Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine

Transcription

1 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine page 1 of 12 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine by Richard J. Petti version of March 18, 2006 Abstract This paper describes Bayesian methods of evaluating programs of diagnostics and therapeutic procedures in clinical medicine. The method identifies the branch in a decision tree of diagnostic and therapeutic procedures that maximizes expected utility for the patient. The utility of a procedure includes cost, morbidity (harm to the patient), and valuation of the final outcome for the patient compared his/her initial state. Like all Bayesian methods, it explicitly models the prior information that is used in making decisio, and how this is combined with information about an individual patient. The paper defines representatio for all required structures and an explicit algorithm that has been implemented in prototype software. Contents 1. Description of Diagnostic Situatio Prior Information about the Patient or Population Characterization of Therapeutic Procedures Description of Diagnostic Tests The Diagnostic and Therapeutic Procedures The Maximum Likelihood Diagnosis The Utility-Maximizing Therapeutic Procedure Diagnostic Testing Strategy Some Common Measures of the Value of Knowing the Result of a Test Receiver Operating Characteristic (ROC) Sequencing of Diagnostic and Therapeutic Procedures Changes in Life Situation Software Implementation The Meanings of a State of the Patient Computation of Prior Probabilities A Simple Example A Second Example - A Parallel Circuit A General Implementation Glossary of Symbols References copyright Richard Petti 2006

2 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine page 2 of Description of Diagnostic Situatio 1.1. Prior Information about the Patient or Population Suppose that the patient 1 can be in one of states, labeled by the index s = 1,..., Utility Values of the States of the Patient Let U(s) denote the utility of a patient in state s. Numerous methods are available for defining and, in principle, measuring utilities and for applying them in decision models for maximization of expected utility (Von Neumann and Morgetern 1953, Henderson and Quandt 1958, Zellner 1971, Copeland and Weston 1979). 2 The utilities {U(s)} are the least easily quantified parameters in the model of diagnostic decision-making presented here. The utilities of various states may vary according to the patient's life situation for example, the value of specific functionality can depend on life situation. Assigning quantitative utilities to the state of a patient involves judgments that are likely to be controversial. Large parts of the decision-making algorithms presented here can be applied without utility functio, namely with maximum likelihood methods Prior Information about the Patient or Population Denote by prior(s) the prior probability of state s, based on prior information about the patient that is available at the start of the diagnostic process. We require that (1) Σ prior(s) = 1. s=1 Prior information may include component lifetime failure distributio, fleet history, and usage data for the individual unit Characterization of Therapeutic Procedures Suppose that the patient can be treated with one of nrp different therapeutic procedures, labeled by rp=1,...,nrp The Effect of Therapeutic Procedures Let PR rp(s,s) denote the probability that therapeutic procedure rp changes the state of a patient from initial state s to state s. Then PR rp(s,s) defines a Markov process, in the see that each PR rp(s,s) is nonnegative and that for each s=1,, (2) Σ PR rp(s, s) = 1. s =1 1 In testing applicatio outside of medicine, the term patient is replaced by system under test. 2 The expected utility model of decision-making has several strengths. First of all, it is the foundation of many of the most successful models of decision-making under uncertainty. Secondly it makes explicit many of the assumptio and tradeoffs that underlie diagnostic decision-making. Although other approaches (such as information-theoretic models) may appear to be more neutral and objective, these models hide the implicit assumptio about utility, and they usually can be restated as Bayesian expected utility models. The main objection often raised agait the expected utility approach to modeling decision making is that it makes explicit assumptio about the utility value of states in a manner in which calibration with the judgment of experts is fairly involved. However, in simple cases the relative utility values may be deduced without much calculation, as in Kundel (1981).

