Academia and Clinic
15 June 1999

Toward Evidence-Based Medical Statistics. 2: The Bayes Factor

Publication: Annals of Internal Medicine
Volume 130, Number 12

Abstract

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.

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References

1.
Goodman SN. Toward evidence-based medical statistics. 1: The P value fallacy. Ann Intern Med. 1999;130:995-1004.
2.
Edwards A. A History of Likelihood. International Statistical Review. 1974;42:9-15.
3.
Fisher LD. Comments on Bayesian and frequentist analysis and interpretation of clinical trials. Control Clin Trials. 1996;17:423-34.
4.
Brophy JMJoseph L. Placing trials in context using Bayesian analysis. GUSTO revisited by Reverend Bayes. JAMA. 1995;273:871-5.
5.
Browne RH. Bayesian analysis and the GUSTO trial. Global Utilization of Streptokinase and Tissue Plasminogen Activator in Occluded Coronary Arteries [Letter]. JAMA. 1995;274:873.
6.
Good I. Probability and the Weighing of Evidence. New York: Charles Griffin; 1950.
7.
Cornfield J. The Bayesian outlook and its application. Biometrics. 1969;25:617-57.
8.
Berger JOBerry DA. Statistical analysis and the illusion of objectivity. American Scientist. 1988;76:159-65.
9.
Berry D. Interim analyses in clinical trials: classical vs. Bayesian approaches. Stat Med. 1985;4:521-6.
10.
Belanger DMoore MTannock I. How American oncologists treat breast cancer: an assessment of the influence of clinical trials. J Clin Oncol. 1991;9:7-16.
11.
Omoigui NASilver MJRybicki LARosenthal MBerdan LGPieper Ket al . Influence of a randomized clinical trial on practice by participating investigators: lessons from the Coronary Angioplasty Versus Excisional Atherectomy Trial (CAVEAT). CAVEAT I and II Investigators. J Am Coll Cardiol. 1998;31:265-72.
12.
Goodman SNRoyall R. Evidence and scientific research. Am J Public Health. 1988;78:1568-74.
13.
Royall R. Statistical Evidence: A Likelihood Primer. Monographs on Statistics and Applied Probability, #71. London: Chapman and Hall; 1997.
14.
Edwards A. Likelihood. Cambridge, UK: Cambridge Univ Pr; 1972.
15.
Goodman SN. Meta-analysis and evidence. Control Clin Trials. 1989; 10:188-204, 435.
16.
Efron B.. Empirical Bayes methods for combining likelihoods. Journal of the American Statistical Association. 1996;91:538-50.
17.
Hardy RJThompson SG. A likelihood approach to meta-analysis with random effects. Stat Med. 1996;15:619-29.
18.
Berger J. Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag; 1985.
19.
Edwards WLindman HSavage L. Bayesian statistical inference for psychological research. Psychol Rev. 1963;70:193-242.
20.
Diamond GAForrester JS. Clinical trials and statistical verdicts: probable grounds for appeal. Ann Intern Med. 1983;98:385-94.
21.
Lilford RBraunholtz D. The statistical basis of public policy: a paradigm shift is overdue. BMJ. 1996;313:603-7.
22.
Peto R. Why do we need systematic overviews of randomized trials? Stat Med. 1987;6:233-44.
23.
Pogue JYusuf S. Overcoming the limitations of current meta-analysis of randomised controlled trials. Lancet. 1998;351:47-52.
24.
Fisher R. Statistical Methods and Scientific Inference. 3d ed. New York: Macmillan; 1973.
25.
Jeffreys H. Theory of Probability. 2d ed. Oxford: Oxford Univ Pr; 1961.
26.
Kass RRaftery A. Bayes Factors. Journal of the American Statistical Association. 1995;90:773-95.
27.
Cornfield J. A Bayesian test of some classical hypotheses—with applications to sequential clinical trials. Journal of the American Statistical Association. 1966;61:577-94.
28.
Kass RGreenhouse J. Comments on “Investigating therapies of potentially great benefit: ECMO” (by JH Ware). Statistical Science. 1989;4:310-7.
29.
Spiegelhalter D, Freedman L, Parmar M. Bayesian approaches to randomized trials. Journal of the Royal Statistical Society, Series A. 1994; 157:357- 87.
30.
Berger JSellke T. Testing a point null hypothesis: the irreconcilability of p-values and evidence. Journal of the American Statistical Association. 1987;82:112-39.
31.
Bayarri M, Berger J. Quantifying surprise in the data and model verification. Proceedings of the 6th Valencia International Meeting on Bayesian Statistics, 1998. 1998:1-18.
32.
