- Email: [email protected]
Prime time for Bayes
Prime Time for Bayes Joseph B. Kadane Department
of Statistics,
Carnegie
Mellon
University,
Pittsburgh,
Pennsylvania
ABSTRACT: This paper reviews what Bayesian statistics is and gives pointers to the literature. The need for a subjectively determined prior distribution, likelihood, and loss function is often taken to be the principal disadvantage of Bayesian statistics. This paper argues that the requirement that these be explicitly stated is a distinct Bayesian advantage. Advances in Bayesian technology make it ready now to be the main inferential tool for clinical trials. KEY WORDS: Elicitation, loss function, prior distribution, posterior distribution, objectivity, subjectivity, utility function
WHAT IS BAYESIAN ANALYSIS? Before discussing the advantages and disadvantages of Bayesian analysis, I think it is useful to present a definition. By a Bayesian analysis I mean one in which the parameters, not having been observed, are treated as random whereas the data, having been observed, are treated as fixed, By contrast, I take a classical analysis to do just the opposite: to treat the data as random even after observation, and to treat the parameter as a fixed unknown constant not having a probability distribution. Some introductory texts in Bayesian analysis are given in Refs. l-4. More advanced books are cited in Refs. 5-7. A good book for helping to appreciate the distinction between classical and Bayesian analysis is that in Ref. 8. WHY BAYESIAN ANALYSIS? To my mind, the most important reason for thinking about statistics in a Bayesian way is that Bayesian statistics is internally consistent. Every Bayesian with the same prior, likelihood function, and data computes the same posterior distribution, and every Bayesian with the same opinion (prior or posterior) and the same utility or loss function will find the same optimal action and the same expected utility or loss. Thus all of the problematic matters-choice of prior, likelihood, and utility (or loss) -are made public and as such are subject to criticism. The situation for classical analysis is very different. There are any number of classical “principles,” each having serious flaws and counterexamples. The defense of using each principle only when it is “sensible” is vague because sensibility is rarely
Address reprint requests to: 1. B. Kadane, Department Pittsburgh, Pennsylvania 15213. Received May 30, 1994; revised March 16, 7995. Controlled Clinical Trials 16:313-318 (199.5) 0 Else&r Science Inc. 1995 655 Avenue of the Americas, New York, NY 10010
of Statistics, Carnegie Mellon University,
0197-2456/95/$9.50 SSDI 0197.2456(95)ooO72-0
J. B. Kadane
314
rigorously defined. Thus classical statistics looks like an unordered collection of ideas, each somewhat flawed, but with little agreement on”principles of principles” to guide their use. Furthermore, the reasons why a classical statistician chose a particular form of analysis, and the alternatives considered and rejected, are rarely made explicit. Hence classical analyses are often opaque. Second, I believe that Bayesian analysis requires that the right, difficult questions be addressed. In addition to the specification of a likelihood (which many classical analyses do as well), Bayesian analysis requires an explicit prior distribution. This is the occasion for analysts to state publicly their beliefs (and prejudices) about the phenomenon at hand. These beliefs guide all analyses, Bayesian or classical, but only Bayesian analysis requires their explicit statement. In an attempt to “justify” a prior (i.e., to persuade readers that their priors might be similar), reasons and data will be cited and their relevance argued and weighed. Far from being a weakness, the requirement of stating a prior is a benefit because it allows much more accurate communication between writer and reader. By the same token, the specification of a loss function (or its negative, a utility function) is an opportunity (and requirement) to state explicitly who is making a decision, what the possible decisions are, and what is the value of the consequences that may ensue from each decision in each possible state of the world. In order to make a decision recommendation understandable and persuasive, it seems to me that an explicit exposition of the underlying values is essential. Hence I regard the requirement to model beliefs (priors and likelihoods) and values (utility or loss), while often difficult and thought provoking, as a step toward honesty in analysis. To obscure them does neither the analysis nor the science behind it a favor. Finally, Bayesian analysis is undergoing very rapid advances technically at present. Computations that would have been impossible until very recently are