A Simple Two-Sample Bayesian<i>t</i>-Test for Hypothesis Testing

Wang, Min; Liu, Guangying

doi:10.1080/00031305.2015.1093027

Cited by 41 publications

(56 citation statements)

References 12 publications

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“…Compare with Gönen et al (2005) and Wang and Liu (2016). When δ = 0, cos θ (ν,δ) = 0, and ρ h (ν,δ=0) (θ ) simplifies to the Fisher-Student's central h-distribution ρ (ν,0) (θ ), as expected.…”

Section: Noncentral Hypersphere H-distributionmentioning

confidence: 99%

“…The latter observation is most pertinent for the biomedical research realm in which "empirical evidence suggests that most medical intervention effects are small or modest" (Pereira, Horwitz, and Ioannidis 2012). This maxent prior avoids Bartlett's and the information paradoxes (Wang and Liu 2016). The equal-tail Bayesian credible interval CI 1−α for the posteriors is given by the integration limits for the integrand (13) which leave out α/2 of the integrand on each tail.…”

Section: Bayesian Hypothesis Testing Frameworkmentioning

confidence: 99%

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Bayesian Analysis on a Noncentral Fisher–Student’s Hypersphere

Blanc

2018

The American Statistician

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Fisher succeeded early on in redefining Student's t-distribution in geometrical terms on a central hypersphere. Intriguingly, a noncentral analytical extension for this fundamental Fisher-Student's central hypersphere h-distribution does not exist. We therefore set to derive the noncentral h-distribution and use it to graphically illustrate the limitations of the Neyman-Pearson null hypothesis significance testing framework and the strengths of the Bayesian statistical hypothesis analysis framework on the hypersphere polar axis, a compact nontrivial one-dimensional parameter space. Using a geometrically meaningful maximal entropy prior, we requalify the apparent failure of an important psychological science reproducibility project. We proceed to show that the Bayes factor appropriately models the two-sample t-test p-value density of a gene expression profile produced by the high-throughput genomic-scale microarray technology, and provides a simple expression for a local false discovery rate addressing the multiple hypothesis testing problem brought about by such a technology.

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Section: Noncentral Hypersphere H-distributionmentioning

confidence: 99%

Section: Bayesian Hypothesis Testing Frameworkmentioning

confidence: 99%

Bayesian Analysis on a Noncentral Fisher–Student’s Hypersphere

Blanc

2018

The American Statistician

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“…Null hypothesis significance tests which employ p values are prone to inflate false-positive error rates if the distributional assumptions are violated (Rochon et al 2012), if optional stopping rules are applied (Kruschke and Liddell 2018b;Berger and Wolpert 1988), or the study conducted is underpowered (McElreath and Smaldino 2015). To mitigate these problems, a lot of research has been carried out in the last decade on developing Bayesian counterparts to popular frequentist two-sample tests like Student's t-test and the Mann-Whitney U test (van Doorn et al 2020;Gönen et al 2005;Wetzels et al 2009;Wang and Liu 2016;Gronau et al 2019). Bayesian versions of such traditional frequentist hypothesis tests have become much more popular recently, in particular, in the biomedical and cognitive sciences (Van De Schoot et al 2017;Wagenmakers et al 2016;Morey et al 2016).…”

Section: Introductionmentioning

confidence: 99%

Analysis of type I and II error rates of Bayesian and frequentist parametric and nonparametric two-sample hypothesis tests under preliminary assessment of normality

Kelter

2020

Comput Stat

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Testing for differences between two groups is among the most frequently carried out statistical methods in empirical research. The traditional frequentist approach is to make use of null hypothesis significance tests which use p values to reject a null hypothesis. Recently, a lot of research has emerged which proposes Bayesian versions of the most common parametric and nonparametric frequentist two-sample tests. These proposals include Student’s two-sample t-test and its nonparametric counterpart, the Mann–Whitney U test. In this paper, the underlying assumptions, models and their implications for practical research of recently proposed Bayesian two-sample tests are explored and contrasted with the frequentist solutions. An extensive simulation study is provided, the results of which demonstrate that the proposed Bayesian tests achieve better type I error control at slightly increased type II error rates. These results are important, because balancing the type I and II errors is a crucial goal in a variety of research, and shifting towards the Bayesian two-sample tests while simultaneously increasing the sample size yields smaller type I error rates. What is more, the results highlight that the differences in type II error rates between frequentist and Bayesian two-sample tests depend on the magnitude of the underlying effect.

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“…Bayes factors have been developed for a variety of testing problems frequently encountered in practice. For example, Gönen et al (2005) proposed a Bayesian two-sample t-test that was later extended by Wang and Liu (2016). Klugkist, Laudy, and Hoijtink (2005) developed a Bayes factor for testing order constrained hypotheses on the means in analysis of (co)variance models.…”

Section: Introductionmentioning

confidence: 99%

Bayes Factors for Testing Order Constraints on Variances of Dependent Outcomes

Böing-Messing

Mulder

2020

The American Statistician

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In statistical practice, researchers commonly focus on patterns in the means of multiple dependent outcomes while treating variances as nuisance parameters. However, in fact, there are often substantive reasons to expect certain patterns in the variances of dependent outcomes as well. For example, in a repeated measures study, one may expect the variance of the outcome to increase over time if the difference between subjects becomes more pronounced over time because the subjects respond differently to a given treatment. Such expectations can be formulated as order constrained hypotheses on the variances of the dependent outcomes. Currently, however, no methods exist for testing such hypotheses in a direct manner. To fill this gap, we develop a Bayes factor for this challenging testing problem. Our Bayes factor is based on the multivariate normal distribution with an unstructured covariance matrix, which is often used to model dependent outcomes. Order constrained hypotheses can then be formulated on the variances on the diagonal of the covariance matrix. To compute Bayes factors between multiple order constrained hypotheses, a prior distribution needs to be specified under every hypothesis to be tested. Here, we use the encompassing prior approach in which priors under order constrained hypotheses are truncations of the prior under the unconstrained hypothesis. The resulting Bayes factor is fully automatic in the sense that no subjective priors need to be specified by the user.

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A Simple Two-Sample Bayesiant-Test for Hypothesis Testing

Cited by 41 publications

References 12 publications

Bayesian Analysis on a Noncentral Fisher–Student’s Hypersphere

Bayesian Analysis on a Noncentral Fisher–Student’s Hypersphere

Analysis of type I and II error rates of Bayesian and frequentist parametric and nonparametric two-sample hypothesis tests under preliminary assessment of normality

Bayes Factors for Testing Order Constraints on Variances of Dependent Outcomes

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