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Bayes' theorem: An idea whose time has come?
EDITORIALS
Bayes’ Theorem: An Idea Whose Time Has Come?*
HENRY
N. WAGNER,
Baltimore,
Jr.,
MD,
FACC
Maryland
Fifty years ago, Paul Dudley White1 wrote: “The capacity to elicit significant symptoms and signs, the ability to analyze these symptoms and signs after they have been found, the knowledge of the best therapy and, not the least important of all, the appraisal of the sort of person to be treated are all essential to the satisfactory practice of cardiology.” These principles are as true today as they were then, but with an important difference: The number of facts that the cardiologist can obtain concerning the structure and function of his patient’s heart and circulatory system has increased beyond the wildest dreams of the cardiologists in Paul Dudley White’s time. The information provides both an opportunity and a challenge: an opportunity to provide better care, a challenge to use the data effectively. Diagnosis assisted by computer: How can all this potentially useful information be transformed into better patient care? The solution to this problem is the computer, not large computers of the type foremost in our minds, but the much smaller personal computer. The computer will not in any sense replace the cardiologist, but act as an assistant. In the foreseeable future, the ability of the human being to perceive patients and their problems is far greater than that of the most advanced computer. What the human being lacks is adequate memory and the ability to handle logically and completely masses of information. The computer supplements the senses and mind of the physician. The time has come to rid ourselves of the image of the computer as an inhuman monster waiting in the wings to replace the physician as a diagnostician, and to begin to look upon the computer as a descendant of the slide rule, as a slave, not a master.
From the Division of Nuclear Medicine, the Johns Hopkins Medical Institutions, Baltimore, Maryland. Manuscript received August 8, 1981; revised manuscript received October 26, 1981, accepted November 5, 1981. Address for reprints: Henry N. Wagner, Jr., MD, Division of Nuclear Medicine, the Johns Hopkins Medical Institutions, 615 North Wolfe Street, Baltimore, Maryland 21205.
The modern cardiologist is aware of the impact of computers on everyday life, including their use in airports, banks and, in some hospitals, in coronary care units. More and more physicians are using computers in managing the business aspects of their practice. What I am talking about is a new use for personalized microcomputers, a process that we call “diagnosis assisted by computer,” a series of programs designated by the suffix -DAC. For example, PEDAC stands for “pulmonary embolism diagnosis assisted by computer.” With these programs, the role of the physician is not diminished, but enhanced, just as the slide rule in no sense diminished the role of the engineer. The time has come for the physician to take the microcomputer to the patient’s bedside. Cardiologists can become pioneers in the application of the computer in everyday practice. Application of Bayes’ theorem in cardiology: In 1959 Ledley and Lusted2 suggested the use of Bayes’ theorem in medicine. What is Bayes’ theorem? Essentially, it states that the probability that a patient with a particular set of manifestations of illness (symptoms, signs or laboratory results) has a particular disease is directly proportional to the probability of occurrence of that particular set of findings (syndrome) in that disease multiplied by the a priori prevalence of that disease, and inversely proportional to the probability of occurrence of this set of manifestations in the general population. In 1968, I3 proposed a model of the entire diagnostic process based on the sequential application of Bayes’ theorem. Implementation of the model was not possible at the time for several reasons: (1) The data base relating manifestations and diseases was not available; (2) computer technology had not developed to the point where every physician could easily afford a computer of his own; and (3) the concept that medical diagnosis was probabilistic was not as well accepted by cardiologists as it is today. The exercise electrocardiographers took the lead in applying probability theory to cardiology in a routine way, particularly in the diagnosis of coronary artery
disease. For example, in 1977, Rifkin and Hood4 pro-
Editorials published by the Journal reflect the views of the authors and do not necessarily represent the views of the Journal or of the American College of Cardiology. ??