now routine. Clinical trials would benefit by borrowing this technology, briefly reviewed in the section, “How?” WHY NOT? There are some poor reasons for doing Bayesian analysis. The first is that it is now becoming fashionable. This can lead to poorly thought out attempts to imitate classical analysis in a Bayesian guise, e.g., by the overly casual use of reference priors. As explained above, I believe that this obviates some of the advantages of doing a Bayesian analysis to begin with. Another poor reason to do Bayesian analysis is the thought that by cleverly choosing a prior one can make the answer come out however one wishes. Generally this is the obverse of the danger discussed above, as it requires using overly opinionated priors. There are also some bad reasons for not doing a Bayesian analysis. One is the argument that people do not behave “rationally” as the Bayesian axioms suggest they should. Certainly there is a large body of literature supporting the claim that people in general depart in systematic ways from the axioms (see, for example, the articles in Ref. 9). However, the conclusion does not follow that Bayesian analysis is inappropriate because people are not able to intuit the Bayesian calculations unaided. Because unaided human beings make systematic errors, the value of such analysis is greater than it might otherwise seem. The Bayesian argument
Prime Time for Bayes
31.5
is a normative one (that a reasonable person would want to make decisions maximizing expected utility if he or she could), not a descriptive one (he or she can). Second, some regard it as a disadvantage that Bayesian analysis is avowedly subjective, as opposed to a classical statistics, which claims to be objective. It would be wonderful if claiming objectivity for an analysis would make it objective. But various purportedly objective classical analyses come to different conclusions (by applying different principles, as explained above). If there are not principles of principles, how can one analysis be proclaimed the objective truth? What is the status of the other analysis? What conclusion can be drawn from the fact that different individuals come to different interpretations of the same classical, and supposedly objective, analysis? I believe that the claim of objectivity for classical analysis is mere propaganda. By contrast, Bayesian analysis, by stressing the subjectivity of the inputs to an analysis, also encourages authors to justify their choices. Since Bayesian analysis admits its subjectivity, it encourages readers to form their own opinion of the import of the data and of the appropriate decisions (therapeutic or otherwise) to be made. Thus Bayesianism gives up the (illegitimate) attempt to coerce readers that a claim of objectivity implies. HOW? I conceive the methodology of Bayesian statistics in two parts: elicitation and computation. Elicitation of prior distributions has some subtleties because often substantive experts do not understand much probability theory. Additionally, even statisticians may find it difficult to state distributions for parameters, e.g., in hierarchical models and in models involving a covariance matrix. For these reasons, methodology has been developed to present questions in a predictive form, asking questions only about quantities of observable variables [lo-14). A recent example of the use of prior elicitation methods for use in a clinical trial is given in Ref. 15. A partial alternative to elicitation of a prior distribution is to choose a class of prior distributions and to see the extent to which the posterior distribution, or some aspect of it, varies within that class. A review of this literature is found in Ref. 16. The use of elicitation for losses (or utilities) is less well developed; I believe that greater effort will be needed on this topic. Some early work has been reported [17,18]. A good general introduction to the area, with emphasis on the psychological aspects, is given in Ref. lo. The second important area of Bayesian methodology is computation, especially of posterior distributions. In recent years there has been an explosion of activity in this area. Building from early ideas based on sophisticated quadrature [20], some of the important methods are Monte Carlo [21], Laplace approximation [22, 23, and references cited therein], and the techniques variously known as Markov chain Monte Carlo, Gibbs sampling, or successive substitution sampling [24-261. FOR WHAT? Bayesian methods are useful both for analysis and for design. Because analysis is the more frequent application, I discuss if first. Unlike classical theories, Bayesian analysis relates naturally to a rational decision calculus. The posterior probabilities calculated as described in the section
316
