Toward Evidence-Based Medical Statistics. 2: The Bayes FactorSteven N. Goodman, MD, PhDBayesian inference is usually presented as a method fordetermining how scientific belief should be modified bydata. Although Bayesian methodology has been one ofthe most active areas of statistical development in the past20 years, medical researchers have been reluctant to em-brace what they perceive as a subjective approach to dataanalysis. It is little understood that Bayesian methods havea data-based core, which can be used as a calculus ofevidence. This core is the Bayes factor, which in its simplestform is also called a likelihood ratio. The minimum Bayesfactor is objective and can be used in lieu of the P value asa measure of the evidential strength. Unlike P values,Bayes factors have a sound theoretical foundation and aninterpretation that allows their use in both inference anddecision making. Bayes factors show that P values greatlyoverstate the evidence against the null hypothesis. Mostimportant, Bayes factors require the addition of backgroundknowledge to be transformed into inferences—probabilitiesthat a given conclusion is right or wrong. They make thedistinction clear between experimental evidence and infer-ential conclusions while providing a framework in which tocombine 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.In the first of two articles on evidence-based sta-tistics (1), I outlined the inherent difficulties ofthe standard frequentist statistical approach to in-ference: problems with using the P value as a mea-sure of evidence, internal inconsistencies of the com-bined hypothesis test–P value method, and how thatmethod inhibits combining experimental results withbackground information. Here, I explore, as non-mathematically as possible, the Bayesian approachto measuring evidence and combining informationand epistemologic uncertainties that affect all statis-tical approaches to inference. Some of this presen-tation may be new to clinical researchers, but mostof it is based on ideas that have existed at least sincethe 1920s and, to some extent, centuries earlier (2).The Bayes Factor AlternativeBayesian inference is often described as amethod of showing how belief is altered by data.Because of this, many researchers regard it as non-scientific; that is, they want to know what the datasay, not what our belief should be after observingthem (3). Comments such as the following, which ap-peared in response to an article proposing a Bayesiananalysis of the GUSTO (Global Utilization of Strep-tokinase and tPA for Occluded Coronary Arteries)trial (4), are typical.When modern Bayesians include a “prior probabilitydistribution for the belief in the truth of a hypothesis,”they are actually creating a metaphysical model ofattitude change...Theresult...cannot be field-testedfor its validity, other than that it “feels” reasonable tothe consumer....The real problem is that neither classical nor Bayesianmethods are able to provide the kind of answers cli-nicians want. That classical methods are flawed is un-deniable—I wish I had an alternative....(5)This comment reflects the widespread mispercep-tion that the only utility of the Bayesian approach isas a belief calculus. What is not appreciated is thatBayesian methods can instead be viewed as an evi-dential calculus. Bayes theorem has two compo-nents—one that summarizes the data and one thatrepresents belief. Here, I focus on the componentrelated to the data: the Bayes factor, which in itssimplest form is also called a likelihood ratio. In Bayestheorem, the Bayes factor is the index through whichthe data speak, and it is separate from the purelysubjective part of the equation. It has also been calledthe relative betting odds, and its logarithm is some-times referred to as the weight of the evidence (6, 7).The distinction between evidence and error is clearwhen 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) isaltered by the data. The equation is as follows:Prior Oddsof Null Hypothesis3BayesFactor5Posterior Oddsof Null Hypothesiswhere Bayes factor 5Prob~Data, given the null hypothesis!Prob~Data, given the alternative hypothesis!The Bayes factor is a comparison of how welltwo hypotheses predict the data. The hypothesisthat predicts the observed data better is the onethat is said to have more evidence supporting it.Unlike the P value, the Bayes factor has a soundtheoretical foundation and an interpretation thatSee related article on pp 995-1004 and editorialcomment on pp 1019-1021.©1999 American College of Physicians–American Society of Internal Medicine 1005allows it to be used in both inference and decisionmaking. It links notions of objective probability, ev-idence, and subjective probability into a coherentpackage and is interpretable from all three perspec-tives. For example, if the Bayes factor for the nullhypothesis compared with another hypothesis is 1/2,the meaning can be expressed in three ways.1. Objective probability: The observed results arehalf as probable under the null hypothesis as theyare under the alternative.2. Inductive evidence: The evidence supports thenull hypothesis half as strongly as it does the alter-native.3. Subjective probability: The odds of the nullhypothesis relative to the alternative hypothesis af-ter the experiment are half what they were beforethe experiment.The Bayes factor differs in many ways from aP value. First, the Bayes factor is not a probabilityitself but a ratio of probabilities, and it can varyfrom zero to infinity. It requires two hypotheses,making it clear that for evidence to be against thenull hypothesis, it must be for some alternative.Second, the Bayes factor depends on the probabilityof the observed data alone, not including unobserved“long run” results that are part of the P value calcu-lation. Thus, factors unrelated to the data that affectthe P value, such as why an experiment was stopped,do not affect the Bayes factor (8, 9).Because we are so accustomed to thinking of“evidence” and the probability of “error” as synon-ymous, it may be difficult to know how to deal witha measure of evidence that is not a probability. It ishelpful to think of it as analogous to the
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