3 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine page 3 of 12 The effect of each therapeutic procedure rp is completely specified by the probabilities PR rp(s,s) The Value of Therapeutic Procedures Let UR(rp s) denote the expected therapeutic value of applying therapeutic procedure rp to a patient who is in state s. Then (3) UR(rp s) := Σ U(s ) PR rp(s,s) s= The Cost of Therapeutic Procedures Let K rp denote the monetary cost of applying therapeutic procedure rp. We express the cost of a therapeutic procedure in the same units of utility that are used to describe states of a patient. Defining a traformation between the scale of costs and the scale of utilities of states involves making some very important assumptio about the relative value of improved outcomes and economic costs. This identification of utility scales is particularly problematic when human lives and welfare are involved. The utility functio UR(rp s) provide the basic information about therapeutic procedures that is needed to implement a therapeutic program Description of Diagnostic Tests Relation Between Test Results and State of the Patient The result of a test is a set of parameter values (e.g. electrical readings, respoe times, fluid pressures) from a range of possible results, RR. That is, a test result is an ordered set of numbers r=(r 1,...,r nr ) in RR. The numbers r 1,r 2,... can be from discrete or continuous ranges of possible values. The dependence of the test result r on the state s of the patient is described by the conditional probabilities PT t (r s) := probability of obtaining result r when performing test t on a patient in state s. Normalization of probability requires that the sum/integral of PT t (r s) over the entire space of results RR is 1, for each state s Cost of Performing a Diagnostic Test Performance of diagnostic test t on a patient incurs two types of costs. C t := monetary cost of performing the test. The cost is expressed in the same units of utility as the utilities of therapeutic procedures, UT t (rp s). The cost of performing a test is assumed to be independent of the state of the patient. Mor t (s):= expected morbidity or disutility of performing test t on patients who are in state s. Morbidity refers to the fact that performing test t may do significant harm to patients in state s. The units of utility for diagnostic procedures are the same as those used for the utilities of therapeutic procedures, UR(rp s). Note that the monetary cost and clinical value of performing a diagnostic or therapeutic procedure are expressed in the same utility units that are used to describe states of the patient. This makes very important and controversial assumptio about the relative value of diagnosis and outcomes and economic costs. The probability function PT t (r s) and the monetary cost C t and the morbidity Mor t (s) provide all the information about diagnostic tests that is needed to implement testing, diagnosis and therapeutic programs.

4 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine page 4 of The Diagnostic and Therapeutic Procedures After result r is obtained from performing test t, we use the new information to improve upon the prior probability distribution of the state of the patient. The posterior probabilities of the state of the patient are: PT t (r s) prior(s) (4) post t (s r) = Σ PT t (r z) prior(z) z=1 This relatiohip is known as Bayes Theorem, which defines the posterior probability distribution of states in terms of the conditional probabilities of obtaining result r for a patient in state s and the prior distribution. The posterior probabilities post t (s r) become the prior probabilities for subsequent diagnostic and therapeutic decisio. Two kinds of decision processes will be coidered: diagnosis of the patient's condition, and determination of the optimal therapeutic procedure. Although these two processes are related, they are distinct, as will become clear below The Maximum Likelihood Diagnosis For some diagnosis s*, post t (s r) takes on its largest value, i.e. post t (s* r) is greater than or equal to post t (s r) for any other value of s. We shall use the following notation for the maximum likelihood diagnosis: ML t (r):= the maximum likelihood diagnosis of the state of the patient, given that test t yielded result r. Given prior information on the state of a patient, and given the result r for test t concerning the state of the patient, the maximum likelihood diagnosis ML t (r) is the one which maximizes the value of the probability post t (s r). Note that the function ML t (r) is completely determined by the characteristic function of the test, PT t (r s), and by the prior probability distribution, prior(s). Here are some important variatio on the maximum likelihood method. A diagnosis of state s* is made only if post t (s* r) is larger than any other post t (s r) by some margin. A diagnosis of state s* is made only if post t (s* r) is larger than some threshold, such as 90%. The most important weakness of maximum likelihood diagnostic processes is that they do not in general determine the optimal therapeutic procedure in the see of expected utility; in fact, they can suggest performing procedures that are significantly suboptimal. For example: Suppose that : nrp, and that patients in state s are best treated with therapy rp=s the maximum likelihood diagnosis is s=1; that is, ML t(r)= 1; that is, post t(1 r) post t(s,r) for any s 1. It does not follow that therapeutic procedure rp=1 maximizes the expected utility. Theraeutic procedure rp may yield a utility (with or without including morbidity) that is far from optimal for a patient in some of the possible states, while the other therapeutic procedures are not far from optimal for a patient in state s=1. So some other therapeutic procedure rp for rp 1 may have higher expected utility for the patient. This situation illustrates that there is a difference between a good diagnostic strategy and a good therapeutic strategy.