J. B. Kadane “How?” are exactly what is needed to find optimal decisions using utilities elicited using the methods of the section ‘Why Not?” This relationship is a major strength of the Bayesian method. By contrast, a typical classical analysis calculates a type I error, i.e., the probability, if the null hypothesis were true, of observing data as or more extreme than those observed. But of what use is this number? The tempting error is to interpret a type I error as if it were the probability of the null hypothesis after observing the data, which it is not. In fact, a classical statistician cannot speak of such a probability because it requires treating the parameters as random. Nonetheless many of our students (and, I would bet, many of our clients) drift into this misinterpretation because it is so natural (and so Bayesian) to do so. Similarly a confidence interval is the set of simple null hypotheses, which, had I been testing them at the specified level, I would not have rejected. Again it is not clear why such a set should be of interest unless one makes the natural error of thinking of the parameter as random and the confidence set as containing the parameter with a specified probability. Again, this is a statement only a Bayesian can make, although confidence intervals are often so misinterpreted. I find the classical quantities useless for making decisions and believe that they are widely misinterpreted as Bayesian because the Bayesian quantities are more natural. There is a growing literature applying Bayesian analysis in clinical trials. An excellent recent review is given in Refs. 27 and 28. Another fast-growing area of application is meta-analysis, in which Bayesian analysis is particularly appealing and intuitive [29,30]. The other major application of Bayesian ideas in clinical trials is in design. This is a natural area for use because obviously prior opinion, whether formally modeled or not, plays a crucial role in design. Traditionally, Bayesian design modeled the designer and the user of the data as the same actor, with the same prior and utility (51. M ore recently, a number of authors have explored models in which the designer and user need not be the same [31]. Clinical trials are even more complicated than this, and they involve patients and attending physicians as well as designers of the trial and consumers of the data. A full model about how the interests of each party enter into such a design has not yet been explored, to my knowledge, but I believe that the theoretical underpinnings for such work are now in place [32]. Perhaps such a model would help in gaining an appreciation for the role that randomization plays in clinical trials 1331. An application of some of these notions is proposed in Ref. 34 and whose implementation is described in Ref. 3.5. In this design, an indicator of patient well-being is chosen and important covariates (in addition to treatment) are identified. Five experts’ priors on this indicator (as a function of treatment and covariates) are elicited and updated using evidence gathered in the trial. The ethical principal proposed is that unless at least one expert would “recommend” the use of a treatment for a patient with the covariates presented, that treatment should not be assigned. Thus the Bayesian aspect of this design is solely to protect patients. This kind of design is useful only for acute conditions in which the data from earlier patients might be substantial, thus aiding in protecting later patients from untoward treatments. A fuller exposition of this trial and the ideas behind it can be found in Ref. 36. It may be regarded as a way of implementing Freedman’s idea of “clinical equipoise” [ 371.
Prime Time for Bayes
317
Bayesian design contrasts with classical methods, particularly power calculations to determine sample size. To determine a sample size, a classical statistician chooses an arbitrary point in the space of the alternative hypothesis and an arbitrary level of type II error to be permitted at that point. Classical statistics gives no guidelines about how these choices are to be made (which is convenient if one has to justify a predetermined budget).
PRIME TIME FOR BAYES! Bayesian ideas are a useful set of tools for clinical trials. They are becoming more widespread in biostatistics generally [38-411. There are both methodologic innovations and many issues of implementation to be resolved. The effort is worth undertaking, though, because of the reward of having a foundation for clinical trials that makes sense. This work was supported by ONR Grant NOOO14-89-J-1851,NSF Grant SES-9123370, and NSF Grant DMS-9302557.
REFERENCES 1. Lindley DV: Making Decisions, 2nd ed. John Wiley and Sons, Chichester, 2. Lee PM: Bayesian Statistics: An Introduction.
