Harold Jeffreys pioneered the development of default Bayes factor hypothesis tests for standard statistical problems. Using Jeffreys’s Bayes factor hypothesis tests, researchers can grade the ...decisiveness of the evidence that the data provide for a point null hypothesis H0 versus a composite alternative hypothesis H1. Consequently, Jeffreys’s tests are of considerable theoretical and practical relevance for empirical researchers in general and for experimental psychologists in particular. To highlight this relevance and to facilitate the interpretation and use of Jeffreys’s Bayes factor tests we focus on two common inferential scenarios: testing the nullity of a normal mean (i.e., the Bayesian equivalent of the t-test) and testing the nullity of a correlation. For both Bayes factor tests, we explain their development, we extend them to one-sided problems, and we apply them to concrete examples from experimental psychology.
•The Bayes factor follows logically from Jeffreys’s philosophy of model selection.•The ideas are illustrated with two examples: the Bayesian t-test and correlation test.•The Bayes factors are adapted to one-sided tests.•The Bayes factors are illustrated with various applications in psychological research.
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
).
Pearson's correlation is one of the most common measures of linear dependence. Recently, Bernardo (11th International Workshop on Objective Bayes Methodology, 2015) introduced a flexible class of ...priors to study this measure in a Bayesian setting. For this large class of priors, we show that the (marginal) posterior for Pearson's correlation coefficient and all of the posterior moments are analytic. Our results are available in the open‐source software package JASP.
This paper introduces JASP, a free graphical software package for basic statistical procedures such as t tests, ANOVAs, linear regression models, and analyses of contingency tables. JASP is ...open-source and differentiates itself from existing open-source solutions in two ways. First, JASP provides several innovations in user interface design; specifically, results are provided immediately as the user makes changes to options, output is attractive, minimalist, and designed around the principle of progressive disclosure, and analyses can be peer reviewed without requiring a "syntax". Second, JASP provides some of the recent developments in Bayesian hypothesis testing and Bayesian parameter estimation. The ease with which these relatively complex Bayesian techniques are available in JASP encourages their broader adoption and furthers a more inclusive statistical reporting practice. The JASP analyses are implemented in R and a series of R packages.
We present a suite of Bayes factor hypothesis tests that allow researchers to grade the decisiveness of the evidence that the data provide for the presence versus the absence of a correlation between ...two variables. For concreteness, we apply our methods to the recent work of Donnellan et al. (
in press
) who conducted nine replication studies with over 3,000 participants and failed to replicate the phenomenon that lonely people compensate for a lack of social warmth by taking warmer baths or showers. We show how the Bayes factor hypothesis test can quantify evidence in favor of the null hypothesis, and how the prior specification for the correlation coefficient can be used to define a broad range of tests that address complementary questions. Specifically, we show how the prior specification can be adjusted to create a two-sided test, a one-sided test, a sensitivity analysis, and a replication test.
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, mostly amorphous, 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.
Researchers in numerical cognition rely on hypothesis testing and parameter estimation to evaluate the evidential value of data. Though there has been increased interest in Bayesian statistics as an ...alternative to the classical, frequentist approach to hypothesis testing, many researchers remain hesitant to change their methods of inference. In this tutorial, we provide a concise introduction to Bayesian hypothesis testing and parameter estimation in the context of numerical cognition. Here, we focus on three examples of Bayesian inference: the t-test, linear regression, and analysis of variance. Using the free software package JASP, we provide the reader with a basic understanding of how Bayesian inference works “under the hood” as well as instructions detailing how to perform and interpret each Bayesian analysis.
Whenever parameter estimates are uncertain or observations are contaminated by measurement error, the Pearson correlation coefficient can severely underestimate the true strength of an association. ...Various approaches exist for inferring the correlation in the presence of estimation uncertainty and measurement error, but none are routinely applied in psychological research. Here we focus on a Bayesian hierarchical model proposed by Behseta, Berdyyeva, Olson, and Kass (2009) that allows researchers to infer the underlying correlation between error-contaminated observations. We show that this approach may be also applied to obtain the underlying correlation between uncertain parameter estimates as well as the correlation between uncertain parameter estimates and noisy observations. We illustrate the Bayesian modeling of correlations with two empirical data sets; in each data set, we first infer the posterior distribution of the underlying correlation and then compute Bayes factors to quantify the evidence that the data provide for the presence of an association.
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