Explore the vast range of titles available.

It is common practice in statistical data analysis to perform data-driven variable selection and derive statistical inference from the resulting model. Such inference enjoys none of the guarantees that classical statistical theory provides for tests and confidence intervals when the model has been chosen a priori. We propose to produce valid \"post-selection inference\" by reducing the problem to one of simultaneous inference and hence suitably widening conventional confidence and retention intervals. Simultaneity is required for all linear functions that arise as coefficient estimates in all submodels. By purchasing \"simultaneity insurance\" for all possible submodels, the resulting post-selection inference is rendered universally valid under all possible model selection procedures. This inference is therefore generally conservative for particular selection procedures, but it is always less conservative than full Scheffé protection. Importantly it does not depend on the truth of the selected submodel, and hence it produces valid inference even in wrong models. We describe the structure of the simultaneous inference problem and give some asymptotic results.

In objective Bayesian model selection, no single criterion has emerged as dominant in defining objective prior distributions. Indeed, many criteria have been separately proposed and utilized to propose differing prior choices. We first formalize the most general and compelling of the various criteria that have been suggested, together with a new criterion. We then illustrate the potential of these criteria in determining objective model selection priors by considering their application to the problem of variable selection in normal linear models. This results in a new model selection objective prior with a number of compelling properties.

This paper studies the multiplicity-correction effect of standard Bayesian variable-selection priors in linear regression. Our first goal is to clarify when, and how, multiplicity correction happens automatically in Bayesian analysis, and to distinguish this correction from the Bayesian Ockham's-razor effect. Our second goal is to contrast empirical-Bayes and fully Bayesian approaches to variable selection through examples, theoretical results and simulations. Considerable differences between the two approaches are found. In particular, we prove a theorem that characterizes a surprising aymptotic discrepancy between fully Bayes and empirical Bayes. This discrepancy arises from a different source than the failure to account for hyperparameter uncertainty in the empirical-Bayes estimate. Indeed, even at the extreme, when the empirical-Bayes estimate converges asymptotically to the true variable-inclusion probability, the potential for a serious difference remains.

Stepwise multiple testing procedures have attracted several statisticians for decades and are also quite popular with statistics users because of their technical simplicity. The Bonferroni procedure has been one of the earliest and most prominent testing rules for controlling the familywise error rate (FWER). A recent article established that the FWER for the Bonferroni method asymptotically (i.e., when the number of hypotheses becomes arbitrarily large) approaches zero under any positively equicorrelated multivariate normal framework. However, similar results for the limiting behaviors of FWER of general stepwise procedures are nonexistent. The present work addresses this gap in a unified manner by elucidating that, under the multivariate normal setups with some correlation structures, the probability of rejecting one or more null hypotheses approaches zero asymptotically for any step-down procedure. Consequently, the FWER and power of the step-down procedures also tend to be asymptotically zero. We also establish similar limiting zero results on FWER of other popular multiple testing rules, e.g., Hochberg’s and Hommel’s procedures. It turns out that, within our chosen asymptotic framework, the Benjamini–Hochberg method can hold the FWER at a strictly positive level asymptotically under the equicorrelated normality.

We investigate the properties of the Benjamini-Hochberg method for multiple testing and of a variant of Storey's generalization of it, extending and complementing the asymptotic and exact results available in the literature. Results are obtained under two different sets of assumptions and include asymptotic and exact expressions and bounds for the proportion of rejections, the proportion of incorrect rejections out of all rejections and two other proportions used to quantify the efficacy of the method.

Closed testing and partitioning are recognized as fundamental principles of familywise error control. In this paper, we argue that sequential rejection can be considered equally fundamental as a general principle of multiple testing. We present a general sequentially rejective multiple testing procedure and show that many well-known familywise error controlling methods can be constructed as special cases of this procedure, among which are the procedures of Holm, Shaffer and Hochberg, parallel and serial gatekeeping procedures, modern procedures for multiple testing in graphs, resampling-based multiple testing procedures and even the closed testing and partitioning procedures themselves. We also give a general proof that sequentially rejective multiple testing procedures strongly control the familywise error if they fulfill simple criteria of monotonicity of the critical values and a limited form of weak familywise error control in each single step. The sequential rejection principle gives a novel theoretical perspective on many well-known multiple testing procedures, emphasizing the sequential aspect. Its main practical usefulness is for the development of multiple testing procedures for null hypotheses, possibly logically related, that are structured in a graph. We illustrate this by presenting a uniform improvement of a recently published procedure.

