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# Null Hypothesis Significance Testing Interpreted and Calibrated by Estimating Probabilities of Sign Errors: A Bayes-Frequentist Continuum

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## Abstract

Hypothesis tests are conducted not only to determine whether a null hypothesis (H0) is true but also to determine the direction or sign of an effect. A simple estimate of the posterior probability of a sign error is PSE = (1 - PH0) p/2 + PH0, depending only on a two-sided p value and PH0, an estimate of the posterior probability of H0. A convenient option for PH0 is the posterior probability derived from estimating the Bayes factor to be its e p ln(1/p) lower bound. In that case, PSE depends only on p and an estimate of the prior probability of H0. PSE provides a continuum between significance testing and traditional Bayesian testing. The former effectively assumes the prior probability of H0 is 0, as some statisticians argue. In that case, PSE is equal to a one-sided p value. (In that sense, PSE is a calibrated p value.) In traditional Bayesian testing, on the other hand, the prior probability of H0 is at least 50%, which usually brings PSE close to PH0.

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... While empirical Bayes estimation is traditionally considered as a multiple comparison procedure (e.g., Efron 2010), its application to estimating false discovery rates applies not only to multiple testing but also to testing a single null hypothesis (e.g., Bickel 2019a, chapter 7;2021). That can be seen in terms of the problem of testing a null hypothesis against the most commonly used type of alternative hypothesis, that with two sides. ...
... where # and P are the parameter of interest and the p value as random variables, h H 0 is the value of h, the scalar parameter of interest under the null hypothesis that # ¼ h H 0 (that # is sufficiently close to h H 0 for practical purposes), and B is the Bayes factor f ðpj# ¼ h H 0 Þ=f ðpj# 6 ¼ h H 0 Þ based on f, the probability density function of P. Any nuisance parameters have been eliminated by integration with respect to their prior distributions. When Prð# ¼ h H 0 jP ¼ pÞ is a frequency-type probability, it is known as the local false discovery rate (LFDR) in the empirical Bayes literature on testing multiple hypotheses (e.g., Efron 2010; Efron et al. 2001) and on testing a single hypothesis (e.g., Bickel 2017Bickel , 2019aBickel , 2021. Under broad assumptions, Sellke, Bayarri, and Berger (2001) derived ...
... Additional arguments for d LFDR ¼ LFDR appear in Bickel (2021). ᭡ Example 3. Different assumptions lead to different versions of B, the lower bound on the Bayes factor (Held and Ott 2018). ...
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... A fiducial distribution is called a confidence distribution if it has the property that there is a 95% probability that ϑ lies within the limits of a 95% confidence interval computed from the fixed data set. A confidence distribution may be interpreted as an estimate of a Bayesian posterior distribution based on a prior distribution that assumes as little as possible (e.g., Bickel, 2021). Regardless of whether ϑ = ϑ 1 or ϑ = ϑ 2 , we see that Pr (28 MY ≤ ϑ ≤ 270 MY) ≥ 95% . ...
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... [31] To deal with inherent biases, of which some researchers may not even be consciously aware, there is need for new kind of design that approaches the hypothesis from different perspectives. It has been suggested that there should be a clear identification of the underlying uncertainty model associated with the scientific study [32] [33][34] [35] and some have even argued that NHST should be abandoned [36] [37]. ...
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The p-value quantifies the discrepancy between the data and a null hypothesis of interest, usually the assumption of no difference or no effect. A Bayesian approach allows the calibration of p-values by transforming them to direct measures of the evidence against the null hypothesis, so-called Bayes factors. We review the available literature in this area and consider two-sided significance tests for a point null hypothesis in more detail. We distinguish simple from local alternative hypotheses and contrast traditional Bayes factors based on the data with Bayes factors based on p-values or test statistics. A well-known finding is that the minimum Bayes factor, the smallest possible Bayes factor within a certain class of alternative hypotheses, provides less evidence against the null hypothesis than the corresponding p-value might suggest. It is less known that the relationship between p-values and minimum Bayes factors also depends on the sample size and on the dimension of the parameter of interest. We illustrate the transformation of p-values to minimum Bayes factors with two examples from clinical research.
