scholarly journals The Pearson Bayes factor: An analytic formula for computing evidential value from minimal summary statistics

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Thomas J. Faulkenberry
Keyword(s):  
Prior Distribution ◽  
Degrees Of Freedom ◽  
Bayes Factor ◽  
Marginal Likelihood ◽  
Original Data ◽  
Bayes Factors ◽  
Model Parameters ◽  
Summary Statistics ◽  
Evidential Value ◽  
Scientific Papers

Summary In Bayesian hypothesis testing, evidence for a statistical model is quantified by the Bayes factor, which represents the relative likelihood of observed data under that model compared to another competing model. In general, computing Bayes factors is difficult, as computing the marginal likelihood of data under a given model requires integrating over a prior distribution of model parameters. In this paper, I capitalize on a particular choice of prior distribution that allows the Bayes factor to be expressed without integral representation, and I develop a simple formula – the Pearson Bayes factor – that requires only minimal summary statistics as commonly reported in scientific papers, such as the t or F score and the degrees of freedom. In addition to presenting this new result, I provide several examples of its use and report a simulation study validating its performance. Importantly, the Pearson Bayes factor gives applied researchers the ability to compute exact Bayes factors from minimal summary data, and thus easily assess the evidential value of any data for which these summary statistics are provided, even when the original data is not available.

scholarly journals Simulation Studies as a Tool to Understand Bayes Factors

2021 ◽  
Vol 4 (1) ◽  
pp. 251524592097262
Author(s):  
Don van Ravenzwaaij ◽  
Alexander Etz
Keyword(s):  
Simulation Study ◽  
Prior Distribution ◽  
Bayes Factor ◽  
R Package ◽  
Bayes Factors ◽  
Simulation Studies ◽  
Social Scientists ◽  
Simple Simulation ◽  
Set Up

When social scientists wish to learn about an empirical phenomenon, they perform an experiment. When they wish to learn about a complex numerical phenomenon, they can perform a simulation study. The goal of this Tutorial is twofold. First, it introduces how to set up a simulation study using the relatively simple example of simulating from the prior. Second, it demonstrates how simulation can be used to learn about the Jeffreys-Zellner-Siow (JZS) Bayes factor, a currently popular implementation of the Bayes factor employed in the BayesFactor R package and freeware program JASP. Many technical expositions on Bayes factors exist, but these may be somewhat inaccessible to researchers who are not specialized in statistics. In a step-by-step approach, this Tutorial shows how a simple simulation script can be used to approximate the calculation of the Bayes factor. We explain how a researcher can write such a sampler to approximate Bayes factors in a few lines of code, what the logic is behind the Savage-Dickey method used to visualize Bayes factors, and what the practical differences are for different choices of the prior distribution used to calculate Bayes factors.

scholarly journals Estimating Bayes factors from minimal summary statistics in repeated measures analysis of variance designs

2020 ◽  
Vol 17 (1) ◽  
Author(s):  
Thomas Faulkenberry
Keyword(s):  
Analysis Of Variance ◽  
Repeated Measures ◽  
Bayes Factor ◽  
Linear Models ◽  
Repeated Measurements ◽  
Bayes Factors ◽  
Complex Method ◽  
Summary Statistics ◽  
Minimal Set ◽  
Repeated Measures Analysis

In this paper, I develop a formula for estimating Bayes factors directly from minimal summary statistics produced in repeated measures analysis of variance designs. The formula, which requires knowing only the F-statistic, the number of subjects, and the number of repeated measurements per subject, is based on the BIC approximation of the Bayes factor, a common default method for Bayesian computation with linear models. In addition to providing computational examples, I report a simulation study in which I demonstrate that the formula compares favorably to a recently developed, more complex method that accounts for correlation between repeated measurements. The minimal BIC method provides a simple way for researchers to estimate Bayes factors from a minimal set of summary statistics, giving users a powerful index for estimating the evidential value of not only their own data, but also the data reported in published studies.

scholarly journals Simulation Studies as a Tool to Understand Bayes Factors

2020 ◽  
Author(s):  
Don van Ravenzwaaij ◽  
Alexander Etz
Keyword(s):  
Simulation Study ◽  
Prior Distribution ◽  
Bayes Factor ◽  
R Package ◽  
Bayes Factors ◽  
Simulation Studies ◽  
Social Scientists ◽  
Simple Simulation ◽  
Set Up

