Sanat K. Sarkar

The Simes Method for Multiple Hypothesis Testing with Positively Dependent Test Statistics

Abstract The Simes method for testing intersection of more than two hypotheses is known to control the probability of type I error only when the underlying test statistics are independent. Although this method is more powerful than the classical Bonferroni method, it is not known whether it is conservative when the test statistics are dependent. This article proves that for multivariate distributions exhibiting a type of positive dependence that arise in many multiple-hypothesis testing situations, the Simes method indeed controls the probability of type I error. This extends some results established very recently in the special case of two hypotheses.

A Continuous Bivariate Exponential Distribution

Abstract Among many bivariate exponential distributions proposed in the statistical literature, the BVE of Marshall and Olkin (1967) is widely accepted. An important property of the BVE is that if X and Y are BVE, then X = Y with positive probability, which makes the BVE inappropriate in many practical situations. For example, in certain types of diseases where the occurrence of simultaneous failure of a pair of organs is rare, an appropriate bivariate exponential distribution to describe the failure of these paired organs should be one that is absolutely continuous. The BVE, however, possesses some other properties that have useful physical interpretations. Therefore, the derivation of an absolutely continuous bivariate exponential distribution retaining the desirable properties of the BVE appears to be a worthwhile objective. To this end, we consider a characterization property of the BVE and modify it suitably to derive an absolutely continuous bivariate distribution. The important characterization pro...

False discovery and false nondiscovery rates in single-step multiple testing procedures

Results on the false discovery rate (FDR) and the false nondiscovery rate (FNR) are developed for single-step multiple testing procedures. In addition to verifying desirable properties of FDR and FNR as measures of error rates, these results extend previously known results, providing further insights, particularly under dependence, into the notions of FDR and FNR and related measures. First, considering fixed configurations of true and false null hypotheses, inequalities are obtained to explain how an FDR- or FNR-controlling single-step procedure, such as a Bonferroni or Sidak procedure, can potentially be improved. Two families of procedures are then constructed, one that modifies the FDR-controlling and the other that modifies the FNR-controlling Sidak procedure. These are proved to control FDR or FNR under independence less conservatively than the corresponding families that modify the FDR- or FNR-controlling Bonferroni procedure. Results of numerical investigations of the performance of the modified Sidak FDR procedure over its competitors are presented. Second, considering a mixture model where different configurations of true and false null hypotheses are assumed to have certain probabilities, results are also derived that extend some of Storeys work to the dependence case.

Stepup procedures controlling generalized FWER and generalized FDR

In many applications of multiple hypothesis testing where more than one false rejection can be tolerated, procedures controlling error rates measuring at least k false rejections, instead of at least one, for some fixed k ≥ 1 can potentially increase the ability of a procedure to detect false null hypotheses. The k-FWER, a generalized version of the usual familywise error rate (FWER), is such an error rate that has recently been introduced in the literature and procedures controlling it have been proposed. A further generalization of a result on the k-FWER is provided in this article. In addition, an alternative and less conservative notion of error rate, the k-FDR, is introduced in the same spirit as the k-FWER by generalizing the usual false discovery rate (FDR). A k-FWER procedure is constructed given any set of increasing constants by utilizing the kth order joint null distributions of the p-values without assuming any specific form of dependence among all the p-values. Procedures controlling the k-FDR are also developed by using the kth order joint null distributions of the p-values, first assuming that the sets of null and nonnull p-values are mutually independent or they are jointly positively dependent in the sense of being multivariate totally positive of order two (MTP2) and then discarding that assumption about the overall dependence among the p-values.

Controlling False Discoveries in Multidimensional Directional Decisions, with Applications to Gene Expression Data on Ordered Categories

Microarray gene expression studies over ordered categories are routinely conducted to gain insights into biological functions of genes and the underlying biological processes. Some common experiments are time-course/dose-response experiments where a tissue or cell line is exposed to different doses and/or durations of time to a chemical. A goal of such studies is to identify gene expression patterns/profiles over the ordered categories. This problem can be formulated as a multiple testing problem where for each gene the null hypothesis of no difference between the successive mean gene expressions is tested and further directional decisions are made if it is rejected. Much of the existing multiple testing procedures are devised for controlling the usual false discovery rate (FDR) rather than the mixed directional FDR (mdFDR), the expected proportion of Type I and directional errors among all rejections. Benjamini and Yekutieli (2005, Journal of the American Statistical Association 100, 71-93) proved that an augmentation of the usual Benjamini-Hochberg (BH) procedure can control the mdFDR while testing simple null hypotheses against two-sided alternatives in terms of one-dimensional parameters. In this article, we consider the problem of controlling the mdFDR involving multidimensional parameters. To deal with this problem, we develop a procedure extending that of Benjamini and Yekutieli based on the Bonferroni test for each gene. A proof is given for its mdFDR control when the underlying test statistics are independent across the genes. The results of a simulation study evaluating its performance under independence as well as under dependence of the underlying test statistics across the genes relative to other relevant procedures are reported. Finally, the proposed methodology is applied to a time-course microarray data obtained by Lobenhofer et al. (2002, Molecular Endocrinology 16, 1215-1229). We identified several important cell-cycle genes, such as DNA replication/repair gene MCM4 and replication factor subunit C2, which were not identified by the previous analyses of the same data by Lobenhofer et al. (2002) and Peddada et al. (2003, Bioinformatics 19, 834-841). Although some of our findings overlap with previous findings, we identify several other genes that complement the results of Lobenhofer et al. (2002).

On a generalized false discovery rate

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 flxed k ‚ 1. A less conservative notion, the k-FDR, has been introduced very recently by Sarkar [19], generalizing the false discovery rate of Banjamini and Hochberg [1]. 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 kFWER and k-FDR methods. Finally, we apply our methods to a real data set.

On the Simes inequality and its generalization

The Simes inequality has received considerable attention recently because of its close connection to some important multiple hypothesis testing procedures. We revisit in this article an old result on this inequality to clarify and strengthen it and a recently proposed generalization of it to offer an alternative simpler proof.

On improving the Shortest Length Confidence interval for the Generalized Variance

A multivariate extension of Cohens (1972, J. Amer. Statist. Assoc. 67 382-387) result on interval estimation of normal variance is made in this article. Based on independent random matrices X : p - m and S : p - p distributed, respectively, as Npm([mu], [Sigma] [circle times operator] Im) and Wp(n, [Sigma]) with [mu] unknown and n >= p, the problem of obtaining confidence interval for [Sigma] is considered. The shortest length invariant confidence interval is obtained and is shown to be improved by some other interval estimators. Some new properties of the noncentral and central distributions of sample generalized variance have been proved for this purpose.

On improving the min test for the analysis of combination drug trials

In the analysis of clinical trials of combination therapies, the min test is often used to demonstrate a combination therapys superiority to its components. Although uniformly most powerful within a class of monotone tests, this test is excessively conservative with low power at certain alternatives. This paperdemonstrates that more powerful tests may be found outside of this class. Some such alternative tests are suggested and compared with the min tests on the basis of their actual significance levels and powers. The proposed tests are observed to be less conservative and uniformly more powerful than the min test.

A generalized model for the analysis of association in ordinal contingency tables

Abstract A generalized ordinal association model for two-way contingency tables is obtained by using the Box-Cox family of power transformations to scale the interaction between row and column classifications. Goodmans RC model and the rank-2 correlation model, which are the most commonly used ordinal association models, are special cases of the proposed generalized model. Several properties of the new model are derived, which lead to meaningful interpretations of the model and the association parameter. The present results also unify the similar results known for the RC model and the rank-2 correlation model.