1985
Edward Arnold, London, 1989
3. Press SJ: Bayesian Statistics. John Wiley and Sons, New York, 1989 4. Gelman A, Carlin JB, Stem HS, Rubin DB: Statistical Methods: A Bayesian Approach. Chapman and Hall, New York, 1995 5. Lindley DV. Bayesian Statistics: A Review. SIAM, Philadelphia, 1972 6. Bernard0 J, Smith AFM: Bayesian Theory. John Wiley and Sons, Chichester, 1994 7. O’Hagan AO: Kendall’s Advanced Theory of Statistics 2B: Bayesian Inference. Edward Arnold, London, 1994 8. Barnett V: Comparative Statistical Inference, John Wiley and Sons, Chichester, 1982 9. Kahneman D, Slavic P, Tversky A: Judgment Under Uncertainty: Heuristics and Biases. Cambridge University Press, Cambridge, 1982 10. Kadane JB: Predictive and structural methods for eliciting prior distributions, in Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys, A Zellner, ed. North Holland, Amsterdam, 1980, pp 89-93 11. Kadane JB, Dickey JM, Winkler R, Smith W, Peters S: Interactive elicitation of opinion for a normal linear model, J Am Stat Assoc 75:845-854, 1980 12. Winkler RL: Prior information, predictive distributions and Bayesian model building, in Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys, A Zellner, ed. North Holland, Amsterdam, 1980 13. Kadane JB, Wolfson L: Priors for the design and analysis of clinical trials, in Bayesian Biostatistics, DA Berry and D Stangl, eds. Marcel Dekker, New York, 1995 14. Chaloner K: The elicitation of prior distributions, in Bayesian Biostatistics, DA Berry and D Stangl, eds. Marcel Dekker, New York, 1995 15. Chaloner K, Church T, Louis A, Matts JP: Graphical elicitation of a prior distribution for a clinical trial, The Statistician 42:341-353, 1993 16. Berger JO: A review of recent developments in robust Bayesian analysis, TEST 35124 (with discussion), 1994 17. Mosteller F, Nogee P: Elicitation of utilities-an J Pol Econ 59:371-404, 1951
experimental measurement of utility,
J. B. Kadane
318 18. Savage LJ: Elicitation of personal probabilities and expectations, 1971
JASA 66:781-801,
19. von Winterfeldt D, Edwards W: Decision Analysis and Behavioral Research. Cambridge University Press, Cambridge, UK, 1986 20. Naylor JC, Smith AFM: Applications of a method for the efficient computation posterior distributions. Appl Stat 31:214-225, 1982
of
21. Geweke J: Bayesian inference in econometric models using Monte Carlo integration, Econometrica 57:1317-1339, 1989 22. Tierney L, Kadane JB: Accurate approximations for posterior moments and marginal densities. J Am Stat Assoc 81:82-86, 1986 23. Kass RE, Tiemey L, Kadane JB: Asymptotics in Bayesian computation, in Bayesian Statistics, 88. 3, J Bemardo, M DeGroot, DV Lindley, and AMF Smith, eds. Oxford University Press, New York, pp 261-278, 1988 24. Gelfand A, Smith AFM: Sampling based approaches to calculating marginal densities. JASA 85:398-409, 1990 25. Casella G, George EL Explaining the Gibbs sampler. Am Statistician 46:167-174,1992 26. Gelman A, Rubin DB: Inference from iterative simulation using multiple sequences. With discussion. Stat Sci 7:457-511, 1992 27. Spiegelhalter DJ, Freedman LS, Parmar MKB: Bayesian approaches to randomized trials. With discussion. JRSS, Series A, 157:357-416, 1994 28. Greenhouse JB: On some applications of Bayesian methods in cancer clinical trials. Stat Med 11:37-53, 1992 29. DuMouchel W: Bayesian metaanalysis. in Statistical Methodology in the Pharmaceutical Sciences, DA Berry, ed. Marcel Dekker, New York, pp 509-529, 1989 30. Lau J, Antman EM, Jimenez-Silva J, Kupelink B, Mosteller SF, Chalmers TC: Cumulative meta-analysis of therapeutic trials for myocardial infarction. N Engl J Med 327: 248-254, 1992 31. Etzioni R, Kadane JB: Optimal experimental design for another’s analysis. J Am Stat Assoc 88:1404-1411, 1993 32. Kadane, JB: Several Bayesians: a review. With discussion. TEST 2(1-2):1-32,
1993
33. Kadane JB, Seidenfeld T: Randomization in a Bayesian perspective. Stat Plann Inf 25: 329-345, 1990 34. Kadane JB, Sedransk N: Toward a more ethical clinical trial, in Bayesian Statistics, JM Bernardo, MH DeGroot, DV Lindley, and AFM Smith, eds., University Press, Valencia, pp 329-338, 1980 35. Kadane J: Progress toward a more ethical method for clinical trials, J Med Phil 11: 385-404, 1986 36. Kadane J (ed.): Clinical Trials: Bayesian Methods and Ethics, John Wiley and Sons, New York, 1995 37. Freedman B: Equipoise and the ethics of medical research. N Engl J Med 317:141-145, 1987 38. Breslow N: Biostatistics and Bayes, Stat Sci 5:267-298,
