Bayes Theorem and diagnostic tests with application to patients with suspected angina. 



Abstract 
Patients with suspected angina often undergo a variety of noninvasive tests to confirm or exclude the presence of obstructive coronary artery disease. Such tests, however, are not always able to accurately identify patients with or without disease. This limitation of these tests gives rise to the concept of false positive (a positive test in a patient who does not have the condition) and false negative (a negative test in a patient who does have the condition) results. A basic understanding of the statistics of diagnostic tests is necessary to enable the clinician to select an appropriate test and interpret the result, whether positive or negative. The purpose of this review is to present a summary of the statistics of diagnostic tests with application to the investigation of patients with suspected angina. 

Introduction
Patients with suspected angina often undergo a variety of noninvasive tests to confirm or exclude the presence of obstructive coronary artery disease. Such tests, however, are not always able to accurately identify patients with or without disease. This limitation of these tests gives rise to the concept of false positive (a positive test in a patient who does not have the condition) and false negative (a negative test in a patient who does have the condition) results. A basic understanding of the statistics of diagnostic tests is necessary to enable the clinician to select an appropriate test and interpret the result, whether positive or negative. The purpose of this review is to present a summary of the statistics of diagnostic tests with application to the investigation of patients with suspected angina.
The statistics of diagnostic tests
Diagnostic tests are often assessed in terms of sensitivity, specificity, positive and negative predictive values, usually expressed as a percentage. Various definitions of these terms are available, which do not always lead to clarity in the clinical setting. High values indicate a good test and low values a poor test. An informative way to interpret these terms is as probabilities. The probability being the quotient of the number of ways an event can occur to the total number of possible outcomes. For example the probability of a ‘head’ when tossing a coin is ½ (one way a head can occur and two possible outcomes). A related concept is that of the odds, which is the quotient of the number of ways an event can occur to the number of ways it cannot occur. For example the odds of getting a head is 1/1 (one way it can occur and one way it cannot occur). Similarly the odds of a three after the roll of a die are 1/5, whereas the probability is 1/6. Probability and odds are related as:
Probability = Odds/(1+Odds)
Odds = Probability/(1Probability)
We see from these relationships that when the probability is close to zero it is approximately equal to the odds. Whereas when the probability is close to one the odds are very large.
Figure 1 shows the possible results of a test when applied to a sample of patients with and without disease. The columns show whether the disease is present or absent and the rows show whether the test is positive or negative. The four cells give the number of patients in each of the four possible categories of disease and test status. For example there are ‘a’ patients who have the disease and a positive test. We then have the following definitions:
Sensitivity = Probability (positive test, given disease present) = a(/a+c)
Specificity = Probability (negative test, given disease absent) = d/(b+d)
Positive predictive value = Probability (disease present, given test positive) = a/(a+b)
Negative predictive value = Probability (disease absent, given test negative) = d/(c+d)
Probability of disease present = (a+c)/(a+b+c+d)
Odds of disease present = (a+c)/(b+d).
In the clinical setting we are presented with a patient with suspected disease and a test result. Sensitivity and specificity are not helpful for they tell us the probability of the test result being positive or negative, given the disease status – which we do not know. Rather it is the predictive values that are potentially of value. They tell us the probability of disease status, given the test result – which we do know. The difficulty with the predictive values is that they depend on the probability (prevalence) of disease. Consequently, if our patient has a probability of disease which is different from the patients on whom the test was evaluated (the data in Figure 1), the predictive values will give an inaccurate estimate of the probability of disease. What is needed is a way of combining the characteristics of the test (which are independent of the characteristics of the patient) and the patient characteristics. This can be achieved using Bayes Theorem.
Bayes Theorem
Bayes Theorem for diagnostic tests can be written as^{1}:
Post test Odds = LR*(Pretest Odds).
Where the pretest Odds are the odds of disease being present prior to undertaking the test, and the post test Odds are the odds of the disease being present once the test result is available. LR is the likelihood ratio, which for a positive test is:
LR(+) = Sensitivity/(1Specificity).
And for a negative test is:
LR() = (1 Sensitivity)/Specificity.
