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We discuss several approaches to defining power in studies designed around the Benjamini-Hochberg (BH) false discovery rate (FDR) procedure. We focus primarily on the \textit{average power} and the $\lambda$-\textit{power}, which are the expected true positive fraction and the probability that the true positive fraction exceeds $\lambda$, respectively. We prove results concerning strong consistency and asymptotic normality for the positive call fraction (PCF), the true positive fraction (TPF) and false discovery fraction (FDF). Convergence of their corresponding expected values, including a convergence result for the average power, follow as a corollaries. After reviewing what is known about convergence in distribution of the errors of the plugin procedure, (Genovese, 2004), we prove central limit theorems for fully empirical versions of the PCF, TPF, and FDF, using a result for stopped stochastic processes. The central limit theorem (CLT) for the TPF is used to obtain an approximate expression for the $\lambda$-power, while the CLT for the FDF is used to introduce an approximate procedure for determining a suitably small nominal FDR that results in a speicified bound on the FDF with stipulated high probability. The paper also contains the results of a large simulation study covering a fairly substantial portion of the space of possible inputs encountered in application of the results in the design of a biomarker study, a micro-array experiment and a GWAS study., Comment: 49 pages, 2 figures, 9 tables

Designed gene expression micro-array experiments, consisting of several treatment levels with a number of replicates per level, are analyzed by applying simple tests for group differences at the per gene level. The gene level statistics are sorted and a criterion for selecting important genes which takes into account multiplicity is applied. A caveat arises in that true signals (genes truly over or under expressed) are "competing" with fairly large type I error signals. False positives near the top of a sorted list can occur when genes having very small fold-change are compensated by small enough variance to yield a large test statistic. One of the first attempts around this caveat as the development of "significance analysis of micro-arrays (SAM)", which used a modified t-type statistic thresholded against its permutation distribution. The key innovation of the modified t-statistic was the addition of a constant to the per gene standard errors in order to stabilize the coefficient of variation of the resulting test statistic. Since then, several authors have proposed the use of shrinkage variance estimators in conjunction with t-type, and more generally, ANOVA type tests at the gene level. Our new approach proposes the use of a shrinkage variance Hotelling T-squared statistic in which the per gene sample covariance matrix is replaced by a shrinkage estimate borrowing strength from across all genes. It is demonstrated that the new statistic retains the F-distribution under the null, with added degrees of freedom in the denominator. Advantages of this class of tests are (i) flexibility in that a whole family of hypothesis tests is possible (ii) the gains of the above-mentioned earlier innovations are enjoyed more fully. This paper summarizes our results and presents a simulation study benchmarking the new statistic against another recently proposed statistic.

In this paper we consider a group sequentially monitored trial on a survival endpoint, monitored using a weighted log-rank (WLR) statistic with deterministic weight function. We introduce a summary statistic in the form of a weighted average logged relative risk and show that if there is no sign change in the instantaneous logged relative risk, there always exists a bijection between the WLR statistic and the weighted average logged relative risk. We show that this bijection can be consistently estimated at each analysis under a suitable shape assumption, for which we have listed two possibilities. We indicate how to derive a design-adjusted p-value and confidence interval and suggest how to apply the bias-correction method. Finally, we document several decisions made in the design of the NLST interim analysis plan and in reporting its results on the primary endpoint.