1990
39. Etzioni RD, Kadane JB: Bayesian statistical methods in public health and medicine, Ann Rev Public Health 16:23-41, 1995 40. Berry DA, Stangl D, (eds.): Bayesian Biostatistics.
Marcel Dekker, New York, 1995
41. Berry DA: A case for Bayesianism in clinical trials (with discussion). Stat Med 21: 1377-1404, 1993
of Statistics,
Carnegie
Mellon
University,
Pittsburgh,
Pennsylvania
ABSTRACT: This paper reviews what Bayesian statistics is and gives pointers to the literature. The need for a subjectively determined prior distribution, likelihood, and loss function is often taken to be the principal disadvantage of Bayesian statistics. This paper argues that the requirement that these be explicitly stated is a distinct Bayesian advantage. Advances in Bayesian technology make it ready now to be the main inferential tool for clinical trials. KEY WORDS: Elicitation, loss function, prior distribution, posterior distribution, objectivity, subjectivity, utility function
WHAT IS BAYESIAN ANALYSIS? Before discussing the advantages and disadvantages of Bayesian analysis, I think it is useful to present a definition. By a Bayesian analysis I mean one in which the parameters, not having been observed, are treated as random whereas the data, having been observed, are treated as fixed, By contrast, I take a classical analysis to do just the opposite: to treat the data as random even after observation, and to treat the parameter as a fixed unknown constant not having a probability distribution. Some introductory texts in Bayesian analysis are given in Refs. l-4. More advanced books are cited in Refs. 5-7. A good book for helping to appreciate the distinction between classical and Bayesian analysis is that in Ref. 8. WHY BAYESIAN ANALYSIS? To my mind, the most important reason for thinking about statistics in a Bayesian way is that Bayesian statistics is internally consistent. Every Bayesian with the same prior, likelihood function, and data computes the same posterior distribution, and every Bayesian with the same opinion (prior or posterior) and the same utility or loss function will find the same optimal action and the same expected utility or loss. Thus all of the problematic matters-choice of prior, likelihood, and utility (or loss) -are made public and as such are subject to criticism. The situation for classical analysis is very different. There are any number of classical “principles,” each having serious flaws and counterexamples. The defense of using each principle only when it is “sensible” is vague because sensibility is rarely
Address reprint requests to: 1. B. Kadane, Department Pittsburgh, Pennsylvania 15213. Received May 30, 1994; revised March 16, 7995. Controlled Clinical Trials 16:313-318 (199.5) 0 Else&r Science Inc. 1995 655 Avenue of the Americas, New York, NY 10010
of Statistics, Carnegie Mellon University,
0197-2456/95/$9.50 SSDI 0197.2456(95)ooO72-0
J. B. Kadane
314
rigorously defined. Thus classical statistics looks like an unordered collection of ideas, each somewhat flawed, but with little agreement on”principles of principles” to guide their use. Furthermore, the reasons why a classical statistician chose a particular form of analysis, and the alternatives considered and rejected, are rarely made explicit. Hence classical analyses are often opaque. Second, I believe that Bayesian analysis requires that the right, difficult questions be addressed. In addition to the specification of a likelihood (which many classical analyses do as well), Bayesian analysis requires an explicit prior distribution. This is the occasion for analysts to state publicly their beliefs (and prejudices) about the phenomenon at hand. These beliefs guide all analyses, Bayesian or classical, but only Bayesian analysis requires their explicit statement. In an attempt to “justify” a prior (i.e., to persuade readers that their priors might be similar), reasons and data will be cited and their relevance argued and weighed. Far from being a weakness, the requirement of stating a prior is a benefit because it allows much more accurate communication between writer and reader. By the same token, the specification of a loss function (or its negative, a utility function) is an opportunity (and requirement) to state explicitly who is making a decision, what the possible decisions are, and what is the value of the consequences that may ensue from each decision in each possible state of the world. In order to make a decision recommendation understandable and