5 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine page 5 of The Utility-Maximizing Therapeutic Procedure The maximum expected utility principle can be used to choose a therapeutic procedure for the patient. After test t has been administered and the posterior probability distribution of states has been determined, the posterior expected utility of therapy rp is: (5) UT t (rp r) := Σ UR(rp s) post t (s r) s=1 For some therapy rp*, the expected utility of the therapy takes on its maximum value. We shall use the following notation for the utility-maximizing therapy: MUT t (r):= utility-maximizing therapy, given that test t yielded result r. Given prior information on the state of a patient, and given the result r obtained from test t concerning the state of the patient, the utility-maximizing therapeutic procedure MUT t (r) is the one which maximizes the value of the expected utility UT t (rp r). This therapeutic strategy determines the therapeutic procedure with the greatest expected value to the patient, and it does not in general require that we obtain an unambiguous diagnosis of the state of the patient Diagnostic Testing Strategy Up to this point, we have examined strategies for determining a diagnosis or selecting a therapeutic procedure, given that we have performed a specified test t on a patient. We are now equipped to explore the question of which test(s) should be administered. The expected incremental utility of diagnostic test t is: nr (6) dut t := Σ{Σ {(UR(rp* s) K rp* ) PT t (r s)} Mor t (s)} prior(s) - C t s=1 r=1 - Σ { UR(rp0 s) - K rp0 } prior(s), s=1 where rp*= MUT t (r), and where rp0 is the utility-maximizing therapy determined using only the prior information; that is, rp=rp0 maximizes (7) Σ UR(rp s) prior(s). s=1 Given prior information on the state of a patient, the utility-maximizing test t* is the one which maximizes the expected incremental utility to the patient, dut t Some Common Measures of the Value of Knowing the Result of a Test When all the necessary information is available, the expected incremental utility dut t is the best measure of the value of the test. However, dut t depends upon the utilities UR(rp s) and Mor t (s). In this section we define some common measures of the value of a test which do not depend on these utility functio, but depend only on the characteristic probabilities PT t (r s) of the test and the prior information prior(s).

6 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine page 6 of Measures of the Value of a Test First define a preliminary concept. (8) PML t (s' s):= Σ delta(s', ML t (r)) PT t (r s) r=1 nr = probability that performing test t yields maximum likelihood diagnosis s', given a patient in state s. (delta(i,j):= if i=j then 1 else 0.) We can define some measures of value of a test in terms of PML t (s' s), i.e. in terms of correctness of diagnosis. Seitivity measures how well a test detects the presence of a given state. (9) Seitivity t (s):= PML t (s s) Specificity measures how reliable is the detection of a given state, i.e. how likely are false-positive results. (10) Specificity t (s):= Σ PML t (s2 s1) = PML t (not s not s) s1 s s2 s Accuracy is the average seitivity over all states in the population. (11) Accuracy t := Σ Seitivity t (s) prior(s) s=1 These measures of the value of knowing the result of a test do not depend on utility or cost estimates for diagnostic or therapeutic procedures. They are limited, in that they evaluate a test in terms of diagnostic correctness itead of therapeutic value. The best estimate of the value of a test is the expected utility of the resulting therapy, dut t. The Receiver Operating Characteristic (ROC) discussed below describes how the optimal therapeutic procedures change as the expected utility estimates for the procedures vary Application: Screening Situation Coider the following diagnostic situation: There are two possible states of the patient, state 1 = diseased state, and state 2 = non-diseased state. There are two possible therapeutic procedures. Therapeutic procedure 1 is best for patients in state 1, and therapeutic procedure 2 is best for patients in state 2. We can identify conditio that characterize a screening situation, and test characteristics that make a test a good screening test. (For example, testing for lung cancer in the general population is a screening situation, and a chest x-ray is a good screening test.) A screening situation is one in which the incidence of disease is low; the cost of a false negative is quite high; and a false positive result is not very harmful. That is, prior(1) << prior(2) UR(rp=1,s=1) >> UR(rp=2,s=1) and UR(rp=2,s=2) >~ UR(rp=1,s=2). A screening test t is one for which the test is very seitive to detection of the diseased state; the number of false positives is fairly small (i.e. the specificity is high); and the cost of performing the test, including morbidity, is low, especially for the larger non-healthy subpopulation. That is,