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This paper proposes a general framework for prediction in which a prediction is presented in the form of a distribution function, called predictive distribution function. This predictive distribution function is well suited for the notion of confidence subscribed in the frequentist interpretation, and it can provide meaningful answers for questions related to prediction. A general approach under this framework is formulated and illustrated by using the so-called confidence distributions (CDs). This CD-based prediction approach inherits many desirable properties of CD, including its capacity for serving as a common platform for connecting and unifying the existing procedures of predictive inference in Bayesian, fiducial and frequentist paradigms. The theory underlying the CD-based predictive distribution is developed and some related efficiency and optimality issues are addressed. Moreover, a simple yet broadly applicable Monte Carlo algorithm is proposed for the implementation of the proposed approach. This concrete algorithm together with the proposed definition and associated theoretical development produce a comprehensive statistical inference framework for prediction. Finally, the approach is applied to simulation studies, and a real project on predicting the incoming volume of application submissions to a government agency. The latter shows the applicability of the proposed approach to dependence data settings.
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Minimum Bayes factors are commonly used to transform two-sided p-values to lower bounds on the posterior probability of the null hypothesis. Several proposals exist in the literature, but none of them depends on the sample size. However, the evidence of a p-value against a point null hypothesis is known to depend on the sample size. In this article, we consider p-values in the linear model and propose new minimum Bayes factors that depend on sample size and converge to existing bounds as the sample size goes to infinity. It turns out that the maximal evidence of an exact two-sided p-value increases with decreasing sample size. The effect of adjusting minimum Bayes factors for sample size is shown in two applications.
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R. A. Fisher, the father of modern statistics, proposed the idea of fiducial inference during the first half of the 20th century. While his proposal led to interesting methods for quantifying uncertainty, other prominent statisticians of the time did not accept Fisher’s approach as it became apparent that some of Fisher’s bold claims about the properties of fiducial distribution did not hold up for multi-parameter problems. Beginning around the year 2000, the authors and collaborators started to re-investigate the idea of fiducial inference and discovered that Fisher’s approach, when properly generalized, would open doors to solve many important and difficult inference problems. They termed their generalization of Fisher’s idea as generalized fiducial inference (GFI). The main idea of GFI is to carefully transfer randomness from the data to the parameter space using an inverse of a data generating equation without the use of Bayes theorem. The resulting generalized fiducial distribution (GFD) can then be used for inference. After more than a decade of investigations, the authors and collaborators have developed a unifying theory for GFI, and provided GFI solutions to many challenging practical problems in different fields of science and industry. Overall, they have demonstrated that GFI is a valid, useful, and promising approach for conducting statistical inference. The goal of this paper is to deliver a timely and concise introduction to GFI, to present some of the latest results, as well as to list some related open research problems. It is the authors’ hope that their contributions to GFI will stimulate the growth and usage of this exciting approach for statistical inference.
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The complete final product of Bayesian inference is the posterior distribution of the quantity of interest. Important inference summaries include point estimation, region estimation and precise hypotheses testing. Those summaries may appropriately be described as the solution to specific decision problems which depend on the particular loss function chosen. The use of a continuous loss function leads to an integrated set of solutions where the same prior distribution may be used throughout. Objective Bayesian methods are those which use a prior distribution which only depends on the assumed model and the quantity of interest. As a consequence, objective Bayesian methods produce results which only depend on the assumed model and the data obtained. The combined use of intrinsic discrepancy, an invariant information-based loss function, and appropriately defined reference priors, provides an integrated objective Bayesian solution to both estimation and hypothesis testing problems. The ideas are illustrated with a large collection of non-trivial examples.
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Concerns about a lack of reproducibility of statistically significant results have recently been raised in many fields, and it has been argued that this lack comes at substantial economic costs. We here report the results from prediction markets set up to quantify the reproducibility of 44 studies published in prominent psychology journals and replicated in the Reproducibility Project: Psychology. The prediction markets predict the outcomes of the replications well and outperform a survey of market participants' individual forecasts. This shows that prediction markets are a promising tool for assessing the reproducibility of published scientific results. The prediction markets also allow us to estimate probabilities for the hypotheses being true at different testing stages, which provides valuable information regarding the temporal dynamics of scientific discovery. We find that the hypotheses being tested in psychology typically have low prior probabilities of being true (median, 9%) and that a "statistically significant" finding needs to be confirmed in a well-powered replication to have a high probability of being true. We argue that prediction markets could be used to obtain speedy information about reproducibility at low cost and could potentially even be used to determine which studies to replicate to optimally allocate limited resources into replications.