When social scientists wish to learn about an empirical phenomenon, they perform an experiment. When they wish to learn about a complex numerical phenomenon,they can perform a simulation study. The goal of this paper is twofold. Firstly, this paper introduces how to set up a simulation study using the relatively simple example of simulating from the prior. Secondly, this paper demonstrates how simulation can be used to learn about the Jeffreys-Zellner-Siow (JZS) Bayes factor: a currently popular implementation of the Bayes factor employed in the BayesFactor R-package and freeware program JASP. Many technical expositions exist on JZS Bayes factors, but these may be somewhat inaccessible to researchers that are not specialized in statistics. This paper aims to show in a step-by-step approach how a simple simulation script can be used to approximate the calculation of the JZS Bayes factor. We explain how a researcher can write such a sampler to approximate JZS Bayes factors in a few lines of code, what the logic is behind the Savage Dickey method used to visualize JZS Bayes factors, and what the practical differences are for different choices of the prior distribution for calculating Bayes factors.

scholarly journals Computing Bayes factors to measure evidence from experiments: An extension of the BIC approximation

2018 ◽  
Vol 55 (1) ◽  
pp. 31-43 ◽  
Author(s):  
Thomas J. Faulkenberry
Keyword(s):  
Bayesian Inference ◽  
Closed Form ◽  
Analysis Of Variance ◽  
Degrees Of Freedom ◽  
Bayes Factor ◽  
T Test ◽  
Bayes Factors ◽  
Computational Tools ◽  
Testing Hypotheses

Summary Bayesian inference affords scientists powerful tools for testing hypotheses. One of these tools is the Bayes factor, which indexes the extent to which support for one hypothesis over another is updated after seeing the data. Part of the hesitance to adopt this approach may stem from an unfamiliarity with the computational tools necessary for computing Bayes factors. Previous work has shown that closed-form approximations of Bayes factors are relatively easy to obtain for between-groups methods, such as an analysis of variance or t-test. In this paper, I extend this approximation to develop a formula for the Bayes factor that directly uses information that is typically reported for ANOVAs (e.g., the F ratio and degrees of freedom). After giving two examples of its use, I report the results of simulations which show that even with minimal input, this approximate Bayes factor produces similar results to existing software solutions.

scholarly journals A Bayesian solution to the Behrens–Fisher problem

2021 ◽  
Vol 115 (4) ◽  
Author(s):  
Fco. Javier Girón ◽  
Carmen del Castillo
Keyword(s):  
Closed Form ◽  
Hierarchical Model ◽  
Bayes Factor ◽  
Bayes Factors ◽  
Simple Solution ◽  
Asymptotic Approximations ◽  
Normal Distributions ◽  
Fisher Distribution ◽  
Leibler Divergence

AbstractA simple solution to the Behrens–Fisher problem based on Bayes factors is presented, and its relation with the Behrens–Fisher distribution is explored. The construction of the Bayes factor is based on a simple hierarchical model, and has a closed form based on the densities of general Behrens–Fisher distributions. Simple asymptotic approximations of the Bayes factor, which are functions of the Kullback–Leibler divergence between normal distributions, are given, and it is also proved to be consistent. Some examples and comparisons are also presented.

Do sexist jokes increase rape proclivity among males high in hostile sexism? Evidence from two pre-registered direct replications of Thomae & Viki (2013)

2021 ◽  
Author(s):  
Neil McLatchie ◽  
Manuela Thomae
Keyword(s):  
Bayes Factor ◽  
Alternative Hypothesis ◽  
Meta Analysis ◽  
Statistical Significance ◽  
Original Study ◽  
Bayes Factors ◽  
Hostile Sexism ◽  
Replication Studies ◽  
Open Research ◽  
Rape Proclivity

Thomae and Viki (2013) reported that increased exposure to sexist humour can increase rape proclivity among males, specifically those who score high on measures of Hostile Sexism. Here we report two pre-registered direct replications (N = 530) of Study 2 from Thomae and Viki (2013) and assess replicability via (i) statistical significance, (ii) Bayes factors, (iii) the small-telescope approach, and (iv) an internal meta-analysis across the original and replication studies. The original results were not supported by any of the approaches. Combining the original study and the replications yielded moderate evidence in support of the null over the alternative hypothesis with a Bayes factor of B = 0.13. In light of the combined evidence, we encourage researchers to exercise caution before claiming that brief exposure to sexist humour increases male’s proclivity towards rape, until further pre-registered and open research demonstrates the effect is reliably reproducible.

