The Weighted Likelihood Ratio, Linear Hypotheses on Normal Location Parameters

a b s t r a c tIn the field of cognitive psychology, the p-value hypothesis test has established a stranglehold on statistical reporting. This is unfortunate, as the p-value provides at best a rough estimate of the evidence that the data provide for the presence of an experimental effect. An alternative and arguably more appropriate measure of evidence is conveyed by a Bayesian hypothesis test, which prefers the model with the highest average likelihood. One of the main problems with this Bayesian hypothesis test, however, is that it often requires relatively sophisticated numerical methods for its computation. Here we draw attention to the Savage-Dickey density ratio method, a method that can be used to compute the result of a Bayesian hypothesis test for nested models and under certain plausible restrictions on the parameter priors. Practical examples demonstrate the method's validity, generality, and flexibility.

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“…''Jimmie" Savage. The result is now generally known as the Savage-Dickey density ratio (e.g., Dickey, 1971;Gamerman & Lopes, 2006, pp. 72-74, pp.…”

Section: The Savage-dickey Density Ratiomentioning

confidence: 99%

Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method

Wagenmakers

Lodewyckx

Kuriyal

et al. 2010

Cognitive Psychology

652

642

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“…The Savage-Dickey density ratio (Dickey 1971) estimate of the Bayes factor for comparing nested models (e.g., H 0 : u = 0 vs. H 1 : u 6 ¼ 0) is given by the ratio of prior to posterior densities at 0,…”

Section: Methodsmentioning

confidence: 99%

Experimental Designs for Robust Detection of Effects in Genome-Wide Case–Control Studies

Ball

2011

Genetics

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In genome-wide association studies hundreds of thousands of loci are scanned in thousands of cases and controls, with the goal of identifying genomic loci underpinning disease. This is a challenging statistical problem requiring strong evidence. Only a small proportion of the heritability of common diseases has so far been explained. This "dark matter of the genome" is a subject of much discussion. It is critical to have experimental design criteria that ensure that associations between genomic loci and phenotypes are robustly detected. To ensure associations are robustly detected we require good power (e.g., 0.8) and sufficiently strong evidence [i.e., a high Bayes factor (e.g., 10 6 , meaning the data are 1 million times more likely if the association is real than if there is no association)] to overcome the low prior odds for any given marker in a genome scan to be associated with a causal locus. Power calculations are given for determining the sample sizes necessary to detect effects with the required power and Bayes factor for biallelic markers in linkage disequilibrium with causal loci in additive, dominant, and recessive genetic models. Significantly stronger evidence and larger sample sizes are required than indicated by traditional hypothesis tests and power calculations. Many reported putative effects are not robustly detected and many effects including some large moderately low-frequency effects may remain undetected. These results may explain the dark matter in the genome. The power calculations have been implemented in R and will be available in the R package ldDesign.T HE goal of genome-wide association studies (GWAS) is to understand the genetic basis of quantitative traits and complex diseases, by relating genotypes of large numbers of SNP markers to observed phenotypes. Results of GWAS to date suggest that the traits of interest are governed by many small effects (e.g., Wellcome Trust Case Control Consortium 2007; Diabetes Genetics Initiative of Broad Institute of Harvard and MIT 2007;Gudbjartsson et al. 2008;Lettre et al. 2008;Weedon et al. 2008;Lango Allen et al. 2010). Previously, we gave Bayesian power calculations for genome-wide association studies for quantitative traits (Ball 2005). This approach ensures power to detect effects of a specified size with a given Bayes factor, where the Bayes factor is chosen large enough to overcome the low prior odds for effects. These were the first results showing that much larger sample sizes were needed (e.g., thousands, even for a modest Bayes factor) than had been used at the time (cf. Luo 1998). However, these calculations do not apply to a binary phenotype such as presence or absence of a disease, e.g., coronary artery disease or type II diabetes, or case-control studies. Case-control studies, where a fixed (e.g., approximately equal) number of cases and controls are sampled, are used to study diseases that may not have high prevalence in the general population. This approach has higher power than a random population sample that may other...

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“…We construct this Bayes factor using the Savage-Dickey density ratio introduced by Dickey (1971), which allows us to easily build a Bayes factor for nested models. Specifically, the Savage-Dickey density ratio can be used to construct the Bayes factor when one model can be converted to the other model by setting a parameter to a specific value.…”

Section: Model Comparisonmentioning

confidence: 99%

Measuring the Natural Rate of Interest: Alternative Specifications

Lewis

Vazquez-Grande

2017

FEDS

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We build on the work of Laubach and Williams (2003) and subsequent studies by analyzing the effect on the estimates of the natural rate of interest (r * ) of accounting for full parameter uncertainty and alternative specifications for the underlying components of the natural rate. Our estimation technique delivers richer time-series dynamics for the median estimate of r * within the Laubach and Williams model. Additionally, we find that models with transitory shocks to the non-growth component of the natural rate have a higher marginal likelihood and produce an upward-sloping post-crisis trajectory of the r * path and thus a higher recent median point estimate (1.8% in 2016:Q3).JEL: C32, E43, E52, O40

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The Weighted Likelihood Ratio, Linear Hypotheses on Normal Location Parameters

Cited by 428 publications

References 23 publications

Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method

Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method

Experimental Designs for Robust Detection of Effects in Genome-Wide Case–Control Studies

Measuring the Natural Rate of Interest: Alternative Specifications

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