Multiple Testing for Exploratory Research

Motivated by the practice of exploratory research, we formulate an approach to multiple testing that reverses the conventional roles of the user and the multiple testing procedure. Traditionally, the user chooses the error criterion, and the procedure the resulting rejected set. Instead, we propose to let the user choose the rejected set freely, and to let the multiple testing procedure return a confidence statement on the number of false rejections incurred. In our approach, such confidence statements are simultaneous for all choices of the rejected set, so that post hoc selection of the rejected set does not compromise their validity. The proposed reversal of roles requires nothing more than a review of the familiar closed testing procedure, but with a focus on the non-consonant rejections that this procedure makes. We suggest several shortcuts to avoid the computational problems associated with closed testing.Comment: Published in at http://dx.doi.org/10.1214/11-STS356 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Simultaneous directional inference

We consider the problem of inference on the signs of $n>1$ parameters. We aim to provide $1-\alpha$ post-hoc confidence bounds on the number of positive and negative (or non-positive) parameters. The guarantee is simultaneous, for all subsets of parameters. Our suggestion is as follows: start by using the data to select the direction of the hypothesis test for each parameter; then, adjust the $p$-values of the one-sided hypotheses for the selection, and use the adjusted $p$-values for simultaneous inference on the selected $n$ one-sided hypotheses. The adjustment is straightforward assuming that the $p$-values of one-sided hypotheses have densities with monotone likelihood ratio, and are mutually independent. We show that the bounds we provide are tighter (often by a great margin) than existing alternatives, and that they can be obtained by at most a polynomial time. We demonstrate the usefulness of our simultaneous post-hoc bounds in the evaluation of treatment effects across studies or subgroups. Specifically, we provide a tight lower bound on the number of studies which are beneficial, as well as on the number of studies which are harmful (or non-beneficial), and in addition conclude on the effect direction of individual studies, while guaranteeing that the probability of at least one wrong inference is at most 0.05.Comment: 59 pages, 11 figures, 7 table

Multi Split Conformal Prediction

Split conformal prediction is a computationally efficient method for performing distribution-free predictive inference in regression. It involves, however, a one-time random split of the data, and the result depends on the particular split. To address this problem, we propose multi split conformal prediction, a simple method based on Markov's inequality to aggregate single split conformal prediction intervals across multiple splits.Comment: 12 pages, 1 figure, 2 tabl

Rejoinder to "Multiple Testing for Exploratory Research"

Rejoinder to "Multiple Testing for Exploratory Research" by J. J. Goeman, A. Solari [arXiv:1208.2841].Comment: Published in at http://dx.doi.org/10.1214/11-STS356REJ the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Only Closed Testing Procedures are Admissible for Controlling False Discovery Proportions

We consider the class of all multiple testing methods controlling tail probabilities of the false discovery proportion, either for one random set or simultaneously for many such sets. This class encompasses methods controlling familywise error rate, generalized familywise error rate, false discovery exceedance, joint error rate, simultaneous control of all false discovery proportions, and others, as well as seemingly unrelated methods such as gene set testing in genomics and cluster inference methods in neuroimaging. We show that all such methods are either equivalent to a closed testing method, or are uniformly improved by one. Moreover, we show that a closed testing method is admissible as a method controlling tail probabilities of false discovery proportions if and only if all its local tests are admissible. This implies that, when designing such methods, it is sufficient to restrict attention to closed testing methods only. We demonstrate the practical usefulness of this design principle by constructing a uniform improvement of a recently proposed method