We describe Bayes factors based on z , t , x2, and F statistics when non-local moment prior distributions are used to define alternative hypotheses. The non-local alternative prior distributions are centered on standardized effects. The prior densities include a dispersion parameter that can be used to model prior precision and the variation of effect sizes across replicated experiments. We examine the convergence rates of Bayes factors under true null and true alternative hypotheses and show how these Bayes factors can be used to construct Bayes factor functions. Examples illustrate the application of resulting Bayes factors functions to psychological experiments.

We provide a correction to an equation in Table 1 of Datta et al. (2025) for two-sided t tests, as well as to the corresponding equation in the supplement that led to the error.

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We describe a modified sequential probability ratio test that can be used to reduce the average sample size required to perform statistical hypothesis tests at specified levels of significance and power. Examples are provided for z tests, t tests, and tests of binomial success probabilities. A description of a software package to implement the test designs is provided. We compare the sample sizes required in fixed design tests conducted at 5% significance levels to the average sample sizes required in sequential tests conducted at 0.5% significance levels, and we find that the two sample sizes are approximately equal.

Efficient variable selection in high-dimensional cancer genomic studies is critical for discovering genes associated with specific cancer types and for predicting response to treatment. Censored survival data is prevalent in such studies. In this article we introduce a Bayesian variable selection procedure that uses a mixture prior composed of a point mass at zero and an inverse moment prior in conjunction with the partial likelihood defined by the Cox proportional hazard model. The procedure is implemented in the R package BVSNLP, which supports parallel computing and uses a stochastic search method to explore the model space. Bayesian model averaging is used for prediction. The proposed algorithm provides better performance than other variable selection procedures in simulation studies and appears to provide more consistent variable selection when applied to actual genomic datasets.

Uniformly most powerful Bayesian tests (UMPBT's) are an objective class of Bayesian hypothesis tests that can be considered the Bayesian counterpart of classical uniformly most powerful tests. Because the rejection regions of UMPBT's can be matched to the rejection regions of classical uniformly most powerful tests (UMPTs), UMPBT's provide a mechanism for calibrating Bayesian evidence thresholds, Bayes factors, classical significance levels and p-values. The purpose of this article is to expand the application of UMPBT's outside the class of exponential family models. Specifically, we introduce sufficient conditions for the existence of UMPBT's and propose a unified approach for their derivation. An important application of our methodology is the extension of UMPBT's to testing whether the non-centrality parameter of a chi-squared distribution is zero. The resulting tests have broad applicability, providing default alternative hypotheses to compute Bayes factors in, for example, Pearson's chi-squared test for goodness-of-fit, tests of independence in contingency tables, and likelihood ratio, score and Wald tests.

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We introduce a new shrinkage prior on function spaces, called the functional horseshoe prior (fHS), that encourages shrinkage towards parametric classes of functions. Unlike other shrinkage priors for parametric models, the fHS shrinkage acts on the shape of the function rather than inducing sparsity on model parameters. We study the efficacy of the proposed approach by showing an adaptive posterior concentration property on the function. We also demonstrate consistency of the model selection procedure that thresholds the shrinkage parameter of the functional horseshoe prior. We apply the fHS prior to nonparametric additive models and compare its performance with procedures based on the standard horseshoe prior and several penalized likelihood approaches. We find that the new procedure achieves smaller estimation error and more accurate model selection than other procedures in several simulated and real examples. The supplementary material for this article, which contains additional simulated and real data examples, MCMC diagnostics, and proofs of the theoretical results, is available online.

We propose an algorithm for clustering high dimensional data. If P features for N objects are represented in an N× P matrix X, where N≪ P, the method is based on exploiting the cluster-dependent structure of the N× N matrix XX^T. Computational burden thus depends primarily on N, the number of objects to be clustered, rather than P, the number of features that are measured. This makes the method particularly useful in high dimensional settings, where it is substantially faster than a number of other popular clustering algorithms. Aside from an upper bound on the number of potential clusters, the method is independent of tuning parameters. When compared to 16 other clustering algorithms on 32 genomic datasets with gold standards, we show that it provides the most accurate cluster configuration more than twice as often than its closest competitors. We illustrate the method on data taken from highly cited genomic studies.