persuasive, it seems to me that an explicit exposition of the underlying values is essential. Hence I regard the requirement to model beliefs (priors and likelihoods) and values (utility or loss), while often difficult and thought provoking, as a step toward honesty in analysis. To obscure them does neither the analysis nor the science behind it a favor. Finally, Bayesian analysis is undergoing very rapid advances technically at present. Computations that would have been impossible until very recently are now routine. Clinical trials would benefit by borrowing this technology, briefly reviewed in the section, “How?” WHY NOT? There are some poor reasons for doing Bayesian analysis. The first is that it is now becoming fashionable. This can lead to poorly thought out attempts to imitate classical analysis in a Bayesian guise, e.g., by the overly casual use of reference priors. As explained above, I believe that this obviates some of the advantages of doing a Bayesian analysis to begin with. Another poor reason to do Bayesian analysis is the thought that by cleverly choosing a prior one can make the answer come out however one wishes. Generally this is the obverse of the danger discussed above, as it requires using overly opinionated priors. There are also some bad reasons for not doing a Bayesian analysis. One is the argument that people do not behave “rationally” as the Bayesian axioms suggest they should. Certainly there is a large body of literature supporting the claim that people in general depart in systematic ways from the axioms (see, for example, the articles in Ref. 9). However, the conclusion does not follow that Bayesian analysis is inappropriate because people are not able to intuit the Bayesian calculations unaided. Because unaided human beings make systematic errors, the value of such analysis is greater than it might otherwise seem. The Bayesian argument
Prime Time for Bayes
31.5
is a normative one (that a reasonable person would want to make decisions maximizing expected utility if he or she could), not a descriptive one (he or she can). Second, some regard it as a disadvantage that Bayesian analysis is avowedly subjective, as opposed to a classical statistics, which claims to be objective. It would be wonderful if claiming objectivity for an analysis would make it objective. But various purportedly objective classical analyses come to different conclusions (by applying different principles, as explained above). If there are not principles of principles, how can one analysis be proclaimed the objective truth? What is the status of the other analysis? What conclusion can be drawn from the fact that different individuals come to different interpretations of the same classical, and supposedly objective, analysis? I believe that the claim of objectivity for classical analysis is mere propaganda. By contrast, Bayesian analysis, by stressing the subjectivity of the inputs to an analysis, also encourages authors to justify their choices. Since Bayesian analysis admits its subjectivity, it encourages readers to form their own opinion of the import of the data and of the appropriate decisions (therapeutic or otherwise) to be made. Thus Bayesianism gives up the (illegitimate) attempt to coerce readers that a claim of objectivity implies. HOW? I conceive the methodology of Bayesian statistics in two parts: elicitation and computation. Elicitation of prior distributions has some subtleties because often substantive experts do not understand much probability theory. Additionally, even statisticians may find it difficult to state distributions for parameters, e.g., in hierarchical models and in models involving a covariance matrix. For these reasons, methodology has been developed to present questions in a predictive form, asking questions only about quantities of observable variables [lo-14). A recent example of the use of prior elicitation methods for use in a clinical trial is given in Ref. 15. A partial alternative to elicitation of a prior distribution is to choose a class of prior distributions and to see the extent to which the posterior distribution, or some aspect of it, varies within that class. A review of this literature is found in Ref. 16. The use of elicitation for losses (or utilities) is less well developed; I believe that greater effort will be needed on this topic. Some early work has been reported [17,18]. A good general introduction to the area, with emphasis on the psychological aspects, is given in Ref. lo. The