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There are a number of available statistical methods for adaptive designs, among which the combination method of Bauer and Kohne's (1994) is well known and widely used. In this work, we revisit the the Bauer-Kohne method in three ways: overall FWER control for single-hypothesis in a two-stage adaptive design, overall FWER control for two-hypothesis in a two-stage adaptive design, and overall FDR control for multiple-hypothesis in a two-stage adaptive design. We first take the Bauer-Kohne method in a more direct manner to have more flexibility in the choice of the early rejection and acceptance boundaries as well as the second stage critical value based on the chosen combination function. Our goal is not to develop a new method, but focus primarily on developing a comprehensive understanding of two-stage designs. Rather than tying up the early rejection and acceptance boundaries by considering the second stage critical value to be the same as that of the level á combination test, as done in the original Bauer-Kohne method, we allow the second-stage critical value to be determined from prefixed early rejection and acceptance boundaries. An explicit formula is derived for the overall Type I error probability to determine the second stage critical value from these stopping boundaries not only for Fisher's combination function but also for other types of combination function. Tables of critical values corresponding to several different choices of early rejection and acceptance boundaries and these combination functions are presented. A dataset from a clinical study is used to apply the different methods based on directly computed second stage critical values from pre fixed stopping boundaries and discuss the outcomes in relation to those produced by the original Bauer-Kohne method. We then extend the Bauer-Kohne method to two-hypothesis setting and propose a stepwise-combination method for a two-stage adaptive design. In particular, we modify Holm's step-down procedure (1979) and suggest a step-down combination method to control the overall FWER at a desired level á. In many scientific studies requiring simultaneous testing of multiple null hypotheses, it is often necessary to carry out the multiple testing in two stages to decide which of the hypotheses can be rejected or accepted at the first stage and which should be followed up for further testing having combined their p-values from both stages. Unfortunately, no multiple testing procedure is available yet to perform this task meeting pre-specified boundaries on the first-stage p-values in terms of the false discovery rate (FDR) and maintaining a control over the overall FDR at a desired level. Our third goal in this work is to present two procedures, extending the classical Benjamini-Hochberg (BH) procedure and its adaptive version incorporating an estimate of the number of true null hypotheses from single-stage to a two-stage setting. These procedures are theoretically proved to control the overall FDR when the pairs of first- and second-stage p-values are independent and those corresponding to the null hypotheses are identically distributed as a pair (p1, p2) satisfying the p-clud property of Brannath, Posch and Bauer (2002, Journal of the American Statistical Association, 97, 236 -244). We consider two types of combination function, Fisher's and Simes', and present explicit formulas involving these functions towards carrying out the proposed procedures based on pre-determined critical values or through estimated FDR's. Simulations were carried to compare the proposed methods with class BH procedure using first stage data only and full data from both stages respectively. Our simulation studies indicate that the proposed procedures can have significant power improvement over the single-stage BH procedure based on the first stage data, at least under independence, and can continue to control the FDR under some dependence situations. Application of the proposed procedures to a real gene expression data set produces more discoveries compared to the single-stage BH procedure using the first stage data and full data as well. / Statistics

The false discovery rate (FDR) introduced by Benjamini and Hochberg (1995) is perhaps the most standard error controlling measure being used in a wide variety of applications involving multiple hypothesis testing. There are two approaches to control the FDR - the fixed error rate approach of Benjamini and Hochberg (BH, 1995) where a rejection region is determined with the FDR below a fixed level and the estimation based approach of Storey (2002) where the FDR is estimated for a fixed rejection region before it is controlled. In this proposal, we concentrate on both these approaches and propose new, improved versions of some FDR controlling procedures available in the literature. A number of adaptive procedures have been put forward in the literature, each attempting to improve the method of Benjamini and Hochberg (1995), the BH method, by incorporating into this method an estimate of number true null hypotheses. Among these, the method of Benjamini, Krieger and Yekutieli (2006), the BKY method, has been receiving lots of attention recently. In this proposal, a variant of the BKY method is proposed by considering a different estimate of number true null hypotheses, which often outperforms the BKY method in terms of the FDR control and power. Storey's (2002) estimation based approach to controlling the FDR has been developed from a class of conservatively biased point estimates of the FDR under a mixture model for the underlying p-values and a fixed rejection threshold for each null hypothesis. An alternative class of point estimates of the FDR with uniformly smaller conservative bias is proposed under the same setup. Numerical evidence is provided to show that the mean squared error (MSE) is also often smaller for this new class of estimates. Compared to Storey's (2002), the present class provides a more powerful estimation based approach to controlling the FDR. / Statistics