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31
Bayes Factors
, 1995
"... In a 1935 paper, and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null ..."
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Cited by 981 (70 self)
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In a 1935 paper, and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null is onehalf. Although there has been much discussion of Bayesian hypothesis testing in the context of criticism of P values, less attention has been given to the Bayes factor as a practical tool of applied statistics. In this paper we review and discuss the uses of Bayes factors in the context of five scientific applications in genetics, sports, ecology, sociology and psychology.
Approximate Bayes Factors and Accounting for Model Uncertainty in Generalized Linear Models
, 1993
"... Ways of obtaining approximate Bayes factors for generalized linear models are described, based on the Laplace method for integrals. I propose a new approximation which uses only the output of standard computer programs such as GUM; this appears to be quite accurate. A reference set of proper priors ..."
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Cited by 96 (28 self)
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Ways of obtaining approximate Bayes factors for generalized linear models are described, based on the Laplace method for integrals. I propose a new approximation which uses only the output of standard computer programs such as GUM; this appears to be quite accurate. A reference set of proper priors is suggested, both to represent the situation where there is not much prior information, and to assess the sensitivity of the results to the prior distribution. The methods can be used when the dispersion parameter is unknown, when there is overdispersion, to compare link functions, and to compare error distributions and variance functions. The methods can be used to implement the Bayesian approach to accounting for model uncertainty. I describe an application to inference about relative risks in the presence of control factors where model uncertainty is large and important. Software to implement the
Bayes factors and model uncertainty
 DEPARTMENT OF STATISTICS, UNIVERSITY OFWASHINGTON
, 1993
"... In a 1935 paper, and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null ..."
Abstract

Cited by 89 (6 self)
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In a 1935 paper, and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null is onehalf. Although there has been much discussion of Bayesian hypothesis testing in the context of criticism of Pvalues, less attention has been given to the Bayes factor as a practical tool of applied statistics. In this paper we review and discuss the uses of Bayes factors in the context of five scientific applications. The points we emphasize are: from Jeffreys's Bayesian point of view, the purpose of hypothesis testing is to evaluate the evidence in favor of a scientific theory; Bayes factors offer a way of evaluating evidence in favor ofa null hypothesis; Bayes factors provide a way of incorporating external information into the evaluation of evidence about a hypothesis; Bayes factors are very general, and do not require alternative models to be nested; several techniques are available for computing Bayes factors, including asymptotic approximations which are easy to compute using the output from standard packages that maximize likelihoods; in "nonstandard " statistical models that do not satisfy common regularity conditions, it can be technically simpler to calculate Bayes factors than to derive nonBayesian significance
Comparing Dynamic Causal Models
 NEUROIMAGE
, 2004
"... This article describes the use of Bayes factors for comparing Dynamic Causal Models (DCMs). DCMs are used to make inferences about effective connectivity from functional Magnetic Resonance Imaging (fMRI) data. These inferences, however, are contingent upon assumptions about model structure, that is, ..."
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Cited by 81 (34 self)
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This article describes the use of Bayes factors for comparing Dynamic Causal Models (DCMs). DCMs are used to make inferences about effective connectivity from functional Magnetic Resonance Imaging (fMRI) data. These inferences, however, are contingent upon assumptions about model structure, that is, the connectivity pattern between the regions included in the model. Given the current lack of detailed knowledge on anatomical connectivity in the human brain, there are often considerable degrees of freedom when defining the connectional structure of DCMs. In addition, many plausible scientific hypotheses may exist about which connections are changed by experimental manipulation, and a formal procedure for directly comparing these competing hypotheses is highly desirable. In this article, we show how Bayes factors can be used to guide choices about model structure, both with regard to the intrinsic connectivity pattern and the contextual modulation of individual connections. The combined use of Bayes factors and DCM thus allows one to evaluate competing scientific theories about the architecture of largescale neural networks and the neuronal interactions that mediate perception and cognition.
LongRun Performance of Bayesian Model Averaging
 Journal of the American Statistical Association
, 2003
"... Hjort and Claeskens (HC) argue that statistical inference conditional on a single selected model underestimates uncertainty, and that model averaging is the way to remedy this; we strongly agree. They point out that Bayesian model averaging (BMA) has been the dominant approach to this, but argue tha ..."
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Cited by 11 (2 self)
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Hjort and Claeskens (HC) argue that statistical inference conditional on a single selected model underestimates uncertainty, and that model averaging is the way to remedy this; we strongly agree. They point out that Bayesian model averaging (BMA) has been the dominant approach to this, but argue that its performance has been inadequately studied, and propose an alternative, Frequentist Model Averaging (FMA). We point out, however, that there is a substantial literature on the performance of BMA, consisting of three main threads: general theoretical results, simulation studies, and evaluation of outofsample performance. The theoretical results are scattered, and we summarize them. The results have been quite consistent: BMA has tended to outperform competing methods for model selection and taking account of model uncertainty. The theoretical results depend on the assumption that the \practical distribution" over which the performance of methods is assessed is the same as the prior distribution used, and we investigate sensitivity of results to this assumption in a simple normal example; they turn out not to be unduly sensitive.