second important area of Bayesian methodology is computation, especially of posterior distributions. In recent years there has been an explosion of activity in this area. Building from early ideas based on sophisticated quadrature [20], some of the important methods are Monte Carlo [21], Laplace approximation [22, 23, and references cited therein], and the techniques variously known as Markov chain Monte Carlo, Gibbs sampling, or successive substitution sampling [24-261. FOR WHAT? Bayesian methods are useful both for analysis and for design. Because analysis is the more frequent application, I discuss if first. Unlike classical theories, Bayesian analysis relates naturally to a rational decision calculus. The posterior probabilities calculated as described in the section
316
J. B. Kadane “How?” are exactly what is needed to find optimal decisions using utilities elicited using the methods of the section ‘Why Not?” This relationship is a major strength of the Bayesian method. By contrast, a typical classical analysis calculates a type I error, i.e., the probability, if the null hypothesis were true, of observing data as or more extreme than those observed. But of what use is this number? The tempting error is to interpret a type I error as if it were the probability of the null hypothesis after observing the data, which it is not. In fact, a classical statistician cannot speak of such a probability because it requires treating the parameters as random. Nonetheless many of our students (and, I would bet, many of our clients) drift into this misinterpretation because it is so natural (and so Bayesian) to do so. Similarly a confidence interval is the set of simple null hypotheses, which, had I been testing them at the specified level, I would not have rejected. Again it is not clear why such a set should be of interest unless one makes the natural error of thinking of the parameter as random and the confidence set as containing the parameter with a specified probability. Again, this is a statement only a Bayesian can make, although confidence intervals are often so misinterpreted. I find the classical quantities useless for making decisions and believe that they are widely misinterpreted as Bayesian because the Bayesian quantities are more natural. There is a growing literature applying Bayesian analysis in clinical trials. An excellent recent review is given in Refs. 27 and 28. Another fast-growing area of application is meta-analysis, in which Bayesian analysis is particularly appealing and intuitive [29,30]. The other major application of Bayesian ideas in clinical trials is in design. This is a natural area for use because obviously prior opinion, whether formally modeled or not, plays a crucial role in design. Traditionally, Bayesian design modeled the designer and the user of the data as the same actor, with the same prior and utility (51. M ore recently, a number of authors have explored models in which the designer and user need not be the same [31]. Clinical trials are even more complicated than this, and they involve patients and attending physicians as well as designers of the trial and consumers of the data. A full model about how the interests of each party enter into such a design has not yet been explored, to my knowledge, but I believe that the theoretical underpinnings for such work are now in place [32]. Perhaps such a model would help in gaining an appreciation for the role that randomization plays in clinical trials 1331. An application of some of these notions is proposed in Ref. 34 and whose implementation is described in Ref. 3.5. In this design, an indicator of patient well-being is chosen and important covariates (in addition to treatment) are identified. Five experts’ priors on this indicator (as a function of treatment and covariates) are elicited and updated using evidence gathered in the trial. The ethical principal proposed is that unless at least one expert would “recommend” the use of a treatment for a patient with the covariates presented, that treatment should not be assigned. Thus the Bayesian aspect of this design is solely to protect patients. This kind of design is useful only for acute conditions in which the data from earlier patients might be substantial, thus aiding in protecting later patients from untoward treatments. A fuller exposition of this trial and the ideas behind it can be found in Ref. 36. It may be regarded as a way of implementing Freedman’s idea of “clinical equipoise” [ 371.