The Bonferroni procedure has been one of the foremost frequentist approaches for controlling the family-wise error rate (FWER) in simultaneous inference. However, many scientific disciplines often require less stringent error rates. One such measure is the generalized family-wise error rate (gFWER) proposed (Lehmann and Romano in Ann Stat 33(3):1138–1154, 2005, https://doi.org/10.1214/009053605000000084). FWER or gFWER controlling methods are considered highly conservative in problems with a moderately large number of hypotheses. Although, the existing literature lacks a theory on the extent of the conservativeness of gFWER controlling procedures under dependent frameworks. In this note, we address this gap in a unified manner by establishing upper bounds for the gFWER under arbitrarily correlated multivariate normal setups with moderate dimensions. Towards this, we derive a new probability inequality which, in turn, extends and sharpens a classical inequality. Our results also generalize a recent related work by the first author.

Consider the problem of simultaneously testing null hypotheses H1,..., Hs. The usual 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. In many applications, particularly if s is large, one might be willing to tolerate more than one false rejection provided the number of such cases is controlled, thereby increasing the ability of the procedure to detect false null hypotheses. This suggests replacing control of the FWER by controlling the probability of k or more false rejections, which we call the k-FWER. We derive both single-step and stepdown procedures that control the k-FWER, without making any assumptions concerning the dependence structure of the p-values of the individual tests. In particular, we derive a stepdown procedure that is quite simple to apply, and prove that it cannot be improved without violation of control of the k-FWER. We also consider the false discovery proportion (FDP) defined by the number of false rejections divided by the total number of rejections (defined to be 0 if there are no rejections). The false discovery rate proposed by Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300] controls E(FDP). Here, we construct methods such that, for any γ and α, $P{FDP>\\gamma}\\leq\\alpha$. Two stepdown methods are proposed. The first holds under mild conditions on the dependence structure of p-values, while the second is more conservative but holds without any dependence assumptions.

Benjamini and Hochberg suggest that the false discovery rate may be the appropriate error rate to control in many applied multiple testing problems. A simple procedure was given there as an FDR controlling procedure for independent test statistics and was shown to be much more powerful than comparable procedures which control the traditional familywise error rate. We prove that this same procedure also controls the false discovery rate when the test statistics have positive regression dependency on each of the test statistics corresponding to the true null hypotheses. This condition for positive dependency is general enough to cover many problems of practical interest, including the comparisons of many treatments with a single control, multivariate normal test statistics with positive correlation matrix and multivariate t. Furthermore, the test statistics may be discrete, and the tested hypotheses composite without posing special difficulties. For all other forms of dependency, a simple conservative modification of the procedure controls the false discovery rate. Thus the range of problems for which a procedure with proven FDR control can be offered is greatly increased.

Equivalence tests are statistical hypothesis testing procedures that aim to establish practical equivalence rather than the usual statistical significant difference. These testing procedures are frequent in “bioequivalence studies,\" where one would wish to show that, for example, an existing drug and a new one under development have comparable therapeutic effects. In this article, we propose a two-stage randomized (RAND2) p-value that depends on a uniformly most powerful (UMP) p-value and an arbitrary tuning parameter c∈[0,1] for testing an interval composite null hypothesis. We investigate the behavior of the distribution function of the two p-values under the null hypothesis and alternative hypothesis for a fixed significance level t∈(0,1) and varying sample sizes. We evaluate the performance of the two p-values in estimating the proportion of true null hypotheses in multiple testing. We conduct a family-wise error rate control using an adaptive Bonferroni procedure with a plug-in estimator to account for the multiplicity that arises from our multiple hypotheses under consideration. The various claims in this research are verified using a simulation study and real-world data analysis.