The LR tells us how much more or less likely the disease is given the test result. A value of one indicates that the test is of no value. There are no particular values of LH to define how good a test is. It is generally accepted, however, that a good test should have LR(+) greater than 10 and/or LH() less than 0.1^{2}. A high value of LR(+) makes the disease more likely after a positive test and a low value of LR() makes the disease less likely after a negative test. For example a test with a sensitivity and specificity both of 91% will have LR(+) = 10.1 and LR()= 0.099.
a fundamental feature of Bayes Theorem is that we do not require an exact value of the pretest Odds; it is only an estimate and could be no more than a guess. The post test odds then give a refinement of this estimate in the light of the test result.
To see how this approach works in practice it is useful to consider some examples. It is first necessary to obtain an estimate of the probability of disease before the test is undertaken (the pretest probability (PTP), and hence the pretest odds of disease) in the patient. This could be from the literature and/or our own personal experience. Suppose we have a patient in whom we have estimated that the PTP of disease is 0.5, giving a pretest odds of 1.0 and we have a test with LH(+) = 10 and LH()= 0.1. Then for a positive test, the post test odds = 10*1 =10, giving a post test probability of disease of 0.91. The test result has enabled us to have much greater confidence of the presence of disease. Now suppose the test is negative then the post test odds = 0.1*1 = 0.1, giving a post test probability of 0.091. The negative test has enabled us to have much greater confidence of the absence of disease.
These examples demonstrate that with a good test and a pretest probability of 0.5, the post test probability for a positive test is greater than 0.9 and for a negative test is less than 0.1. Thus helping the clinician to establish the presence or absence of disease.
Now consider the situation where there is a high PTP e.g. 0.9, then with a positive test and LR(+) =10 we obtain a post test probability of 0.99 and we are even more confident of the presence of disease. When the test is negative, however, and with LR()=0.1 we obtain a post test probability of 0.47, which is not helpful in excluding disease. Consequently when there is a high PTP even a good test is not particularly useful. Similarly when the PTP is very low e.g. 0.1 the post test probability for a positive test is 0.53, which is not helpful in establishing the presence of disease. These two examples demonstrate the importance of interpreting a test result in the context of the PTP. Specifically a positive test may not be sufficient to conclude that disease is present and a negative test may not be sufficient to conclude that disease is absent. If we were to have a near perfect test with say LR(+)= 1000, then even with a low PTP of say 0.1 the post test probability is 0.99, i.e. we are virtually certain that the disease is present. In this situation the information contained in the test result has swamped the information contained in the PTP.
The use of the likelihood ratio allows us to separate the characteristics of the test (as it only depends on sensitivity and specificity and not the PTP) from those of the patient, which are expressed as the PTP. It should be noted, however, that sensitivity and specificity may depend on the type of disease present. For example tests for coronary artery disease may be better at establishing the presence or absence of severe disease than minor disease. The values for sensitivity and specificity (and hence LR) are not usually given for the severity or category of a particular disease, but rather overall values are given.
Application of Bayes Theorem to patients with suspected angina
The determination of whether patients with chest pain have angina is a common problem in cardiological practice. The diagnosis of angina is based on the history, consequently when it is clear that a patient does or does not have angina, no further diagnostic testing is necessary. There is, however, an important group of patients in which the history is not sufficiently clear to confirm or refute the diagnosis of angina. For this type of patient the usual practice is to undertake a test for obstructive coronary artery disease, and conclude that if positive the patient has angina. The gold standard test for the determination of the presence of coronary artery disease is coronary angiography, but this has small but important risks, including exposure to radiation. There is a variety of non invasive ischaemia based imaging tests that can be undertake as alternatives to coronary angiography. These tests typically have sensitivities and specificities of around 85%^{3} giving LR(+)=5.7 and LR()=0.18. These values of LR indicate that these noninvasive tests would not usually be considered good. Such tests should therefore be chosen and interpreted carefully.
The 2013 ESC Guidelines on stable coronary artery disease have placed greater emphasis on the use of the PTP of disease^{3}. It is recommended that patients with a PTP either greater than 0.85 or less than 0.15 do not require further diagnostic evaluation. It is assumed that the diagnosis is established for a PTP above 0.85 and excluded for a PTP below 0.15. These values are arbitrary and are based on the sensitivities and specificities of the noninvasive imaging tests currently available. Whether such an approach is reasonable is debatable. Nevertheless these limits are consistent with the approach that there is little value in undertaking testing in patients with particularly high or low PTP. Further, the choice of these particular values makes the diagnostic work up of patients with suspected angina particularly straightforward.
