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Choosing among Competing Specifications
Published in Douglas D. Gunzler, Adam T. Perzynski, Adam C. Carle, Structural Equation Modeling for Health and Medicine, 2021
Douglas D. Gunzler, Adam T. Perzynski, Adam C. Carle
BIC can be derived from Bayesian theory. BIC provides a large-sample approximation of Bayes factor. Bayes factor is the Bayesian alternative to frequentist hypothesis testing; it is the ratio of the likelihood of two competing hypothesis (e.g. null hypothesis and alternative hypothesis). We give a brief overview of Bayesian SEM in Chapter 15.
Combining Evidence Over Multiple Individual Analyses
Published in Rens van de Schoot, Milica Miočević, Small Sample Size Solutions, 2020
Bayesian statistics is well suited to compare multiple hypotheses, whether they are null hypotheses, unconstrained or informative, like those introduced in the previous section. A Bayes factor quantifies the relative evidence in the data for two hypotheses (Kass & Raftery, 1995). More specifically, a Bayes factor is the rate with which the prior beliefs are updated into posterior beliefs, as shown in Equation 9.1. That is, prior to data collection, a researcher already has knowledge about the probability of two hypotheses that can be quantified to express their relative probability. For example, if the researcher expects both hypotheses to be equally probable before observing the data, the prior ratio is .5/.5. The prior ratio is updated with data and the resulting Bayes factor then quantifies how the data influenced this prior knowledge, summing up an updated ratio. Bayes factors are mostly used to evaluate hypotheses on population effects (i.e., there are no differences in the average reaction times between the conditions). In this chapter, the interest is in describing the relative evidence for two hypotheses for a specific individual. For this purpose, the can be computed per subject. The section ‘Aggregating Bayes factors’ demonstrates how this individual-level evidence can be synthesized.
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Published in Filomena Pereira-Maxwell, Medical Statistics, 2018
The prior belief may be an estimate of the overall prevalence of disease, if the usefulness of a diagnostic test is being evaluated, or it may be an estimate of treatment or exposure effect that was obtained from a previous study. The weight given to the prior belief depends on the precision with which it is estimated. For very vague prior beliefs, Bayesian and frequentist approaches yield comparable results. The posterior distribution may be used to obtain, for example, a 95% credible interval (CrI), which is i nterpreted as having 95% probability of containing the true value of the parameter (cf. interpretation of confidence intervals). Bayes’ factor is used to calculate the posterior probability of the null hypothesis being true, and to compare alternative models for the data observed. KIRKWOOD & STERNE point out the usefulness of Bayesian inference when carrying out interim analyses (specifically, where treatment effects have not been clinically significant), and also in the analysis of equivalence trials. See also BLAND (2015), CLAYTON & HILLS (1993), GREENLAND (2008), and SPIEGELHALTER, ABRAMS & MYLES (2004), ARMITAGE, BERRY & MATHEWS (2002).
Top-down Inhibitory Motor Control Is Preserved in Adults with Developmental Coordination Disorder
Published in Developmental Neuropsychology, 2021
William Mayes, Judith Gentle, Irene Parisi, Laura Dixon, José van Velzen, Ines Violante
All variables were examined for normality violations prior to analysis. All analyses were run using R version 4.0.2 and R Studio version 1.3.1073 (R Core Team, 2019) and data manipulation was achieved using functions from dplyr version 0.8.5 (Wickham, François, Henry., & Müller, 2020). For frequentist and Bayesian group comparison analysis on normally distributed data, independent samples t-tests were run using JASP software for use in R (JASP Team, 2020). If normality violations were present, Wilcoxon rank-sum tests were run instead, again using functions from JASP. A Bayes factor indicates the strength of evidence suggesting that one hypothesis is more likely than another (Rouder, Speckman, Sun, Morey, & Iverson, 2009). Factors of less than 1 indicate no evidence against the null hypothesis, while factors of above 10 indicate strong evidence in favor of the alternate hypothesis. The strengths of evidence is proportional to the Bayes Factor, for example, a factor of 0.5 indicates that the data is twice as likely to be observed under the null hypothesis, while a factor of 5 may indicate the likelihood of this observation to be 5 times as likely under the alternate hypothesis.
Bayesian t-tests for correlations and partial correlations
Published in Journal of Applied Statistics, 2020
Min Wang, Fang Chen, Tao Lu, Jianping Dong
We develop a Bayes factor based restricted most powerful Bayesian test (RMPBT) [3] for testing the presence of a correlation or a partial correlation. In an RMPBT, we choose the prior distribution under the alternative hypothesis by maximizing the power of the Bayesian test with respect to a restricted class of priors. This technique allows us to narrow the search of alternative hypothesis to a class of prior distributions, preferably to the priors that could result in closed-form expressions for the Bayes factors; such as Zellner's g-prior [17]. Analogous to its frequentist counterpart, the resulting Bayes factor with an appropriate choice of the evidence threshold can control the Type I error probability. In addition, the proposed Bayes factor enjoys other appealing properties: (i) it depends simply on the usual frequentist t-statistics and their associated critical values from the standard Student-t distributions; (ii) it can be easily computed using either a pocket calculator or spreadsheets, so long as applied researchers are familiar with the frequentist paradigm; (iii) it allows researchers to simultaneously analyze the results from both frequentist and Bayesian perspectives, and more importantly, (iv) we observe that by matching their rejection regions, the proposed Bayesian test could lead to an identical decision to its frequentist counterpart and that the evidence reflected by the Bayes factor usually requires strong evidence in the frequentist settings; see, also [5].
Bayes factor testing of equality and order constraints on measures of association in social research
Published in Journal of Applied Statistics, 2023
Joris Mulder, John P. T. M. Gelissen
The Bayes factor is a Bayesian criterion that quantifies the relative evidence in the data between two hypotheses. The Bayes factor of hypothesis A1) and (A2), and