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Despite the increasing popularity of Bayesian inference in empirical research, few practical guidelines provide detailed recommendations for how to apply Bayesian procedures and interpret the results. Here we offer specific guidelines for four different stages of Bayesian statistical reasoning in a research setting: planning the analysis, executing the analysis, interpreting the results, and reporting the results. The guidelines for each stage are illustrated with a running example. Although the guidelines are geared towards analyses performed with the open-source statistical software JASP, most guidelines extend to Bayesian inference in general.

Bayesian parameter estimation and Bayesian hypothesis testing present attractive alternatives to classical inference using confidence intervals and p values. In part I of this series we outline ten prominent advantages of the Bayesian approach. Many of these advantages translate to concrete opportunities for pragmatic researchers. For instance, Bayesian hypothesis testing allows researchers to quantify evidence and monitor its progression as data come in, without needing to know the intention with which the data were collected. We end by countering several objections to Bayesian hypothesis testing. Part II of this series discusses JASP, a free and open source software program that makes it easy to conduct Bayesian estimation and testing for a range of popular statistical scenarios (Wagenmakers et al. this issue ).

The ‘MAlB’ phases are nanolaminated, ternary transition metal borides that consist of a transition metal boride sublattice interleaved by monolayers or bilayers of pure aluminum. However, their synthesis and properties remain largely unexplored. Herein, we synthesized dense, predominantly single-phase samples of one such compound, MoAlB, using a reactive hot pressing method. High-resolution scanning transmission electron microscopy confirmed the presence of two Al layers in between a Mo-B sublattice. Unique among the transition metal borides, MoAlB forms a dense, alumina scale when heated in air. Like other alumina formers, the oxidation kinetics follow a cubic time-dependence. At room temperature, its resistivity is low (0.36–0.49 μΩm) and – like a metal – drops linearly with decreasing temperatures. It is also a good thermal conductor (35 Wm −1 K −1 at 26 °C). In the 25–1300 °C temperature range, its thermal expansion coefficient is 9.5 × 10 −6 K −1 . Preliminary results suggest the compound is stable to at least 1400 °C in inert atmospheres. Moderately low Vickers hardness values of 10.6 ± 0.3 GPa, compared to other transition metal borides, and ultimate compressive strengths up to 1940 ± 103 MPa were measured at room temperature. These results are encouraging and warrant further study of this compound for potential use at high temperatures.

People with higher IQ scores also tend to perform better on elementary cognitive-perceptual tasks, such as deciding quickly whether an arrow points to the left or the right Jensen ( 2006 ). The worst performance rule (WPR) finesses this relation by stating that the association between IQ and elementary-task performance is most pronounced when this performance is summarized by people’s slowest responses. Previous research has shown that the WPR can be accounted for in the Ratcliff diffusion model by assuming that the same ability parameter—drift rate—mediates performance in both elementary tasks and higher-level cognitive tasks. Here we aim to test four qualitative predictions concerning the WPR and its diffusion model explanation in terms of drift rate. In the first stage, the diffusion model was fit to data from 916 participants completing a perceptual two-choice task; crucially, the fitting happened after randomly shuffling the key variable, i.e., each participant’s score on a working memory capacity test. In the second stage, after all modeling decisions were made, the key variable was unshuffled and the adequacy of the predictions was evaluated by means of confirmatory Bayesian hypothesis tests. By temporarily withholding the mapping of the key predictor, we retain flexibility for proper modeling of the data (e.g., outlier exclusion) while preventing biases from unduly influencing the results. Our results provide evidence against the WPR and suggest that it may be less robust and less ubiquitous than is commonly believed.

Across the empirical sciences, few statistical procedures rival the popularity of the frequentist -test. In contrast, the Bayesian versions of the -test have languished in obscurity. In recent years, however, the theoretical and practical advantages of the Bayesian -test have become increasingly apparent and various Bayesian t-tests have been proposed, both objective ones (based on general desiderata) and subjective ones (based on expert knowledge). Here, we propose a flexible t-prior for standardized effect size that allows computation of the Bayes factor by evaluating a single numerical integral. This specification contains previous objective and subjective t-test Bayes factors as special cases. Furthermore, we propose two measures for informed prior distributions that quantify the departure from the objective Bayes factor desiderata of predictive matching and information consistency. We illustrate the use of informed prior distributions based on an expert prior elicitation effort. Supplementary materials for this article are available online.

Bayesian hypothesis testing presents an attractive alternative to p value hypothesis testing. Part I of this series outlined several advantages of Bayesian hypothesis testing, including the ability to quantify evidence and the ability to monitor and update this evidence as data come in, without the need to know the intention with which the data were collected. Despite these and other practical advantages, Bayesian hypothesis tests are still reported relatively rarely. An important impediment to the widespread adoption of Bayesian tests is arguably the lack of user-friendly software for the run-of-the-mill statistical problems that confront psychologists for the analysis of almost every experiment: the t -test, ANOVA, correlation, regression, and contingency tables. In Part II of this series we introduce JASP ( http://www.jasp-stats.org ), an open-source, cross-platform, user-friendly graphical software package that allows users to carry out Bayesian hypothesis tests for standard statistical problems. JASP is based in part on the Bayesian analyses implemented in Morey and Rouder’s BayesFactor package for R. Armed with JASP, the practical advantages of Bayesian hypothesis testing are only a mouse click away.

This article outlines a Bayesian methodology to estimate and test the Kendall rank correlation coefficient τ. The nonparametric nature of rank data implies the absence of a generative model and the lack of an explicit likelihood function. These challenges can be overcome by modeling test statistics rather than data. We also introduce a method for obtaining a default prior distribution. The combined result is an inferential methodology that yields a posterior distribution for Kendall's τ.

One of the most common statistical analyses in experimental psychology concerns the comparison of two means using the frequentist t  test. However, frequentist t  tests do not quantify evidence and require various assumption tests. Recently, popularized Bayesian t  tests do quantify evidence, but these were developed for scenarios where the two populations are assumed to have the same variance. As an alternative to both methods, we outline a comprehensive t  test framework based on Bayesian model averaging. This new t  test framework simultaneously takes into account models that assume equal and unequal variances, and models that use t -likelihoods to improve robustness to outliers. The resulting inference is based on a weighted average across the entire model ensemble, with higher weights assigned to models that predicted the observed data well. This new t  test framework provides an integrated approach to assumption checks and inference by applying a series of pertinent models to the data simultaneously rather than sequentially. The integrated Bayesian model-averaged t  tests achieve robustness without having to commit to a single model following a series of assumption checks. To facilitate practical applications, we provide user-friendly implementations in JASP and via the RoBTT package in R . A tutorial video is available at https://www.youtube.com/watch?v=EcuzGTIcorQ