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The concept of k-FWER has received much attention lately as an appropriate error rate for multiple testing when one seeks to control at least k false rejections, for some fixed k ≥ 1. A less conservative notion, the k-FDR, has been introduced very recently by Sarkar [Ann. Statist. 34 (2006) 394-415], generalizing the false discovery rate of Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300]. In this article, we bring newer insight to the k-FDR considering a mixture model involving independent p-values before motivating the developments of some new procedures that control it. We prove the k-FDR control of the proposed methods under a slightly weaker condition than in the mixture model. We provide numerical evidence of the proposed methods' superior power performance over some k-FWER and k-FDR methods. Finally, we apply our methods to a real data set.

We propose a method for the statistical analysis of fMRI data that tests cluster units rather than voxel units for activation. The advantages of this analysis over previous ones are both conceptual and statistical. Recognizing that the fundamental units of interest are the spatially contiguous clusters of voxels that are activated together, we set out to approximate these cluster units from the data by a clustering algorithm especially tailored for fMRI data. Testing the cluster units has a two-fold statistical advantage over testing each voxel separately: the signal to noise ratio within the unit tested is higher, and the number of hypotheses tests compared is smaller. We suggest controlling FDR on clusters, i.e., the proportion of clusters rejected erroneously out of all clusters rejected and explain the meaning of controlling this error rate. We introduce the powerful adaptive procedure to control the FDR on clusters. We apply our cluster-based analysis (CBA) to both an event-related and a block design fMRI vision experiment and demonstrate its increased power over voxel-by-voxel analysis in these examples as well as in simulations.

Improved procedures, in terms of smaller missed discovery rates (MDR), for performing multiple hypotheses testing with weak and strong control of the family-wise error rate (FWER) or the false discovery rate (FDR) are developed and studied. The improvement over existing procedures such as the Šidák procedure for FWER control and the Benjamini—Hochberg (BH) procedure for FDR control is achieved by exploiting possible differences in the powers of the individual tests. Results signal the need to take into account the powers of the individual tests and to have multiple hypotheses decision functions which are not limited to simply using the individual p-values, as is the case, for example, with the Šidák, Bonferroni, or BH procedures. They also enhance understanding of the role of the powers of individual tests, or more precisely the receiver operating characteristic (ROC) functions of decision processes, in the search for better multiple hypotheses testing procedures. A decision-theoretic framework is utilized, and through auxiliary randomizers the procedures could be used with discrete or mixed-type data or with rank-based nonparametric tests. This is in contrast to existing p-value based procedures whose theoretical validity is contingent on each of these p-value statistics being stochastically equal to or greater than a standard uniform variable under the null hypothesis. Proposed procedures are relevant in the analysis of high-dimensional \"large M, small n\" data sets arising in the natural, physical, medical, economic and social sciences, whose generation and creation is accelerated by advances in high-throughput technology, notably, but not limited to, microarray technology.

Performance evaluation of any classification method is fundamental to its acceptance in practice. Evaluation should consider the dependence of a classifier's accuracy on relevant covariates in addition to its overall accuracy. When developing a classifier with a continuous output that allocates units into one of two groups, receiver operating characteristic (ROC) curve analysis is appropriate. The partial area under the ROC curve (pAUC) is a summary measure of the ROC curve used to make statistical inference when only a region of the ROC space is of interest. We propose a new pAUC regression method to evaluate covariate effects on the diagnostic accuracy. We provide asymptotic distribution theory and inference procedures that allow for correlated observations. Graphical methods and goodness-of-fit statistics for model checking are also developed. Simulation studies demonstrate that the large-sample theory provides reasonable inference in small samples and the new estimator is considerably more efficient than the estimator proposed by Dodd and Pepe (2003a). Application to an analysis of prostate-specific antigen (PSA), a biomarker for early detection of prostate cancer, demonstrates the utility of the method in practice.