Carlin CLouis T. Bayes and Empirical Bayes Methods for Data Analysis. London: Chapman and Hall; 1996.
33.
Casella GBerger R. Reconciling Bayesian and frequentist evidence in the one-sided testing problem. Journal of the American Statistical Association. 1987;82:106-11.
34.
Howard J. The 2 × 2 table: a discussion from a Bayesian viewpoint. Statistical Science. 1999;13:351-67.
35.
Cornfield J. Sequential trials, sequential analysis and the likelihood principle. American Statistician. 1966;20:18-23.
36.
Savitz DAOlshan AF. Multiple comparisons and related issues in the interpretation of epidemiologic data. Am J Epidemiol. 1995;142:904-8.
37.
Perneger T. What's wrong with Bonferroni adjustments. BMJ. 1998;316:1236-8.
38.
Goodman SN. Multiple comparisons, explained. Am J Epidemiol. 1998;147:807-12.
39.
Thomas DCSiemiatycki JDewar RRobins JGoldberg MArmstrong BG. The problem of multiple inference in studies designed to generate hypotheses. Am J Epidemiol. 1985;122:1080-95.
40.
Greenland SRobins JM. Empirical-Bayes adjustments for multiple comparisons are sometimes useful. Epidemiology. 1991;2:244-51.
41.
Rothman KJ. No adjustments are needed for multiple comparisons. Epidemiology. 1990;11:43-6.
42.
Berry DA. A case for Bayesianism in clinical trials. Stat Med. 1993;12:1377-93.
43.
Chaloner KChurch TLouis TMatts J. Graphical elicitation of a prior distribution for a clinical trial. The Statistician. 1993;42:341-53.
44.
Chaloner K. Elicitation of prior distributions. In: Berry D, Stangl D, eds. Bayesian Biostatistics. New York: Marcel Dekker; 1996.
45.
Freedman L. Bayesian statistical methods [Editorial]. BMJ. 1996;313:569-70.
46.
Fayers PMAshby DParmar MK. Tutorial in biostatistics: Bayesian data monitoring in clinical trials. Stat Med. 1997;16:1413-30.
47.
Etzioni RDKadane JB. Bayesian statistical methods in public health and medicine. Annu Rev Public Health. 1995;16:23-41.
48.
Berry DA. Benefits and risks of screening mammography for women in their forties: a statistical appraisal. J Natl Cancer Inst. 1998;90:1431-9.
49.
Hughes MD. Reporting Bayesian analyses of clinical trials. Stat Med. 1993;12:1651-64.
50.
Berry DA, Stangl D, eds. Bayesian Biostatistics. New York: Marcel Dekker; 1996.
51.
Berry DA. Decision analysis and Bayesian methods in clinical trials. Cancer Treat Res. 1995;75:125-54.
52.
Spiegelhalter D, Thomas A, Best N, Gilks W. BUGS: Bayesian Inference Using Gibbs Sampling. Cambridge, UK: MRC Biostatistics Unit; 1998. Available at www.mrc-bsu.cam.ac.uk/bugs.
53.
Rubin D. Bayesianly justifiable and relevant frequency calculations for the applied statistician. Annals of Statistics. 1984;12:1151-72.
54.
Shafer G. Savage revisited. Statistical Science. 1986;1:463-501.
55.
Walley P. Statistical Reasoning with Imprecise Probabilities. London: Chapman and Hall; 1991.
56.
Tversky AKahneman D. Judgment under uncertainty: heuristics and biases. In: Slovic P, Tversky A, Kahneman D, eds. Judgment under Uncertainty: Heuristics and Biases. Cambridge: Cambridge Univ Pr; 1982:1-20.
57.
Bacon F. De Augmentis Scientarium, Book I (1605). In: Curtis C, Greenslet F, eds. The Practical Cogitator. Boston: Houghton Mifflin; 1962.

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Published In

cover image Annals of Internal Medicine
Annals of Internal Medicine
Volume 130Number 1215 June 1999
Pages: 1005 - 1013

History

Published in issue: 15 June 1999
Published online: 15 August 2000

Keywords

Authors

Affiliations

Steven N. Goodman, MD, PhD
From Johns Hopkins University School of Medicine, Baltimore, Maryland.
Acknowledgments: The author thanks Dan Heitjan, Russell Localio, Harold Lehmann, and Michael Berkwitz for helpful comments on earlier versions of this article. The views expressed are the sole responsibility of the author.
Corresponding Author: Steven N. Goodman, MD, PhD, Johns Hopkins University, 550 North Broadway, Suite 409, Baltimore, MD 21205; e-mail, [email protected].

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Steven N. Goodman. Toward Evidence-Based Medical Statistics. 2: The Bayes Factor. Ann Intern Med.1999;130:1005-1013. doi:10.7326/0003-4819-130-12-199906150-00019

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