False Rejection Rate Type 1 Error
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Introductory Business Statistics is designed to meet the scope and sequence requirements of the one-semester statistics course for business, economics, and related majors. Core statistical concepts and skills have been augmented with practical business examples, scenarios, and exercises. The result is a meaningful understanding of the discipline, which will serve students in their business careers and real-world experiences.
Multiple hypothesis testing is concerned with appropriately controlling the rate of false positives when testing a large number of hypotheses simultaneously, while maintaining the power of each test as much as possible. For testing multiple null hypotheses, the classical approach to dealing with the multiplicity problem is to restrict attention to procedures that control the familywise error rate (FWER), the probability of even one false rejection. However, quite often, especially when a large number of hypotheses are simultaneously tested, the notion of FWER turns out to be too stringent, allowing little chance to detect many false null hypotheses. Therefore, researchers have focused in the last decade on defining alternative less stringent error rates and developing methods that control them. The false discovery rate (FDR), the expected proportion of falsely rejected null hypotheses, due to Benjamini and Hochberg (1995), is the first of these alternative error rates that has received considerable attention. Recently, the ideas of controlling the probabilities of falsely rejecting at least k null hypotheses, which is the k-FWER, and the false discovery proportion (FDP) exceeding a certain threshold y have been introduced as alternatives to the FWER and methods controlling these new error rates have been suggested. Very recently, following the idea similar to that of the k-FWER, Sarkar (2006) generalized the FDR to the k-FDR, the expected ratio of k or more false rejections to the total number of rejections, which is a less conservative notion of error rate than the FDR and k-FWER. In this work, we develop multiple testing theory and methods for controlling the new type I error rates. Specifically, it consists of four parts: (1) We develop a new stepdown FDR controlling procedure under no assumption on dependency of the underlying p-values, which has much smaller critical constants than that of the existing Benjamini-Yekutieli stepup procedure; (2) We develop new k-FWER and FDP stepdown procedures under the assumption of independence, which are much more powerful than the existing k-FWER and FDP procedures and show that under certain condition, the k-FWER stepdown procedure is unimprovable; (3) We offer a unified approach for construction of k-FWER controlling procedures by generalizing the closure principle in the context of the FWER to the case of the k-FWER; (4) We develop new Benjamini-Hochberg type k-FDR stepup and stepdown procedures in different settings and apply them to one real microarray data analysis.
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