7 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine page 7 of 12 PT t (r=1 s=1) >> PT t (r=2 s=1), and PT t (r=1 s=2) is preferably small, C t + Mor t (s=2) is very small, and C t + Mor t (s=1) is small Application: Confirmation of Diagnoses We can identify conditio that characterize a disease confirmation situation, and test characteristics that make a test a good disease confirmation test. (For example, confirming a preliminary diagnosis of lung cancer with a tissue biopsy is a confirmation situation.) A disease confirmation situation is one in which the patient has a higher probability of being in the diseased state than the general population; and false positives are quite costly. That is, prior(1) > general_population_prior(1) UR(rp=1,s=2) << UR(rp=2,s=2). A disease confirmation test t is one for which the probability of false positives is fairly small (i.e. the specificity is high); and the cost of performing the test, including morbidity, is low, especially for the larger non-diseased subpopulation. That is, PT t (2 2) is large, C t + Mor t (2) is very small, and C t + Mor t (1) is small Receiver Operating Characteristic (ROC) Let (12) PMUT t (rp s):= Σ delta(rp, MUT t (r)) PT t (r s) nr r=1 = probability that the utility-maximizing therapy is rp, given that test t is performed on a patient in state s (where delta(i,j) := if i=j then 1 else 0.) The Receiver Operating Characteristic (ROC) awers the question, As the utilities UR(rp s) of therapeutic procedures assume all possible values, what possible combinatio of conditional probabilities PMUT t (rp s) can result? Example: Suppose there are two states, denoted s=1 and s=2, and two therapies, denoted rp=1 and rp=2. Then for test t there are two independent probabilities, PMUT t (rp=1 s=1) and PMUT t (rp=2 s=2). Assume further that therapy 1 (2) is optimal for patients in state 1 (2). As the utilities UR(rp s) vary, the possible values for PMUT t (rp=1 s=1) and PMUT t (rp=2 s=2) are shown in Figure 1 below. Some limiting values of the ROC curve in this example are: When UR(rp=1 s=1) >> other UR(rp s), then PMUT t (rp=1 s=1)~1 and PMUT t (rp=2 s=2)~0. When UR(rp=2 s=2) >> other UR(rp s), then PMUT t (rp=2 s=2)~1 and PMUT t (rp=1 s=1)~0.

8 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine page 8 of 12 1 PMUT t (rp=2 s=2) 0 0 PMUT t (rp=1 s=1) A Typical Receiver Operating Characteristic Figure 1 1 In the case where there are states, nr test results and nrp therapeutic procedures, the ROC is harder to visualize. There are (nt-1) independent probabilities PMUT t (rp s). As the utilities UR(rp s) are varied, the ROC traced out is generally a surface of dimeion (nt-1)-1 in a space of dimeion (nt-1). The ROC always passes through the points 1, if rp = rp* (13) PMUT t (rp s) = 0, if rp rp*, which corresponds to administering therapy rp* regardless of the test results. There are nt points of this type, corresponding to extreme cases where one therapy is more beneficial for patients in all states. Note that if nt=, and if each state has a different optimal therapy, then there is a 1-to-1 correspondence between states and therapies. In this case, we may interpret PMUT t (rp s) to be the probability of correct diagnosis of the state s Sequencing of Diagnostic and Therapeutic Procedures This model of diagnosis and treatment focuses on performing one test at a time, and one choice of therapeutic procedure. Different levels of utility may result from different sequences of tests and therapeutic procedures. A sequential decision tree model can be used to choose the actio that maximize the expected utility for the patient, using all available diagnostic information at each decision point. In the case where sequential decisio are made, each therapeutic alternative rp may be a different branch of the decision tree. The expected utilities UR(rp s) (referring to applying therapeutic procedure rp when the patient is in state s) are then the expected utilities for following an entire branch of the therapeutic decision tree Changes in Life Situation Bayesian expected utility decision models provide an effective framework for accommodating requirements of different life situatio. For example, the relative value of procedure clock time, professional staff time, cost of materials and equipment, and RTOK rates 3 are reflected in differing choices for the expected utilities of the tests and therapeutic procedures. 3 The RTOK rate, or Return Test OK rate, is the percentage of replaced parts that pass all functional tests after removal.