For patients recommended to undergo noninvasive testing it is first necessary to estimate the PTP. The guidelines^{3} reproduce such estimates from observational data from patients with suspected angina based on age, gender and type of chest pain^{4}. Three categories of chest pain are considered: Definite angina, ‘atypical angina’ (by which is meant probable angina) and nonanginal chest pain. The estimates of PTP for nonanginal chest pain are much higher than that for the general population. Further, since all patients in the dataset had suspected angina, this category, might be better described as possible angina. These estimates of PTP provide a starting point from which to estimate the PTP for a particular patient taking into account additional risk factors and the nature of the chest pain (noting that there is a spectrum of symptoms from probable to possible angina).
The next step is to choose an appropriate noninvasive test (the guidelines recommend noninvasive testing rather than initial coronary angiography). In addition to the ischaemia based imaging tests, there are two other noninvasive tests to be considered: the exercise ECG and CT coronary angiography (CTA).
The exercise ECG has traditionally been the preferred first choice diagnostic test for coronary artery disease. The 2013 ESC guidelines^{3}, however, while still giving a Class I indication for the exercise ECG, restrict its use to patients with a PTP of less than 0.65. The rational for this is that for patients with a PTP greater than 0.65 the exercise ECG will give more incorrect than correct results. This is a rather simplistic approach that does not distinguish between positive and negative tests. Alternatively the exercise ECG can be considered in the context of Bayes Theorem. The guidelines suggest that the exercise ECG has an overall sensitivity of 50% and specificity of 90%, giving LR(+)=5.0 and LR()=0.55. Thus for a positive test the exercise ECG is comparable to ischaemia based imaging tests which have LR(+)=5.7. For negative tests, however, the exercise ECG is very much worse and is unlikely provide any additional information. Therefore when selecting a noninvasive test this limitation of the exercise ECG should be balanced by its low cost, lack of exposure to radiation and its ready availability.
CAD:
Coronary artery disease.
a The 2nd test must be independent of the 1st.
b If the exercise test is negative a 2nd imaging test should be performed.
c If the 1st test is strongly positive, consider coronary angiography rather than a 2nd non invasive test.
CT coronary angiography (CTA) is another imaging test that is now becoming more widely available. It is unreliable in patients with high calcium scores (Agatston score>400), which typically occur in older patients. Consequently the guidelines recommend its use be restricted to patients with a PTP of less than 0.5. A woman of 80 years with probable angina has a PTP of 0.47^{3} and is very likely to have a high calcium score. Thus the use of CTA in older patients irrespective of the PTP may not be advisable. For patients with a low calcium score it has a very good sensitivity of approximately 97% and a specificity of approximately 74% giving LR(+)=3.73 and LR()=0.04. Thus for a positive test it is substantially worse than ischaemia imaging tests and the exercise ECG, but for negative tests it is very good and out performs all other non invasive tests.
Table 1 gives the post test probabilities for the noninvasive tests for PTP between 0.15 and 0.85 (the range for which the guidelines recommend non invasive testing). For the ischaemia imaging tests with a PTP=0.5, positive and negative tests give post test probabilities of 0.85 and 0.15 respectively, thus confirming or refuting the diagnosis with a single test. This is not a coincidence but a consequence of choosing these limits based on the sensitivity and specificity of the tests. We can use the information from the table to guide the choice of test, depending on the PTP. We do not require a precise estimate of the PTP (which in any event we are unlikely to have), all that is necessary is to determine whether the PTP is greater or less than 0.5 (noting that testing should not be undertaken for particularly high or low values).
For PTP<0.5 most patients will not have disease, so the required test should be able to exclude disease when negative, i.e. have a post test probability of less than 0.15. All ischaemia imaging tests and CTA will achieve this, whereas the exercise ECG will not. For patients with a PTP <0.5 and a positive test, the post test probability is greater than 0.5 (not quite achieved for CTA and a PTP<0.2), but not sufficient to establish the diagnosis (greater than 0.85 according to the guidelines). These patients therefore require further evaluation either with coronary angiography or a second noninvasive test (see below). The guidelines, however, suggest that a single positive test in such patients is sufficient to establish the diagnosis, which is clearly not the case.