March 1962
The American Journal of CARDIOLOGY
Volume 49
875
EDITORIALS
posed the use of a Bayesian analysis. In the interpretation of exercise electrocardiograms, the cardiologist considers the a priori probability of coronary artery disease based on the age and sex of the patient, the presence and type of chest pain, and other factors. The degree of S-T segment depression is then combined with the a priori probability to yield the a posteriori or final probability estimate of coronary artery disease. If the patient were an asymptomatic middle-aged man with no risk factors, the pretest estimate of the probability of coronary artery disease would be 20 percent. If exercise resulted in 1.5 to 1.99 mm S-T segment depression, this fact combined with the a priori probability would yield a post-test probability of coronary artery disease of 70 percent. In 1979, Diamond and Forrester extended the idea of using Bayes’ theorem in cardiology by combining the pretest likelihood of coronary artery disease based on age, sex and symptoms with the data from four diagnostic tests: stress electrocardiography, cardiomyography, thallium scintigraphy and cardiac fluoroscopy. The program that they devised has been expanded to include other data such as gated blood pool imaging and has been incorporated into a commercially available microprocessor computer system that is inexpensive enough for use in the cardiologist’s office.6 Today, whenever a cardiologic procedure is performed in our nuclear medicine department, the nuclear physician or cardiologist uses this program to aid in the planning of the type of nuclear study to be performed and the interpretation of the results. Although the program is not yet perfected, it has proved very helpful in day to day practice. Application to nuclear cardiology studies: The essence of Bayes’ theorem is that equal importance is given to the a priori probabilities and to the results of the individual tests themselves. Each contributes to the estimation of the certainty of the final diagnosis. We assess the probability of various diseases being present at several stages of the diagnostic process, making “working diagnoses” after obtaining the medical history, after the physical examination and before and after each diagnostic test. What data do we need to apply Bayes’ theorem in practice? The most important characteristic of a diag-
nostic test is the likelihood that a given result will be found in patients with a particular disease and in normal persons. This probability is referred to as sensitivity. For example, suppose a manifestation such as a clear-cut regional defect in a thallium-201 myocardial perfusion image is observed in 40 percent of patients with transmural infarction. The sensitivity of the manifestation for that state would be 0.40. A manifestation that occurs in all patients with a given disease has a sensitivity of 1.00. The probability of a manifestation being present in all diseases (or normal persons) other than the one of interest is (1 - specificity). For example, if a manifestation occurs in 5 percent of healthy persons or persons with diseases other than the one being considered, its specificity would be 0.95.
876
March 1992
The American Journal of CARDIOLOGY
We always interpret our nuclear studies initially without knowledge of any a priori information. Then
we combine the interpretation of the nuclear data with all other information by means of Bayes’ theorem to yield the probability of a disease, such as coronary artery disease, being present. The nuclear data themselves are expressed as a likelihood ratio, defined as the ratio of the sensitivity (probability) of the particular findings of the nuclear study in the disease of interest to the probability of its occurrence in normal persons or in those with all other diseases. The likelihood ratio can also be defined as sensitivity divided by (1 - specificity). Some define sensitivity, specificity and likelihood ratio in terms of true and false positive results. I believe the time has come to go beyond this approach, which presents the problem that test results expressed only as positive or negative often represent an unnecessary and often troublesome oversimplification. A final example. Suppose we find an ejection fraction of 0.30 and a clear-cut regional wall motion abnormality of the anterior wall of the left ventricle in a gated blood pool study. If the probability of finding these abnormalities is 0.40 in patients with anterior myocardial infarction and 0.10 in normal persons or persons with other types of heart disease, then the likelihood of myocardial infarction is 4:l. Stated another way, the probability of myocardial infarction on the basis of the nuclear studies alone is 0.80. Our next step is to combine the nuclear data with the a priori data. If, for example, the patient had elevation of the S-T segments in the precordial leads, this fact combined with the nuclear data would increase the certainty to nearly 1.00. This is an example of a simple problem. A more difficult problem is presented by a woman with vague chest pain and 0.5 mm S-T segment depression in an exercise test. In this case, all the data need to be considered with proper weighting factors. Clinical approach: Most people tend to approach complicated problems by oversimplification. Many physicians order a large number of tests hoping to find a highly specific abnormality. I call this the “holy grail” or “eureka” approach to medical diagnosis. Experienced physicians pick two or three of the most important findings. around which to build a differential diagnosis. The advantage of the computer, properly programmed, is that it can remember and analyze all the data, considering every finding with the appropriate weighting factors. The physician is infinitely better than the computer in perceiving the manifestations of the patient’s illness. The computer provides information that the physician can not possibly remember and helps with the logical analysis. To be sure, Bayes’ theorem has not been universally accepted as a model for the diagnostic process. For example, Feinstein7 has stated that anyone who has practiced clinical medicine will recognize that Bayes’ approach does not resemble even a weird parody of clinical reasoning. He suggests that as clinicians we would answer a question-such as whether a patient with hemoptysis has lung cancer-not by estimating proba-
Volume 49
EDITORIALS
bilities, but by getting more data. In 1968 I1 suggested an alternative approach: “Every question that the physician asks in obtaining a medical history, every maneuver that he performs in the physical examination and every subsequent laboratory procedure that he orders should be selected because of the likelihood that the new fact will alter the estimate of the probability that the patient has a particular disease or diseases. An essential feature of the diagnostic process is its statistical or probabilistic nature. Rarely can a
diagnosis be made with absolute certainty, even by pathologists looking at histological sections. We in medicine have a tendency to assume erroneously that the best test is perfect. It almost never is, a fact revealed when a better test comes along.” Diamond (personal communication) has stated that we may be standing at the threshold of a perceptual upheaval in medicine, based on the use of the microcomputer to apply Bayes’ theorem in the cardiologist’s office or even at the bedside. In my judgment, within 10 years the computer is likely to join the stethoscope as a symbol of the cardiologist.
