Toward Evidence Based Medical Statistics 2 The Bayes Factor Steven N Goodman MD PhD Bayesian inference is usually presented as a method for determining how scientific belief should be modified by data Although Bayesian methodology has been one of the most active areas of statistical development in the past 20 years medical researchers have been reluctant to embrace what they perceive as a subjective approach to data analysis It is little understood that Bayesian methods have a data based core which can be used as a calculus of evidence This core is the Bayes factor which in its simplest form is also called a likelihood ratio The minimum Bayes factor is objective and can be used in lieu of the P value as a measure of the evidential strength Unlike P values Bayes factors have a sound theoretical foundation and an interpretation that allows their use in both inference and decision making Bayes factors show that P values greatly overstate the evidence against the null hypothesis Most important Bayes factors require the addition of background knowledge to be transformed into inferences probabilities that a given conclusion is right or wrong They make the distinction clear between experimental evidence and inferential conclusions while providing a framework in which to combine prior with current evidence This paper is also available at http www acponline org Ann Intern Med 1999 130 1005 1013 From Johns Hopkins University School of Medicine Baltimore Maryland For the current author address see end of text I n the first of two articles on evidence based statistics 1 I outlined the inherent difficulties of the standard frequentist statistical approach to inference problems with using the P value as a measure of evidence internal inconsistencies of the combined hypothesis test P value method and how that method inhibits combining experimental results with background information Here I explore as nonmathematically as possible the Bayesian approach to measuring evidence and combining information and epistemologic uncertainties that affect all statistical approaches to inference Some of this presentation may be new to clinical researchers but most of it is based on ideas that have existed at least since the 1920s and to some extent centuries earlier 2 The Bayes Factor Alternative Bayesian inference is often described as a method of showing how belief is altered by data Because of this many researchers regard it as nonscientific that is they want to know what the data say not what our belief should be after observing them 3 Comments such as the following which ap peared in response to an article proposing a Bayesian analysis of the GUSTO Global Utilization of Streptokinase and tPA for Occluded Coronary Arteries trial 4 are typical When modern Bayesians include a prior probability distribution for the belief in the truth of a hypothesis they are actually creating a metaphysical model of attitude change The result cannot be field tested for its validity other than that it feels reasonable to the consumer The real problem is that neither classical nor Bayesian methods are able to provide the kind of answers clinicians want That classical methods are flawed is undeniable I wish I had an alternative 5 This comment reflects the widespread misperception that the only utility of the Bayesian approach is as a belief calculus What is not appreciated is that Bayesian methods can instead be viewed as an evidential calculus Bayes theorem has two components one that summarizes the data and one that represents belief Here I focus on the component related to the data the Bayes factor which in its simplest form is also called a likelihood ratio In Bayes theorem the Bayes factor is the index through which the data speak and it is separate from the purely subjective part of the equation It has also been called the relative betting odds and its logarithm is sometimes referred to as the weight of the evidence 6 7 The distinction between evidence and error is clear when it is recognized that the Bayes factor evidence is a measure of how much the probability of truth that is 1 2 prob error where prob is probability is altered by the data The equation is as follows Prior Odds Bayes Posterior Odds of Null Hypothesis 3 Factor 5 of Null Hypothesis where Bayes factor 5 Prob Data given the null hypothesis Prob Data given the alternative hypothesis The Bayes factor is a comparison of how well two hypotheses predict the data The hypothesis that predicts the observed data better is the one that is said to have more evidence supporting it Unlike the P value the Bayes factor has a sound theoretical foundation and an interpretation that See related article on pp 995 1004 and editorial comment on pp 1019 1021 1999 American College of Physicians American Society of Internal Medicine 1005 Table 1 Final Posterior Probability of the Null Hypothesis after Observing Various Bayes Factors as a Function of the Prior Probability of the Null Hypothesis Strength of Evidence Bayes Factor Decrease in Probability of the Null Hypothesis From To No Less Than Weak 1 5 90 50 25 64 17 6 Moderate 1 10 90 50 25 47 9 3 Moderate to strong 1 20 90 50 25 31 5 2 Strong to very strong 1 100 90 50 25 8 1 0 3 Calculations were performed as follows A probability Prob of 90 is equivalent to an odds of 9 calculated as Prob 1 2 Prob Posterior odds 5 Bayes factor 3 prior odds thus 1 5 3 9 5 1 8 Probability 5 odds 1 1 odds thus 1 8 2 8 5 0 64 allows it to be used in both inference and decision making It links notions of objective probability evidence and subjective probability into a coherent package and is interpretable from all three perspectives For example if the Bayes factor for the null hypothesis compared with another hypothesis is 1 2 the meaning can be expressed in three ways 1 Objective probability The observed results are half as probable under the null hypothesis as they are under the alternative 2 Inductive evidence The evidence supports the null hypothesis half as strongly as it does the alternative 3 Subjective probability The odds of the null hypothesis relative to the alternative hypothesis after the experiment are half what they were before the experiment The Bayes factor differs in many ways from a P value First the Bayes factor is not a probability itself but a ratio of probabilities and it can vary from zero to infinity It requires two hypotheses making it clear that for evidence to be against the null hypothesis it must be for some
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