For PTP>0.5 most patients will have disease, we therefore require a test, that when positive will give a post test probability of greater than 0.85 and the diagnosis will be established with a single test. All ischaemia imaging tests and the exercise ECG achieve this. When ischaemia imaging tests are negative the posttest probability will be less than 0.5, but greater than 0.15 so disease cannot be excluded. The guidelines seem to suggest that in this situation obstructive coronary artery disease can be excluded, which clearly it cannot. These patients therefore require further evaluation either with coronary angiography or a second noninvasive test (see below). It is important to note that the post test probability for a negative exercise ECG test may not be less than 0.5.
Thus with an appropriate choice of test the majority of patients will have the diagnosis of coronary artery disease (CAD) confirmed or refuted after a single test. A minority will require further testing, the nature of which depends on the overall clinical situation. For, example, if the first test in a patient with PTP <0.5 was a positive SPECT with greater than 10% of the myocardium affected, then coronary angiography might be more appropriate than a second noninvasive test. To determine the most appropriate second non invasive test it is necessary to understand how to apply Bayes Theorem to sequential testing.
Sequential testing
We can apply Bayes Theorem for each test. Thus the PTP for the second test is the posttest probability from the first test. Bayes Theorem requires the best estimate we have for the PTP, which for the second test is the posttest probability from the first test. For example, consider a patient who had a PTP<0.5 prior to the first test. If this is positive then, the posttest probability will be greater than 0.5. We then apply a suitable second test with a PTP>0.5. If this is positive then the posttest probability will be greater than 0.85 and the diagnosis is established. If the result from the second test conflicts with that from the first, then coronary angiography will probably be required to make a definitive diagnosis. A similar approach is applied for a PTP>0.5.
To apply Bayes Theorem sequentially requires that the tests be independent. Clearly it would be pointless to repeat the first test. It is not possible to determine theoretically whether tests are independent, but we can make informed judgements. Stress echo (whether exercise for pharmacological) and stress CMR all rely on identifying wall motion abnormalities and are therefore not likely to be independent. In contrast methods that identify ischaemia by the use of nuclear tracers are likely to be independent of stress echo/CMR, but not of each other. The
exercise ECG identifies ischaemia by changes of the surface ECG and is therefore likely to be independent of the ischaemia imaging tests. Finally, CTA assess the anatomy of the coronary arteries and is therefore likely to be independent of all the other noninvasive tests.
Thus when considering which is the most appropriate initial test, it is also important to consider which test might be necessary if a second noninvasive test is necessary. The ultimate selection depends on local availability/expertise, patient/clinician preference and to limit exposure to radiation, bearing in mind that even after a second noninvasive test coronary angiography may be necessary.
The exercise ECG and the diagnosis of coronary artery disease
The exercise ECG has the advantage of ready availability and low cost but has important limitations. Provided these limitations are appreciated, it can play an important part of the assessment of patients with suspected CAD. If an exercise ECG is chosen as the first test for patients with a PTP <0.5 and is positive the posttest probability will be greater than 0.5. An imaging test will then be necessary to clarify the diagnosis, in the same way as if the first test was an imaging test. If the exercise ECG is negative, however, no useful information will be obtained and one or two imaging tests will be required to clarify the diagnosis. If an imaging test is chosen as the first test and is positive an exercise ECG could be chosen for the second test. If it were positive the diagnosis would be established and no further testing would be required. If, however, it were negative a second imaging test would be required. Similarly if the initial PTP>0.5 the exercise ECG could be used as the first test and if positive the post test probability would be greater than 0.85 and the diagnosis would be established, thus avoiding the need for any imaging tests. If the test was negative, however, no useful information would be obtained and one or two imaging tests would be required to clarify the diagnosis, in just the same way as if an imaging test had been used as the first test. There is no place for an exercise ECG as a second test for patients with an initial PTP>0.5. Thus a positive exercise ECG can replace an imaging test. If the exercise ECG is negative, however, we need to proceed with imaging test(s). Thus by selecting and interpreting the exercise ECG appropriately, in accordance with Bayes Theorem, it is possible to reduce the number of imaging tests undertaken and reduce patient exposure to radiation. This approach to the use of the exercise ECG is substantially different to that adopted by the Guidelines^{3}.