Uncertainty of prior and posterior model probability: Implications for interpreting Bayes factors

2021 ◽  
Author(s):  
John K. Kruschke
Keyword(s):  
Posterior Distribution ◽  
Bayesian Model ◽  
Model Comparison ◽  
Bayes Factor ◽  
Bayes Factors ◽  
Prior Model ◽  
Bayesian Hypothesis Testing ◽  
Bayesian Model Comparison ◽  
Posterior Model Probability ◽  
Model Probability

In most applications of Bayesian model comparison or Bayesian hypothesis testing, the results are reported in terms of the Bayes factor only, not in terms of the posterior probabilities of the models. Posterior model probabilities are not reported because researchers are reluctant to declare prior model probabilities, which in turn stems from uncertainty in the prior. Fortunately, Bayesian formalisms are designed to embrace prior uncertainty, not ignore it. This article provides a novel derivation of the posterior distribution of model probability, and shows many examples. The posterior distribution is useful for making decisions taking into account the uncertainty of the posterior model probability. Benchmark Bayes factors are provided for a spectrum of priors on model probability. R code is posted at https://osf.io/36527/. This framework and tools will improve interpretation and usefulness of Bayes factors in all their applications.

scholarly journals Computing Bayes factors from data with missing values

2021 ◽  
Author(s):  
Herbert Hoijtink ◽  
Xin Gu ◽  
Joris Mulder ◽  
Yves Rosseel
Keyword(s):  
Missing Values ◽  
Bayes Factor ◽  
Equality Constraints ◽  
Bayes Factors ◽  
Multiple Imputations

The Bayes factor is increasingly used for the evaluation of hypotheses. These may betraditional hypotheses specified using equality constraints among the parameters of thestatistical model of interest or informative hypotheses specified using equality andinequality constraints. So far no attention has been given to the computation of Bayesfactors from data with missing values. A key property of such a Bayes factor should bethat it is only based on the information in the observed values. This paper will show thatsuch a Bayes factor can be obtained using multiple imputations of the missing values.

scholarly journals An In-Class Demonstration of Bayesian Inference

2019 ◽  
Author(s):  
Johnny van Doorn ◽  
Dora Matzke ◽  
Eric-Jan Wagenmakers
Keyword(s):  
Bayesian Inference ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Sequential Analysis ◽  
Bayes Factor ◽  
Classroom Setting ◽  
Key Concepts

Sir Ronald Fisher's venerable experiment "The Lady Tasting Tea'' is revisited from a Bayesian perspective. We demonstrate how a similar tasting experiment, conducted in a classroom setting, can familiarize students with several key concepts of Bayesian inference, such as the prior distribution, the posterior distribution, the Bayes factor, and sequential analysis.

scholarly journals A comparison of conflict diffusion models in the flanker task through pseudo-likelihood Bayes factors

2018 ◽  
Author(s):  
Nathan J. Evans ◽  
Mathieu Servant
Keyword(s):  
Probability Density ◽  
Marginal Likelihood ◽  
Flanker Task ◽  
Approximation Error ◽  
Bayes Factors ◽  
Choice Response ◽  
Density Approximation ◽  
Sources Of Information ◽  
Pseudo Likelihood ◽  
Probability Density Approximation

Conflict tasks are one of the most widely studied paradigms within cognitive psychology, where participants are required to respond based on relevant sources of information while ignoring conflicting irrelevant sources of information. The flanker task, in particular, has been the focus of considerable modeling efforts, with only three models being able to provide a complete account of empirical choice response time distributions: the dual- stage two-phase model (DSTP), the shrinking spotlight model (SSP), and the diffusion model for conflict tasks (DMC). Although these models are grounded in different theoretical frameworks, can provide diverging measures of cognitive control, and are quantitatively distinguishable, no previous study has compared all three of these models in their ability to account for empirical data. Here, we perform a comparison of the precise quantitative predictions of these models through Bayes factors, using probability density approximation to generate a pseudo-likelihood estimate of the unknown probability density function, and thermodynamic integration via differential evolution to approximate the analytically intractable Bayes factors. We find that for every participant across three data sets from three separate research groups, DMC provides an inferior account of the data to DSTP and SSP, which has important theoretical implications regarding cognitive processes engaged in the flanker task, and practical implications for applying the models to flanker data. More generally, we argue that our combination of probability density approximation with marginal likelihood approximation – which we term pseudo-likelihood Bayes factors – provides a crucial step forward for the future of model comparison, where Bayes factors can be calculated between any models that can be simulated. We also discuss the limitations of simulation-based methods, such as the potential for approximation error, and suggest that researchers should use analytically or numerically computed likelihood functions when they are available and computationally tractable.

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