9 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine page 9 of Software Implementation 3.1. The Meanings of a State of the Patient There are several meanings of the state of the patient: A functional goodness state of the patient is characterized by which functio and subfunctio are faulty. Let sfunc i denote the functional state in which function i is faulty and all others are good; let sfunc i,j denote the state in which functio i and j are faulty and all others are good, and so forth. A physical goodness state of the patient is characterized by which orga 4 are functionally impaired. Let sphys i denote the physical state in which function i is faulty and all others are good; let sphys i,j denote the state in which functio i and j are faulty and all others are good, and so forth. A configurational state of the patient is characterized by which parts of the anatomy are opened and which test equipment is attached. Both the functional goodness state and the physical goodness state of the patient describe the patient in terms of goodness of its cotituents. The configurational state is needed only to specify the cost of performing a test or procedure, and does not appear explicitly in the computatio in this summary of Bayesian diagnostic technology. The physical and functional goodness states are related by the cotraint that the state of the leaf functio completely determines the state of the leaf physical assemblies, and vice versa. Normally this is accomplished by having a one-to-one correspondence between the leaf functio and the leaf assemblies. Therefore, the functional goodness state and the physical goodness state are different representatio of the same information about the patient. Normally diagnostics are conducted using functional goodness states, i.e. the diagnostic process determines which functio are good and which are faulty Computation of Prior Probabilities Prior information about the goodness state of the vehicle can be stated in either functional or physical representation. Information about the patient fleet, the individual patient unit usage and therapeutic history can be combined with LRU component lifetime failure distributio or lifetime functional failure information. This information yields probability estimates of the failure of each function or LRU. This tralates into prior probability estimates that the patient is in each (functional and physical) state s: prior(s) = probability that the patient is in state s, given information available prior to performing test t. Prior information about functional or physical failure rates can be combined in specifying prior fault probabilities, so long as they define a coistent goodness state at the leaf node level A Simple Example We can sketch a fairly simple and realistic diagnostic situation as a starting point for discussion of software implementation. In this example, all states are functional goodness states. We make the following assumptio. 4 The analog of an organ in equipment testing is the line replaceable unit or logistic repair unit (LRU), which mea a part that can be replaced at or near an operational zone. For example, most circuit boxes are LRUs, while the circuit boards iide generally are not.

10 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine page 10 of 12 The test t to be performed is associated with a particular patient function. The test has only two outcomes: either the function works or it does not. For a given state s (i.e. a given set of faulty functio), the probability that the tested function will work is either zero or one: PT t (pass s)= 0 if the state s contai faulty subfunction which cause the test t to fail; 1 if the state s has no faulty subfunctio which would cause test t to fail. Assume that only four subfunctio can cause the given test t to fail, and a fault in any one of them will cause a failure. The associated states which cause failure shall be denoted s 1, s 2, s 3, and s 4. Thus, the only states of the patient which need be coidered are s 0 (no faults), s 1, s 2, s 3, s 4, s i,j, s i,j,k, and s 1,2,3,4 (where the subscripts take on the values 1,2,3,4). The fault probabilities for the four primary subfunctio are independent. Thus, (14) prior(s i.j ) = prior(s i ) * prior(s j ). In the situation above, the output of the diagnostic process is the probabilities that each of the four individual suspected subfunctio are faulty. PT t (fail s i ) * prior(s i ) (15) post t (s i fail) = i=1,2,3,4 15 Σ PT t (fail z) * prior(z) z=0 (16) post t (s 0 pass) = 1, and post t (s i pass) = 0, i=1,2,3,4. Applying the assumptio above, the first result can be simplified to prior(s i ) (17) post t (s i fail) = i=1,2,3,4 4 Π (1 + prior(s i )) - 1 i=1 Note that post t (s 0 fail) = 0, whereas generally prior(s 0 ) > 0. If the patient passes the test, then post t (s 0 ) = 1, and post t (s i pass) = 0. When a second test is performed, the posterior probability distribution of the first test becomes the prior probability distribution for the next test. A very simple algorithm for determining which test to perform next is this: Identify which subfunction has the highest probability of being faulty. Identify those tests that can prove that the selected subfunction is faulty, and choose the one with the lowest cost to perform. This test sequence strategy has the desirable properties that it always tries to prove the most likely fault hypothesis, and that it is simple. A realistic example of this type of test situation is a headlight circuit, which contai four major assemblies: the battery, the light switch, the headlamps, and the wiring. (These four subfunctio are leaf assemblies, viewed as leaf subfunctio.)

