A default Bayesian hypothesis test for correlations and partial correlations
TL;DR: A default Bayesian hypothesis test for the presence of a correlation or a partial correlation is proposed, which can quantify evidence in favor of the null hypothesis and allows researchers to monitor the test results as the data come in.
Abstract: We propose a default Bayesian hypothesis test for the presence of a correlation or a partial correlation. The test is a direct application of Bayesian techniques for variable selection in regression models. The test is easy to apply and yields practical advantages that the standard frequentist tests lack; in particular, the Bayesian test can quantify evidence in favor of the null hypothesis and allows researchers to monitor the test results as the data come in. We illustrate the use of the Bayesian correlation test with three examples from the psychological literature. Computer code and example data are provided in the journal archives.
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TL;DR: It is argued Bayes factors allow theory to be linked to data in a way that overcomes the weaknesses of the other approaches, and provides a coherent approach to determining whether non-significant results support a null hypothesis over a theory, or whether the data are just insensitive.
Abstract: No scientific conclusion follows automatically from a statistically non-significant result, yet people routinely use non-significant results to guide conclusions about the status of theories (or the effectiveness of practices). To know whether a non-significant result counts against a theory, or if it just indicates data insensitivity, researchers must use one of: power, intervals (such as confidence or credibility intervals), or else an indicator of the relative evidence for one theory over another, such as a Bayes factor. I argue Bayes factors allow theory to be linked to data in a way that overcomes the weaknesses of the other approaches. Specifically, Bayes factors use the data themselves to determine their sensitivity in distinguishing theories (unlike power), and they make use of those aspects of a theory’s predictions that are often easiest to specify (unlike power and intervals, which require specifying the minimal interesting value in order to address theory). Bayes factors provide a coherent approach to determining whether non-significant results support a null hypothesis over a theory, or whether the data are just insensitive. They allow accepting and rejecting the null hypothesis to be put on an equal footing. Concrete examples are provided to indicate the range of application of a simple online Bayes calculator, which reveal both the strengths and weaknesses of Bayes factors.
1,496 citations
Cites background from "A default Bayesian hypothesis test ..."
...…deciding on a default representation of the predictions of the alternative to use in all or most cases (e.g., Jeffreys et al., 1939/1961; Box and Tiao, 1973; Liang et al., 2008; Rouder et al., 2009; Rouder, 2012; Rouder and Morey, 2011; Wetzels and Wagenmakers, 2012; Wetzels et al., 2012)....
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TL;DR: Ten prominent advantages of the Bayesian approach are outlined, and several objections to Bayesian hypothesis testing are countered.
Abstract: 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).
940 citations
Cites methods from "A default Bayesian hypothesis test ..."
...A slightly different and less transparent Bayesian model for the Pearson correlation coefficient is presented in Wetzels and Wagenmakers (2012). learned....
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TL;DR: In this article, the authors describe how regularization techniques can be used to efficiently estimate a parsimonious and interpretable network structure in psychological data, and demonstrate the method in an empirical example on post-traumatic stress disorder data.
Abstract: Recent years have seen an emergence of network modeling applied to moods, attitudes, and problems in the realm of psychology. In this framework, psychological variables are understood to directly affect each other rather than being caused by an unobserved latent entity. In this tutorial, we introduce the reader to estimating the most popular network model for psychological data: the partial correlation network. We describe how regularization techniques can be used to efficiently estimate a parsimonious and interpretable network structure in psychological data. We show how to perform these analyses in R and demonstrate the method in an empirical example on post-traumatic stress disorder data. In addition, we discuss the effect of the hyperparameter that needs to be manually set by the researcher, how to handle non-normal data, how to determine the required sample size for a network analysis, and provide a checklist with potential solutions for problems that can arise when estimating regularized partial correlation networks.
839 citations
TL;DR: This study indicates that an SI increase in the DN and GP on T1-weighted images is caused by serial application of the linear GBCA gadopentetate dimeglumine but not by the macrocyclic GBCAs gadoterate meglumines.
Abstract: In this study, we found increased signal intensity (SI) in the dentate nucleus and globus pallidus on unenhanced T1-weighted images after serial applications of the linear gadolinium-based contrast agent (GBCA) gadopentetate dimeglumine, while no significant SI increase could be demonstrated for the macrocyclic GBCA gadoterate meglumine.
516 citations
Cites methods from "A default Bayesian hypothesis test ..."
...Correlation analyses were used to test whether the number of contrast-enhanced MR imaging examinations, the mean interval between GBCA administrations, and the accumulated dose of GBCA had a different influence in each patient group, supplementing the analysis with Jeffreys-Zellner-Siow Bayes factors to quantify the strength of evidence (10)....