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Medical and scientific advances are predicated on new knowledge that is robust and reliable and that serves as a solid foundation on which further advances can be built. In biomedical research, we are in the midst of a revolution with the generation of new data and scientific publications at a previously unprecedented rate. However, unfortunately, there is compelling evidence that the majority of these discoveries will not stand the test of time. To a large extent, this reproducibility crisis in basic and preclinical research may be as a result of failure to adhere to good scientific practice and the desperation to publish or perish. This is a multifaceted, multistakeholder problem. No single party is solely responsible, and no single solution will suffice. Here we review the reproducibility problems in basic and preclinical biomedical research, highlight some of the complexities, and discuss potential solutions that may help improve research quality and reproducibility. © 2015 American Heart Association, Inc.
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The controversy concerning the fundamental principles of statistics still remains unresolved. It is suggested that one key to resolving the conflict lies in recognizing that inferential probability derived from observational data is inherently noncoherent, in the sense that their inferential implications cannot be represented by a single probability distribution on the parameter space (except in the Objective Bayesian case). More precisely, for a parameter space R1, the class of all functions of the parameter comprise equivalence classes of invertibly related functions, and to each such class a logically distinct inferential probability distribution pertains. (There is an additional cross‐coherence requirement for simultaneous inference.) The non‐coherence of these distributions flows from the nonequivalence of the relevant components of the data for each. Noncoherence is mathematically inherent in confidence and fiducial theory, and provides a basis for reconciling the Fisherian and Neyman–Pearsonian viewpoints. A unified theory of confidence‐based inferential probability is presented, and the fundamental incompatibility of this with Subjective Bayesian theory is discussed.
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x is a one‐dimensional random variable whose distribution depends on a single parameter θ. It is the purpose of this note to establish two results: (i) The necessary and sufficient condition for the fiducial distribution of θ, given x, to be a Bayes' distribution is that there exist transformations of x to u, and of θ to τ, such that τ is a location parameter for u. The condition will be referred to as (A). This extends some results of Grundy's (1956). (ii) If, for a random sample of any size from the distribution for x, there exists a single sufficient statistic for θ then the fiducial argument is inconsistent unless condition (A) obtains: And when it does, the fiducial argument is equivalent to a Bayesian argument with uniform prior distribution for τ. The note concludes with an investigation of (A) in the case of the exponential family.
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A class of one‐parameter distributions is specified, for which the sample total is a sufficient statistic in samples of arbitrary size. It is proved that the resulting fiducial distribution of the parameter does not coincide with the distribution, a posteriori, given by Bayes' theorem, for any prior distribution whatever.
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Preface Introduction Introduction The Problem of Regions Some Example Applications About This Book Single Parameter Problems Introduction The General Case Smooth Function Model Asymptotic Comparisons Empirical Comparisons Examples Computation Using R Exercises Multiple Parameter Problems Introduction Smooth Function Model Asymptotic Accuracy Empirical Comparisons Examples Computation Using R Exercises Linear Models and Regression Introduction Statistical Framework Asymptotic Accuracy Empirical Comparisons Examples Further Issues in Linear Regression Computation Using R Exercises Nonparametric Smoothing Problems Introduction Nonparametric Density Estimation Density Estimation Examples Solving Density Estimation Problems Using R Nonparametric Regression Nonparametric Regression Examples Solving Nonparametric Regression Problems Using R Exercises Further Applications Classical Nonparametric Methods Generalized Linear Models Multivariate Analysis Survival Analysis Exercises Connections and Comparisons Introduction Statistical Hypothesis Testing Multiple Comparisons Attained Confidence Levels Bayesian Confidence Levels Exercises Appendix: Review of Asymptotic Statistics Taylor's Theorem Modes of Convergence Central Limit Theorem Convergence Rates Exercises References INDEX
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In this book, an integrated introduction to statistical inference is provided from a frequentist likelihood-based viewpoint. Classical results are presented together with recent developments, largely built upon ideas due to R.A. Fisher. The term “neo-Fisherian” highlights this.After a unified review of background material (statistical models, likelihood, data and model reduction, first-order asymptotics) and inference in the presence of nuisance parameters (including pseudo-likelihoods), a self-contained introduction is given to exponential families, exponential dispersion models, generalized linear models, and group families. Finally, basic results of higher-order asymptotics are introduced (index notation, asymptotic expansions for statistics and distributions, and major applications to likelihood inference).The emphasis is more on general concepts and methods than on regularity conditions. Many examples are given for specific statistical models. Each chapter is supplemented with problems and bibliographic notes. This volume can serve as a textbook in intermediate-level undergraduate and postgraduate courses in statistical inference.
