How Strong is the Confirmation of a Hypothesis by Significant Data?

The aim of the article is to determine how much a hypothesis H is actually confirmed if it has successfully passed a classical significance test. Bayesians have already raised many serious objections against significance testing, but in doing so they have always had to rely on epistemic probabilities and a further Bayesian analysis, which are rejected by classical statisticians. Therefore, I will suggest a purely frequentist evaluation procedure for significance tests, that should also be accepted by a classical statistician. This procedure likewise indicates some additional problems of significance tests. Such tests generally offer only incremental support of a hypothesis, although an absolute confirmation is necessary, and they overestimate positive results for small effects, since the confirmation of H in these cases is often rather marginal. This phenomenon leads in specific cases, for example, in cases of ESP-hypotheses, such as precognition, too easily to a significant confirmation. I will propose a method of how to evaluate and supplement significance tests so that we can avoid their epistemic deficits

Second-generation p-values: improved rigor, reproducibility, & transparency in statistical analyses

Verifying that a statistically significant result is scientifically meaningful is not only good scientific practice, it is a natural way to control the Type I error rate. Here we introduce a novel extension of the p-value - a second-generation p-value - that formally accounts for scientific relevance and leverages this natural Type I Error control. The approach relies on a pre-specified interval null hypothesis that represents the collection of effect sizes that are scientifically uninteresting or are practically null. The second-generation p-value is the proportion of data-supported hypotheses that are also null hypotheses. As such, second-generation p-values indicate when the data are compatible with null hypotheses, or with alternative hypotheses, or when the data are inconclusive. Moreover, second-generation p-values provide a proper scientific adjustment for multiple comparisons and reduce false discovery rates. This is an advance for environments rich in data, where traditional p-value adjustments are needlessly punitive. Second-generation p-values promote transparency, rigor and reproducibility of scientific results by a priori specifying which candidate hypotheses are practically meaningful and by providing a more reliable statistical summary of when the data are compatible with alternative or null hypotheses.Comment: 29 pages, 29 page Supplemen

Two simple sufficient conditions for FDR control

We show that the control of the false discovery rate (FDR) for a multiple testing procedure is implied by two coupled simple sufficient conditions. The first one, which we call ``self-consistency condition'', concerns the algorithm itself, and the second, called ``dependency control condition'' is related to the dependency assumptions on the $p$-value family. Many standard multiple testing procedures are self-consistent (e.g. step-up, step-down or step-up-down procedures), and we prove that the dependency control condition can be fulfilled when choosing correspondingly appropriate rejection functions, in three classical types of dependency: independence, positive dependency (PRDS) and unspecified dependency. As a consequence, we recover earlier results through simple and unifying proofs while extending their scope to several regards: weighted FDR, $p$-value reweighting, new family of step-up procedures under unspecified $p$-value dependency and adaptive step-up procedures. We give additional examples of other possible applications. This framework also allows for defining and studying FDR control for multiple testing procedures over a continuous, uncountable space of hypotheses.Comment: Published in at http://dx.doi.org/10.1214/08-EJS180 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org