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TL;DR: This tutorial introduces the reader to estimating the most popular network model for psychological data: the partial correlation network and describes how regularization techniques can be used to efficiently estimate a parsimonious and interpretable network structure in psychological data.
Abstract: Recent years have seen an emergence of network modeling applied to moods, attitudes, and problems in the realm of psychology. In this framework, psychological variables are understood to directly affect each other rather than being caused by an unobserved latent entity. In this tutorial, we introduce the reader to estimating the most popular network model for psychological data: the partial correlation network. We describe how regularization techniques can be used to efficiently estimate a parsimonious and interpretable network structure in psychological data. We show how to perform these analyses in R and demonstrate the method in an empirical example on posttraumatic stress disorder data. In addition, we discuss the effect of the hyperparameter that needs to be manually set by the researcher, how to handle non-normal data, how to determine the required sample size for a network analysis, and provide a checklist with potential solutions for problems that can arise when estimating regularized partial correlation networks. (PsycINFO Database Record (c) 2018 APA, all rights reserved)
453 citations
References
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Book•
01 Jan 1966
TL;DR: In this article, the Straight Line Case is used to fit a straight line by least squares, and the Durbin-Watson Test is used for checking the straight line fit.
Abstract: Basic Prerequisite Knowledge. Fitting a Straight Line by Least Squares. Checking the Straight Line Fit. Fitting Straight Lines: Special Topics. Regression in Matrix Terms: Straight Line Case. The General Regression Situation. Extra Sums of Squares and Tests for Several Parameters Being Zero. Serial Correlation in the Residuals and the Durbin--Watson Test. More of Checking Fitted Models. Multiple Regression: Special Topics. Bias in Regression Estimates, and Expected Values of Mean Squares and Sums of Squares. On Worthwhile Regressions, Big F's, and R 2 . Models Containing Functions of the Predictors, Including Polynomial Models. Transformation of the Response Variable. "Dummy" Variables. Selecting the "Best" Regression Equation. Ill--Conditioning in Regression Data. Ridge Regression. Generalized Linear Models (GLIM). Mixture Ingredients as Predictor Variables. The Geometry of Least Squares. More Geometry of Least Squares. Orthogonal Polynomials and Summary Data. Multiple Regression Applied to Analysis of Variance Problems. An Introduction to Nonlinear Estimation. Robust Regression. Resampling Procedures (Bootstrapping). Bibliography. True/False Questions. Answers to Exercises. Tables. Indexes.
18,952 citations
Book•
01 Jan 1939
TL;DR: In this paper, the authors introduce the concept of direct probabilities, approximate methods and simplifications, and significant importance tests for various complications, including one new parameter, and various complications for frequency definitions and direct methods.
Abstract: 1. Fundamental notions 2. Direct probabilities 3. Estimation problems 4. Approximate methods and simplifications 5. Significance tests: one new parameter 6. Significance tests: various complications 7. Frequency definitions and direct methods 8. General questions
7,086 citations
TL;DR: It is shown that despite empirical psychologists’ nominal endorsement of a low rate of false-positive findings, flexibility in data collection, analysis, and reporting dramatically increases actual false- positive rates, and a simple, low-cost, and straightforwardly effective disclosure-based solution is suggested.
Abstract: In this article, we accomplish two things. First, we show that despite empirical psychologists' nominal endorsement of a low rate of false-positive findings (≤ .05), flexibility in data collection, analysis, and reporting dramatically increases actual false-positive rates. In many cases, a researcher is more likely to falsely find evidence that an effect exists than to correctly find evidence that it does not. We present computer simulations and a pair of actual experiments that demonstrate how unacceptably easy it is to accumulate (and report) statistically significant evidence for a false hypothesis. Second, we suggest a simple, low-cost, and straightforwardly effective disclosure-based solution to this problem. The solution involves six concrete requirements for authors and four guidelines for reviewers, all of which impose a minimal burden on the publication process.
4,727 citations
09 Apr 1998
3,103 citations
"A default Bayesian hypothesis test ..." refers methods in this paper
...Hence, the correlation test can be recast as a comparison between two linear regression models, M0 and M1 (e.g., Draper & Smith, 1998)....
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...Hence, we compare the two models (e.g., Draper & Smith, 1998): Y X Y X X ð14Þ The Bayes factor BF10 using the JZS prior setup can then be calculated as follows: BF10 ¼ p YjM1ð Þp YjM0ð Þ ¼ R1 0 1 þ gð Þ n 1 p1ð Þ=2 1 þ 1 R21ð Þg½ n 1ð Þ=2g 3=2ð Þe n= 2gð ÞdgR1 0 1 þ gð Þ n 1 p0ð Þ=2 1 þ 1 R20ð Þg½…...
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