References 1. White PD. Errors in interpretation of cardiovascular symptoms and signs. Ann intern Med 1936;1:1703-13. 2. Ledley RGF, Lusted LB. Reasoning foundations of medical diagnosis. Science 1959;130:9-13. 3. Wagner HN Jr. Principles of Nuclear Medicine, chap II. The Diagnostic Process. Philadelphia: WB Saunders, 1968. 4. Rifkln RD, Hood WB. Bayesian analysis of electrocardiographic exercise stress testing. N Engl J Med 1977;297:681-6. 5. Diamond GH, Forrester JS. Analysis of probability as an aid in the
clinical diagnosis of coronary artery disease. N Engl J Med 1979; 300: 1350-a. 6. Dlamond GA, Forrester JS, Hirsch M, et al. Applications of conditional probability analysis to the clinical diagnosis of coronary artery disease. J Clin Invest 1980;65:1210-21. 7. Feinstein AR. Clinical biostatistics. 39. The haze of Bayes, the aerial palaces of decision analysis, and the computerized Ouija board. Clin Pharmacol Ther 1977;21:482-95.
March 1982
The American Journal of CARDIOLOGY
Volume 49
877
Bayes’ Theorem: An Idea Whose Time Has Come?*
HENRY
N. WAGNER,
Baltimore,
Jr.,
MD,
FACC
Maryland
Fifty years ago, Paul Dudley White1 wrote: “The capacity to elicit significant symptoms and signs, the ability to analyze these symptoms and signs after they have been found, the knowledge of the best therapy and, not the least important of all, the appraisal of the sort of person to be treated are all essential to the satisfactory practice of cardiology.” These principles are as true today as they were then, but with an important difference: The number of facts that the cardiologist can obtain concerning the structure and function of his patient’s heart and circulatory system has increased beyond the wildest dreams of the cardiologists in Paul Dudley White’s time. The information provides both an opportunity and a challenge: an opportunity to provide better care, a challenge to use the data effectively. Diagnosis assisted by computer: How can all this potentially useful information be transformed into better patient care? The solution to this problem is the computer, not large computers of the type foremost in our minds, but the much smaller personal computer. The computer will not in any sense replace the cardiologist, but act as an assistant. In the foreseeable future, the ability of the human being to perceive patients and their problems is far greater than that of the most advanced computer. What the human being lacks is adequate memory and the ability to handle logically and completely masses of information. The computer supplements the senses and mind of the physician. The time has come to rid ourselves of the image of the computer as an inhuman monster waiting in the wings to replace the physician as a diagnostician, and to begin to look upon the computer as a descendant of the slide rule, as a slave, not a master.
From the Division of Nuclear Medicine, the Johns Hopkins Medical Institutions, Baltimore, Maryland. Manuscript received August 8, 1981; revised manuscript received October 26, 1981, accepted November 5, 1981. Address for reprints: Henry N. Wagner, Jr., MD, Division of Nuclear Medicine, the Johns Hopkins Medical Institutions, 615 North Wolfe Street, Baltimore, Maryland 21205.