11 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine page 11 of A Second Example - A Parallel Circuit Suppose everything is the same as in the first example, except that there are only two relevant subfunctio, and both of them must be diseased to obtain a test failure. Then PT t (fail s i ) = 0, for i = 0,1,2. PT t (fail s 1,2 ) = 1. The important probabilities computed from the test results are: post t (s i fail) = 0, for i = 0,1,2. post t (s 1,2 fail) = 1 prior(s i ) post t (s i pass) = 2 Σ prior(s j ) j=0 This mea that if the patient passes the test, then the patient cannot be in state s 1,2. A realistic example of this situation is a system with two redundant components, where one acts as a backup for the other A General Implementation The general implementation is cotructed from the following data: For each test t and result r, and state s of the patient, the diagnostic system must know PT t (r s). The diagnostic system must have access to estimates of the prior fault probabilities prior(s). From this data, a Bayesian model of test results and fault probabilities can be cotructed as outlined previously. Addition of utility and cost data is needed to implement Bayesian decision models discussed previously. The full Bayesian model can generate significant computational complexity. The challenge is to design the implementation with a useful blend of computational efficiency and accuracy.

12 Bayesian Analysis of Diagnostic and Therapeutic Procedures in Clinical Medicine page 12 of 12 C t dut t K rp nr nrp ML t (r) Mor t (s) Glossary of Symbols The cost of performing test t on a patient, expressed in utility units comparable to those used for UR(rp s). Expected incremental utility of diagnostic test t after its maximum likelihood therapeutic procedure has been applied. The cost of applying therapeutic procedure rp to a patient, expressed in utility units comparable to those used for UR(rp s). The number of possible results of performing a particular test. The number of possible states of the patient that need to be coidered. The number of possible therapeutic procedures which are being coidered for the patient. The maximum likelihood diagnosis of the state of the patient, given that test t yielded result r. The expected morbidity or disutility of performing test t on patients in state s. Morbidity refers to the fact that performing test t may do significant damage to patients which are in state s. The units of utility are the same as those used for the utility of therapies, UR(rp s). MUT t (r) utility-maximizing therapy, given that test t yielded result r. PR rp(s,s) Probability that therapeutic procedure rp converts a patient in initial state s to state s. PT t (r s) Probability of obtaining result r when performing test t on a patient who is in state s. PML t (s' s) probability that the maximum likelihood diagnosis derived from applying test t is s', given a patient in state s. PMUT t (rp s) probability that the utility-maximizing therapy is rp, given a patient in state s. post t (s r) prior(s) The probability distribution over the possible states s of the patient, given that test t was applied, and yielded result r. The a priori probability distribution of states of the patient, before the current diagnostic tests are applied. U(s) The utility of a patient in state s. UR(rp s) The utility of applying the therapeutic procedure rp to a patient who is in state s. UT t (rp r) The utility of applying the therapeutic procedure rp to a patient for whom test t returned result r. References Copeland, T. and Weston, J. (1979) Financial Theory and Corporate Policy, Addison Wesley. Henderson, J. and Quandt, R. (1958) Microeconomic Theory, McGraw Hill, Inc. Kundel, Harold (1981) Radiology, 139 p Von Neumann, J. and Morgetern, O. (1953) The Theory of Games and Economic Behavior, third edition, Princeton University Press. Zellner, A. (1971) Introduction to Bayesian Inference in Econometrics, John Wiley and So.