Statistical model selection methods applied to biological networks
 Transactions in Computational Systems Biology
, 2005
"... Abstract. Many biological networks have been labelled scalefree as their degree distribution can be approximately described by a powerlaw distribution. While the degree distribution does not summarize all aspects of a network it has often been suggested that its functional form contains important c ..."
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Cited by 7 (2 self)
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Abstract. Many biological networks have been labelled scalefree as their degree distribution can be approximately described by a powerlaw distribution. While the degree distribution does not summarize all aspects of a network it has often been suggested that its functional form contains important clues as to underlying evolutionary processes that have shaped the network. Generally determining the appropriate functional form for the degree distribution has been fitted in an adhoc fashion. Here we apply formal statistical model selection methods to determine which functional form best describes degree distributions of protein interaction and metabolic networks. We interpret the degree distribution as belonging to a class of probability models and determine which of these models provides the best description for the empirical data using maximum likelihood inference, composite likelihood methods, the Akaike information criterion and goodnessoffit tests. The whole data is used in order to determine the parameter that best explains the data under a given model (e.g. scalefree or random graph). As we will show, present protein interaction and metabolic network data from different organisms suggests that simple scalefree models do not provide an adequate description of real network data. 1
The Effect of External Representations on Numeric Tasks
"... This article explores the effect of external representations on numeric tasks. Through several minor modifications on the previously reported twodigit number comparison task, we obtained different results. Rather than a holistic comparison, we found parallel comparison. We argue that this differenc ..."
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Cited by 6 (0 self)
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This article explores the effect of external representations on numeric tasks. Through several minor modifications on the previously reported twodigit number comparison task, we obtained different results. Rather than a holistic comparison, we found parallel comparison. We argue that this difference was a reflection of different representational forms: the comparison was based on internal representations in previous studies but on external representations in our present study. This representational effect was discussed under a framework of distributed number representations. We propose that in numerical tasks involving external representations, numbers should be considered as distributed representations and the behavior in these tasks should be considered as the interactive processing of internal and external information through the interplay of perceptual and cognitive processes. We suggest that theories of number representations and process models of numerical tasks should consider external representations as an essential component.
How money buys happiness: Genetic and environmental processes
 Journal of Personality and Social Psychology
, 2006
"... Measures of wealth such as income and assets are commonly considered to be objective measures of environmental circumstances, making direct contributions to life satisfaction. Here, the authors explored the accuracy of this assumption. Using a nationwide sample of 719 twin pairs from the National Su ..."
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Cited by 6 (0 self)
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Measures of wealth such as income and assets are commonly considered to be objective measures of environmental circumstances, making direct contributions to life satisfaction. Here, the authors explored the accuracy of this assumption. Using a nationwide sample of 719 twin pairs from the National Survey of Midlife Development in the United States, the authors first noted the relative independence of most perceptions about financial status from measures of actual wealth. They then demonstrated that perceived financial situation and control over life completely mediated the association between measures of actual wealth and life satisfaction. Finally, they showed that financial resources appeared to protect life satisfaction from environmental shocks. In addition, control appeared to act as a mechanism translating life circumstances into life satisfaction.
Higher perceived life control decreases genetic variance in physical health: Evidence from a National Twin Study
 Journal of Personality and Social Psychology
, 2004
"... Physical health has been linked consistently with both income and sense of control, and the authors previously demonstrated that genetic variation in physical health measures decreased with increasing income (see W. Johnson & R. F. Krueger, 2004). Using a nationwide sample of 719 twin pairs from the ..."
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Cited by 5 (2 self)
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Physical health has been linked consistently with both income and sense of control, and the authors previously demonstrated that genetic variation in physical health measures decreased with increasing income (see W. Johnson & R. F. Krueger, 2004). Using a nationwide sample of 719 twin pairs from the MacArthur Foundation National Survey of Midlife Development in the United States, in this study the authors show that genetic variation in physical health measures (number of chronic illnesses and body mass index) also decreases with increasing sense of control. The authors integrate findings for income and control by demonstrating an interaction between genetic influences on sense of control and income in explaining physical health. They hypothesize that the mechanism underlying the interaction is the known biological relationship between metabolic efficiency and adaptation to stressful environments. Physical health has consistently been linked with income, wealthier people being healthier. The relationship holds over time, in a variety of geographic settings, and for almost every disease and condition (Adler et al., 1994). The association is monotonic across the full range of income; it is not limited to a comparison of those with incomes below and above poverty levels. The relationship