Summary
Figure 2 summarises this approach to the investigation of patients with suspected CAD. It can be seen that the result from a test must be interpreted in conjunction with the PTP of CAD. Specifically, if the PTP<0.5 we cannot conclude from a single positive test that the patient has CAD. Conversely, if the PTP>0.5 then we cannot conclude with a single negative test that the patient does not have CAD. The guidelines^{3} suggest that a single test in these contexts is sufficient to establish or refute a diagnosis of CAD. If a non invasive test is strongly positive coronary angiography would generally be indicated rather than undertaking a second non invasive test, not specifically for diagnostic purposes, but for risk stratification (as recommended by the guidelines). The strategy suggested here should always be considered in conjunction with the clinical situation. If the results of noninvasive testing are not consistent with the patient’s symptoms, coronary angiography should be considered. To exemplify how this approach can be used in practice, two case histories are described.
Case History 1
A 45 year old woman presents with a 3 month history of chest pain which only occurs with exertion. For example undertaking housework or carrying shopping, but not when walking. She is unable to describe the character of the pain, but it is established that it is not pleuritic. She has a strong family history of premature CAD, with a brother aged 48 having had CABG and her father experiencing a myocardial infarction at the age of 50 years. She is an exsmoker, has a BMI of 29Kgm^{2} and a total cholesterol of 6 mmol.l^{1}. It was felt that she probably did not have angina, but that the history was not sufficiently clear to exclude it and so required further diagnostic testing. The first step is to determine whether the PTP is greater or less than 0.5, in this case the PTP is clearly less than 0.5 (the table in the ESC guidelines gives a PTP of 0.08 based on age and gender, which is clearly too low given her additional risk factors). Any noninvasive imaging diagnostic test would be suitable or an exercise test could be undertaken with the caveat that if negative an imaging test would be required (ESC guidelines suggest a negative exercise test in this patient would be sufficient to exclude CAD). An exercise test was performed which was stopped at 7 minutes due to the development of chest pain and dyspnoea. There were no ECG changes to suggest myocardial ischaemia. She went on to have a dipyridamole stress echo, which was negative. Hence CAD was excluded. The important points here are that a negative exercise test is not sufficient to exclude CAD and that if the PTP was as small as 0.08 there would be no point in pursing noninvasive testing.
Case History 2
A 65 year old man presents with a long history of worsening chest pain which is now occurring on a daily basis. The pain is described as gripping and was retrosternal but not consistently occurring with exertion. It was felt that he probably had angina, but that the history was insufficiently clear to be conclusive without further diagnostic testing. The PTP for this man is clearly greater than 0.5 so any ischaemia imaging tests or an exercise test would be appropriate. He went on to have an exercise test, which was positive at 5 minutes of the Bruce protocol, subsequent coronary angiography demonstrated three vessel disease and he was referred for CABG. The important point here is that a positive exercise test can eliminate the need for any further diagnostic testing.
Conclusion
The ESC 2013 Guidelines^{3} provides the basis for a more objective assessment of patients with suspected angina than the traditional more subjective approach. There is more emphasis on the PTP, but unfortunately the results of noninvasive testing are not interpreted in the context of the PTP or the post test probability. The Bayesian approach to diagnostic testing reviewed here and applied to the assessment of patients with suspected angina demonstrates that this is crucially important. It has been shown that a positive test in the context of a PTP<0.5 is not sufficient to conclude that CAD is present, and conversely a negative test in the context of a PTP>0.5 is not sufficient to conclude that CAD is absent. Further the idea that an exercise test should only be undertaken if the PTP<0.65 is based on rather simplistic reasoning and could result in a valuable test being under utilised. The Bayesian approach demonstrates that an exercise test can be used for all PTP, but the results have to be interpreted properly. Specifically a negative test is of little if any value in establishing a diagnosis of CAD, whereas a positive test is of similar value to noninvasive imaging tests.
References
Copyright (c) 2015 Andrew Owen
This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.