Bayesian model selection procedures based on nonlocal alternative prior densities are extended to ultrahigh dimensional settings and compared to other variable selection procedures using precision-recall curves. Variable selection procedures included in these comparisons include methods based on g-priors, reciprocal lasso, adaptive lasso, scad, and minimax concave penalty criteria. The use of precision-recall curves eliminates the sensitivity of our conclusions to the choice of tuning parameters. We find that Bayesian variable selection procedures based on nonlocal priors are competitive to all other procedures in a range of simulation scenarios, and we subsequently explain this favorable performance through a theoretical examination of their consistency properties. When certain regularity conditions apply, we demonstrate that the nonlocal procedures are consistent for linear models even when the number of covariates p increases sub-exponentially with the sample size n. A model selection procedure based on Zellner's g-prior is also found to be competitive with penalized likelihood methods in identifying the true model, but the posterior distribution on the model space induced by this method is much more dispersed than the posterior distribution induced on the model space by the nonlocal prior methods. We investigate the asymptotic form of the marginal likelihood based on the nonlocal priors and show that it attains a unique term that cannot be derived from the other Bayesian model selection procedures. We also propose a scalable and efficient algorithm called Simplified Shotgun Stochastic Search with Screening (S5) to explore the enormous model space, and we show that S5 dramatically reduces the computing time without losing the capacity to search the interesting region in the model space, at least in the simulation settings considered. The S5 algorithm is available in an R package BayesS5 on CRAN.

"We propose to change the default P-value threshold forstatistical significance for claims of new discoveries from 0.05 to 0.005."

Uniformly most powerful Bayesian tests (UMPBTs) are a new class of Bayesian tests in which null hypotheses are rejected if their Bayes factor exceeds a specified threshold. The alternative hypotheses in UMPBTs are defined to maximize the probability that the null hypothesis is rejected. Here, we generalize the notion of UMPBTs by restricting the class of alternative hypotheses over which this maximization is performed, resulting in restricted most powerful Bayesian tests (RMPBTs). We then derive RMPBTs for linear models by restricting alternative hypotheses to g priors. For linear models, the rejection regions of RMPBTs coincide with those of usual frequentist F-tests, provided that the evidence thresholds for the RMPBTs are appropriately matched to the size of the classical tests. This correspondence supplies default Bayes factors for many common tests of linear hypotheses. We illustrate the use of RMPBTs for ANOVA tests and t-tests and compare their performance in numerical studies.

We propose a Bayesian phase I/II dose-finding trial design that simultaneously accounts for toxicity and efficacy. We model the toxicity and efficacy of investigational doses using a flexible Bayesian dynamic model, which borrows information across doses without imposing stringent parametric assumptions on the shape of the dose-toxicity and dose-efficacy curves. An intuitive utility function that reflects the desirability trade-offs between efficacy and toxicity is used to guide the dose assignment and selection. We also discuss the extension of this design to handle delayed toxicity and efficacy. We conduct extensive simulation studies to examine the operating characteristics of the proposed method under various practical scenarios. The results show that the proposed design possesses good operating characteristics and is robust to the shape of the dose-toxicity and dose-efficacy curves.

AbstractBayes' factors play a pivotal role in Bayesian hypothesis testing, representing the factor through which the prior odds between competing hypotheses are transformed to the posterior odds between the hypotheses. Mathematically, Bayes' factors are defined as the ratio of the marginal probability assigned to the data by one hypothesis to the marginal probability assigned to data by the other hypothesis. In many parametric statistical tests, hypotheses are defined by the specification of prior densities on unknown parameters. When the prior density on a parameter concentrates on a single value under both the null and alternative hypotheses (resulting in two ‘simple’ or ‘point’ hypotheses), the Bayes factor equals the likelihood ratio. More generally, Bayes' factors represent the ratio of an averaged likelihood function, averaged with respect to the different prior densities assigned to the unknown parameter under each hypothesis. Inconsistencies in the limiting behaviour of Bayes' factors may arise when hypotheses are defined with respect to prior densities that have overlapping support.Key ConceptsBayes' factors are used in Bayesian hypothesis tests and are the key factors through which experimental data determine the posterior probability assigned to scientific hypotheses.Bayes' factors represent the ratio of the probability assigned by competing hypotheses to a common set of data.When testing whether one of the two simple hypotheses is true, the Bayes factor equals the likelihood ratio between the hypotheses.Bayes' factors equal the posterior odds divided by the prior odds between hypotheses.The natural logarithm of a Bayes' factor is called the weight of evidence.

In recent years, substantial effort has been devoted to methods for analyzing data containing mixed response types, but such techniques typically do not include rank data among the response types. Some unique challenges exist in analyzing rank data, particularly when ties are prevalent. We present techniques for jointly modeling binomial and rank data using Bayesian latent variable models. We apply these techniques to compare the cognitive abilities of nonhuman primates based on their performance on 17 cognitive tasks scored on either a rank or binomial scale. To jointly model the rank and binomial responses, we assume that responses are implicitly determined by latent cognitive abilities. We then model the latent variables using random effects models, with identifying restrictions chosen to promote parsimonious prior specification and model inferences. Results from the primate cognitive data are presented to illustrate the methodology. Our results suggest that the ordering of the cognitive abilities of species varies significantly across tasks, suggesting a partially independent evolution of cognitive abilities in primates. Supplementary materials for this article are available online.