Prime Time for Bayes
317
Bayesian design contrasts with classical methods, particularly power calculations to determine sample size. To determine a sample size, a classical statistician chooses an arbitrary point in the space of the alternative hypothesis and an arbitrary level of type II error to be permitted at that point. Classical statistics gives no guidelines about how these choices are to be made (which is convenient if one has to justify a predetermined budget).
PRIME TIME FOR BAYES! Bayesian ideas are a useful set of tools for clinical trials. They are becoming more widespread in biostatistics generally [38-411. There are both methodologic innovations and many issues of implementation to be resolved. The effort is worth undertaking, though, because of the reward of having a foundation for clinical trials that makes sense. This work was supported by ONR Grant NOOO14-89-J-1851,NSF Grant SES-9123370, and NSF Grant DMS-9302557.
REFERENCES 1. Lindley DV: Making Decisions, 2nd ed. John Wiley and Sons, Chichester, 2. Lee PM: Bayesian Statistics: An Introduction.
1985
Edward Arnold, London, 1989
3. Press SJ: Bayesian Statistics. John Wiley and Sons, New York, 1989 4. Gelman A, Carlin JB, Stem HS, Rubin DB: Statistical Methods: A Bayesian Approach. Chapman and Hall, New York, 1995 5. Lindley DV. Bayesian Statistics: A Review. SIAM, Philadelphia, 1972 6. Bernard0 J, Smith AFM: Bayesian Theory. John Wiley and Sons, Chichester, 1994 7. O’Hagan AO: Kendall’s Advanced Theory of Statistics 2B: Bayesian Inference. Edward Arnold, London, 1994 8. Barnett V: Comparative Statistical Inference, John Wiley and Sons, Chichester, 1982 9. Kahneman D, Slavic P, Tversky A: Judgment Under Uncertainty: Heuristics and Biases. Cambridge University Press, Cambridge, 1982 10. Kadane JB: Predictive and structural methods for eliciting prior distributions, in Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys, A Zellner, ed. North Holland, Amsterdam, 1980, pp 89-93 11. Kadane JB, Dickey JM, Winkler R, Smith W, Peters S: Interactive elicitation of opinion for a normal linear model, J Am Stat Assoc 75:845-854, 1980 12. Winkler RL: Prior information, predictive distributions and Bayesian model building, in Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys, A Zellner, ed. North Holland, Amsterdam, 1980 13. Kadane JB, Wolfson L: Priors for the design and analysis of clinical trials, in Bayesian Biostatistics, DA Berry and D Stangl, eds. Marcel Dekker, New York, 1995 14. Chaloner K: The elicitation of prior distributions, in Bayesian Biostatistics, DA Berry and D Stangl, eds. Marcel Dekker, New York, 1995 15. Chaloner K, Church T, Louis A, Matts JP: Graphical elicitation of a prior distribution for a clinical trial, The Statistician 42:341-353, 1993 16. Berger JO: A review of recent developments in robust Bayesian analysis, TEST 35124 (with discussion), 1994 17. Mosteller F, Nogee P: Elicitation of utilities-an J Pol Econ 59:371-404, 1951
experimental measurement of utility,
J. B. Kadane
318 18. Savage LJ: Elicitation of personal probabilities and expectations, 1971
JASA 66:781-801,
19. von Winterfeldt D, Edwards W: Decision Analysis and Behavioral Research. Cambridge University Press, Cambridge, UK, 1986 20. Naylor JC, Smith AFM: Applications of a method for the efficient computation posterior distributions. Appl Stat 31:214-225, 1982
of
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