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We live in a new age for statistical inference, where modern scientific technology such as microarrays and fMRI machines routinely produce thousands and sometimes millions of parallel data sets, each with its own estimation or testing problem. Doing thousands of problems at once is more than repeated application of classical methods. Taking an empirical Bayes approach, Bradley Efron, inventor of the bootstrap, shows how information accrues across problems in a way that combines Bayesian and frequentist ideas. Estimation, testing, and prediction blend in this framework, producing opportunities for new methodologies of increased power. New difficulties also arise, easily leading to flawed inferences. This book takes a careful look at both the promise and pitfalls of large-scale statistical inference, with particular attention to false discovery rates, the most successful of the new statistical techniques. Emphasis is on the inferential ideas underlying technical developments, illustrated using a large number of real examples.
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A review is provided of the concept confidence distributions. Material covered include: fundamentals, extensions, applications of confidence distributions and available computer software. We expect that this review could serve as a source of reference and encourage further research with respect to confidence distributions.
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Il est courant, en inférence fréquentielle, d'utiliser un point unique (une estimation ponctuelle) ou un intervalle (intervalle de confiance) dans le but d'estimer un paramètre d'intér^t. Une question très simple se pose: peut-on également utiliser, dans le même but, et dans la même optique fréquentielle, à la façon dont les Bayésiens utilisent une loi a posteriori, une distribution de probabilité? La réponse est affirmative, et les distributions de confiance apparaissent comme un choix naturel dans ce contexte. Le concept de distribution de confiance a une longue histoire, longtemps associée, à tort, aux théories d'inférence fiducielle, ce qui a compromis son développement dans l'optique fréquentielle. Les distributions de confiance ont récemment attiré un regain d'intérêt, et plusieurs résultats ont mis en évidence leur potentiel considérable en tant qu'outil inférentiel. Cet article présente une définition moderne du concept, et examine les ses évolutions récentes. Il aborde les méthodes d'inférence, les problèmes d'optimalité, et les applications. A la lumière de ces nouveaux développements, le concept de distribution de confiance englobe et unifie un large éventail de cas particuliers, depuis les exemples paramétriques réguliers (distributions fiducielles), les lois de rééchantillonnage, les p-valeurs et les fonctions de vraisemblance normalisées jusqu'aux a priori et posteriori bayésiens. La discussion est entièrement menée d'un point de vue fréquentiel, et met l'accent sur les applications dans lesquelles les solutions fréquentielles sont inexistantes ou d'une application difficile. Bien que nous attirions également l'attention sur les similitudes et les différences que présentent les approches fréquentielle, fiducielle, et Bayésienne, notre intention n'est pas de rouvrir un débat philosophique qui dure depuis près de deux cents ans. Nous espérons bien au contraire contribuer à combler le fossé qui existe entre les différents points de vue.
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A simple example shows that the classical theory of probability implies more than one can deduce via Kolmogorov's calculus of probability. Developing Dawid's ideas I propose a new calculus of probability which is free from this drawback. This calculus naturally leads to a new interpretation of probability. I argue that attempts to create a general empirical theory of probability should be abandoned and we should content ourselves with the logic of probability establishing relations between probabilistic theories and observations. My approach to the logic of probability is based on a variant of Ville's principle of the excluded gambling strategy. In addition to the classical theory of probability this approach is applied to the probabilistic theories provided by the problem of testing validity of probability forecasts and by statistical models.