The modern cardiologist is aware of the impact of computers on everyday life, including their use in airports, banks and, in some hospitals, in coronary care units. More and more physicians are using computers in managing the business aspects of their practice. What I am talking about is a new use for personalized microcomputers, a process that we call “diagnosis assisted by computer,” a series of programs designated by the suffix -DAC. For example, PEDAC stands for “pulmonary embolism diagnosis assisted by computer.” With these programs, the role of the physician is not diminished, but enhanced, just as the slide rule in no sense diminished the role of the engineer. The time has come for the physician to take the microcomputer to the patient’s bedside. Cardiologists can become pioneers in the application of the computer in everyday practice. Application of Bayes’ theorem in cardiology: In 1959 Ledley and Lusted2 suggested the use of Bayes’ theorem in medicine. What is Bayes’ theorem? Essentially, it states that the probability that a patient with a particular set of manifestations of illness (symptoms, signs or laboratory results) has a particular disease is directly proportional to the probability of occurrence of that particular set of findings (syndrome) in that disease multiplied by the a priori prevalence of that disease, and inversely proportional to the probability of occurrence of this set of manifestations in the general population. In 1968, I3 proposed a model of the entire diagnostic process based on the sequential application of Bayes’ theorem. Implementation of the model was not possible at the time for several reasons: (1) The data base relating manifestations and diseases was not available; (2) computer technology had not developed to the point where every physician could easily afford a computer of his own; and (3) the concept that medical diagnosis was probabilistic was not as well accepted by cardiologists as it is today. The exercise electrocardiographers took the lead in applying probability theory to cardiology in a routine way, particularly in the diagnosis of coronary artery
disease. For example, in 1977, Rifkin and Hood4 pro-
Editorials published by the Journal reflect the views of the authors and do not necessarily represent the views of the Journal or of the American College of Cardiology. ??
March 1962
The American Journal of CARDIOLOGY
Volume 49
875
EDITORIALS
posed the use of a Bayesian analysis. In the interpretation of exercise electrocardiograms, the cardiologist considers the a priori probability of coronary artery disease based on the age and sex of the patient, the presence and type of chest pain, and other factors. The degree of S-T segment depression is then combined with the a priori probability to yield the a posteriori or final probability estimate of coronary artery disease. If the patient were an asymptomatic middle-aged man with no risk factors, the pretest estimate of the probability of coronary artery disease would be 20 percent. If exercise resulted in 1.5 to 1.99 mm S-T segment depression, this fact combined with the a priori probability would yield a post-test probability of coronary artery disease of 70 percent. In 1979, Diamond and Forrester extended the idea of using Bayes’ theorem in cardiology by combining the pretest likelihood of coronary artery disease based on age, sex and symptoms with the data from four diagnostic tests: stress electrocardiography, cardiomyography, thallium scintigraphy and cardiac fluoroscopy. The program that they devised has been expanded to include other data such as gated blood pool imaging and has been incorporated into a commercially available microprocessor computer system that is inexpensive enough for use in the cardiologist’s office.6 Today, whenever a cardiologic procedure is performed in our nuclear medicine department, the nuclear physician or cardiologist uses this program to aid in the planning of the type of nuclear study to be performed and the interpretation of the results. Although the program is not yet perfected, it has proved very helpful in day to day practice. Application to nuclear cardiology studies: The essence of Bayes’ theorem is that equal importance is given to the a priori probabilities and to the results of the individual tests themselves. Each contributes to the estimation of the certainty of the final diagnosis. We assess the probability of various diseases being present at several stages of the diagnostic process, making “working diagnoses” after obtaining the medical history, after the physical examination and before and after each diagnostic test. What data do we need to apply Bayes’ theorem in practice? The most important characteristic of a diag-
nostic test is the likelihood that a given result will be found in patients with a particular disease and in normal persons. This probability is referred to as sensitivity. For example, suppose a manifestation such as a clear-cut regional defect in a thallium-201 myocardial perfusion image is observed in 40 percent of patients with transmural infarction. The sensitivity of the manifestation for that state would be 0.40. A manifestation that occurs in all patients with a given disease has a sensitivity of 1.00. The probability of a manifestation being present in all diseases (or normal persons) other than the one of interest is (1 - specificity). For example, if a manifestation occurs in 5 percent of healthy persons or persons with diseases other than the one being considered, its specificity would be 0.95.
