very unlikely.Using the region and cluster information, we combine testing of these for association with a clinical variable in a hierarchical multiple testing approach. This allows for interpreting the significance of both regions and clusters while controlling the Family-Wise Error Rate simultaneously. We prove that in the context of permutation tests and permutation-invariant clusters it is allowed to perform clustering and testing on the same data set. Our procedures are illustrated on two cancer data sets.

Testing over a continuum of null hypotheses with False Discovery Rate control

We consider statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses, under the assumption that a suitable single test (and corresponding

Resampling-based confidence regions and multiple tests for a correlated random vector

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Continuous testing for Poisson process intensities: a new perspective on scanning statistics

-value) is known for each individual hypothesis. We extend to this setting the notion of false discovery rate (FDR) as a measure of type I error. Our main result studies specific procedures based on the observation of the

Adaptive False Discovery Rate Control under Independence and Dependence

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Two simple sufficient conditions for FDR control

-value process. Control of the FDR at a nominal level is ensured either under arbitrary dependence of

Exact calculations for false discovery proportion with application to least favorable configurations

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