876
March 1992
The American Journal of CARDIOLOGY
We always interpret our nuclear studies initially without knowledge of any a priori information. Then
we combine the interpretation of the nuclear data with all other information by means of Bayes’ theorem to yield the probability of a disease, such as coronary artery disease, being present. The nuclear data themselves are expressed as a likelihood ratio, defined as the ratio of the sensitivity (probability) of the particular findings of the nuclear study in the disease of interest to the probability of its occurrence in normal persons or in those with all other diseases. The likelihood ratio can also be defined as sensitivity divided by (1 - specificity). Some define sensitivity, specificity and likelihood ratio in terms of true and false positive results. I believe the time has come to go beyond this approach, which presents the problem that test results expressed only as positive or negative often represent an unnecessary and often troublesome oversimplification. A final example. Suppose we find an ejection fraction of 0.30 and a clear-cut regional wall motion abnormality of the anterior wall of the left ventricle in a gated blood pool study. If the probability of finding these abnormalities is 0.40 in patients with anterior myocardial infarction and 0.10 in normal persons or persons with other types of heart disease, then the likelihood of myocardial infarction is 4:l. Stated another way, the probability of myocardial infarction on the basis of the nuclear studies alone is 0.80. Our next step is to combine the nuclear data with the a priori data. If, for example, the patient had elevation of the S-T segments in the precordial leads, this fact combined with the nuclear data would increase the certainty to nearly 1.00. This is an example of a simple problem. A more difficult problem is presented by a woman with vague chest pain and 0.5 mm S-T segment depression in an exercise test. In this case, all the data need to be considered with proper weighting factors. Clinical approach: Most people tend to approach complicated problems by oversimplification. Many physicians order a large number of tests hoping to find a highly specific abnormality. I call this the “holy grail” or “eureka” approach to medical diagnosis. Experienced physicians pick two or three of the most important findings. around which to build a differential diagnosis. The advantage of the computer, properly programmed, is that it can remember and analyze all the data, considering every finding with the appropriate weighting factors. The physician is infinitely better than the computer in perceiving the manifestations of the patient’s illness. The computer provides information that the physician can not possibly remember and helps with the logical analysis. To be sure, Bayes’ theorem has not been universally accepted as a model for the diagnostic process. For example, Feinstein7 has stated that anyone who has practiced clinical medicine will recognize that Bayes’ approach does not resemble even a weird parody of clinical reasoning. He suggests that as clinicians we would answer a question-such as whether a patient with hemoptysis has lung cancer-not by estimating proba-
Volume 49
EDITORIALS
bilities, but by getting more data. In 1968 I1 suggested an alternative approach: “Every question that the physician asks in obtaining a medical history, every maneuver that he performs in the physical examination and every subsequent laboratory procedure that he orders should be selected because of the likelihood that the new fact will alter the estimate of the probability that the patient has a particular disease or diseases. An essential feature of the diagnostic process is its statistical or probabilistic nature. Rarely can a
diagnosis be made with absolute certainty, even by pathologists looking at histological sections. We in medicine have a tendency to assume erroneously that the best test is perfect. It almost never is, a fact revealed when a better test comes along.” Diamond (personal communication) has stated that we may be standing at the threshold of a perceptual upheaval in medicine, based on the use of the microcomputer to apply Bayes’ theorem in the cardiologist’s office or even at the bedside. In my judgment, within 10 years the computer is likely to join the stethoscope as a symbol of the cardiologist.
References 1. White PD. Errors in interpretation of cardiovascular symptoms and signs. Ann intern Med 1936;1:1703-13. 2. Ledley RGF, Lusted LB. Reasoning foundations of medical diagnosis. Science 1959;130:9-13. 3. Wagner HN Jr. Principles of Nuclear Medicine, chap II. The Diagnostic Process. Philadelphia: WB Saunders, 1968. 4. Rifkln RD, Hood WB. Bayesian analysis of electrocardiographic exercise stress testing. N Engl J Med 1977;297:681-6. 5. Diamond GH, Forrester JS. Analysis of probability as an aid in the
clinical diagnosis of coronary artery disease. N Engl J Med 1979; 300: 1350-a. 6. Dlamond GA, Forrester JS, Hirsch M, et al. Applications of conditional probability analysis to the clinical diagnosis of coronary artery disease. J Clin Invest 1980;65:1210-21. 7. Feinstein AR. Clinical biostatistics. 39. The haze of Bayes, the aerial palaces of decision analysis, and the computerized Ouija board. Clin Pharmacol Ther 1977;21:482-95.
March 1982
The American Journal of CARDIOLOGY
Volume 49
877