Chapter 3.The diagnostic test evaluation and application of Bayes’ theorem to
3.3 Bayesian methods for test accuracy
This section will introduce an application of Bayes’ Rule in the evaluation of diagnostic tests, and show how Bayes’ Rule can be used to switch between conditional and unconditional test accuracy.
3.3.1 Fundamentals of Bayesian
Bayes’ theorem, is a logical consequence of the product rule of probability (Khamis, 1990). The theory of conditional probability and Bayes’ theorem are found in various applications in formulating mathematical models in all sciences (Hardeo, 1992). Bayes’
rule in medical diagnosis is applied and discussed in various texts, including Lusted (1968) and Sox et al (1998) (Parmigiani, 2002). Furthermore, to evaluate laboratory tests and the principles and techniques of medical decision analysis, Bayes’ rule is commonly applied.
Bayes’ Rule is a way of calculating conditional probabilities. Bayes was the first to use probability theory inductively, which developed the mathematical basis for probability inference (Lesaffre et al., 2007). The concept of conditional probability provides information about how the occurrence of one event predicts the probability of another event. The essential fundamentals of the Bayesian method are their estimated unknown probability, and making decisions on the basis of new (sample) information (Okeh and Ugwu, 2008). In other words, Bayesian data analysis is a practical method for making inferences from data, using probability models for quantities we observe and for quantities about which we wish to learn.
3.3.2 Simple statement of theorem
Conditional probability is a very helpful method and is used in many ways. Bayes’ theorem associates the conditional and marginal probabilities of event A and B. Bayes’ theorem in this form indicates a mathematical representation of how the conditional probability of event A given B is associated with the converse conditional probability of B given A. We symbolize conditionality by using a vertical slash ‘│’ , which can be referred to as ‘ given’.
P B A P A P A B =
P B
Each term in Bayes’ theorem states:
P(A│B) Shows the conditional probability of A given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.
P(B│A) States the conditional probability of B given A. It is also called the likelihood.
P(A) Indicates the prior probability or marginal probability of A.
It is “prior” in the sense that it does not take into account any information about B.
P(B) Indicates the prior or marginal probability of B, and acts as a normalizing constant.
P A B P B P B A =
P A Each term in Bayes’ theorem states:
P(B│A) Shows the conditional probability of B given A. It is also called the posterior probability because it is derived from or depends upon the specified value of A.
P(A│B) States the conditional probability of A given B. It is also called the likelihood.
P(B) Indicates the prior probability or marginal probability of B.
It is “prior” in the sense that it does not take into account any information about A.
P(A) Indicates the prior or marginal probability of A, and acts as a normalizing constant.
3.3.3 An application of Bayes’ Rule to screening and diagnostic test evaluation
The function of a diagnostic test is to make a diagnosis. Clinicians have to know the probability that the test results will give the correct diagnosis. In fact, medical diagnosis tests commonly yield mathematic test results such as sensitivity and specificity. However, the sensitivity and specificity of test results alone do not provide strong enough guidance
to make clinical decisions. They cannot be converted into clinically relevant quantities without information on disease prevalence. On the other hand, probabilities related to screening and diagnosis tests are based on Bayes’ theorem, and is applied to diagnostic test evaluation. Bayes’ rule shows that positive and negative predictive values can be calculated from the sensitivity and specificity of diagnostic tests, which are values that a clinician needs in order to make appropriate diagnoses or decisions. (Khamis, 1990) (Anthony, 2007) (Okeh and Ugwu, 2008). Suppose you test positive for a disease. What is the probability that you actually have the disease? It depends on the accuracy and sensitivity of the test, and on the background (prior) probability of the disease. Let “D+“
stand for “disease present” and “D-” stand for “disease absent”. Let “T” indicate test results and “+” the event of a positive test and “-“ the event of a negative test. In particular, the prevalence rate in the population is represented as P(D+), the sensitivity as P(T+│D+) and the specificity as P(T-│D-).
There are two possible decision tree maps which involve diagnostic tests. Firstly, the initial branching of the decision tree starts with testing. According to a decision tree map, a patient visiting the clinic undergoes tests and examinations to distinguish whether the patient is with or without disease. Treatment or further tests will be offered according to those test results. The final results represent whether disease is present or absent in the patient. Such, Bayes theory uses conditional probability to assess decision making under uncertainty and is particularly applicable to decision trees. Bayes’ rule is therefore applied to determine the probability of a particular diagnosis, given the appearance of specific signs, symptoms and test outcomes (MedicineNet, 2004). Figure 3.3 presents a common example involving one diagnostic test where treatment is given following the test result. The beginning of the decision tree map starts with receiving a positive or negative test result. The probabilities are denoted by P(T+) and P(T-). The disease outcomes must be shown in the right hand of the test results as the disease status. The subsequent branching probabilities are the likelihood of disease in each outcome. The probabilities are denoted by P(D+│T+), P(D-│T+), P(D+│T-), and P(D-│T-).
Figure 3.3 Tests approach to decision problem with testing
The four possible final outcomes associated with this decision tree are then calculated True Positive = P(T+) * P(D+│T+)
False Negative = P(T+) * P(D-│T+) False Positive = P(T-) * P(D+│T-) True Negative = P(T-) * P(D-│T-)
We can compute the posterior probability P(D+│T+) of a patient who tested positive while actually having the disease. In this context, Bayes’ rule takes the from
Positive predictive value = P(D+│T+) =
P T+ D+ P D+
P T+
Positive predictive value = P(D+│T+) =
P T+ D+ P D+
P T+ D+ P D+ +P T+ P
D-Positive Predictive Value =
Sensitivity Prevalence
Sensitivity Prevalence + 1-Specificity 1-Prevalence
P(D+) The probability that the disease is present in the patient, regardless of any other information. This is the prior probability of D.
P(T+) The probability of a positive event, which is found by adding the probability that a true positive result will appear with the probability that a false positive will
Disease T+ D+ True Positive (TP) P(D+│T +)
Test Positive P(T+)
No Disease T+ D- False Positive (FP)
Test P(D- │T +)
Disease T- D+ False Negative (FN) P(D+│T-)
Test Negative P(T-)
No Disease T- D- True Negative (TN) P(D-│T-)
appear.
P(T+│D+) This is the probability of true positive rate, that is, that the test is positive and the disease present.
P(T+│D-) This is the probability of false positive rate, that is, that the test is positive, though the disease is absent. The prior probability of P(T+│D-) is 1 - P(T+│D+).
Similarly, we can also compute the posterior probability P(D-│T-) of a patient who tests negative while actually being without the disease, from Bayes’s theorem.
Negative predictive value = P(D-│T-) =
P T- P D-P
T-Negative predictive value = P(D-│T-) =
P T- D- P D-
P T- D- P D- +P T- D+ P D+
Negative Predictive Value=
Specificity 1-Prevalence
Sepcificity 1-Prevalence + 1-Sencitivity Prevalence
P(D-) The probability that the disease is absent in the patient.
The prior probability of D- is 1- P(D).
P(T-) The probability of a negative event, which is found by adding the probability that a true negative result will appear with the probability that a false negative will appear.
P (T-│D-) This is the probability of true negative rate, that is, that the test is negative and the disease absent.
P(T-│D+) This is the probability of false negative rate, that is, that the test is negative, though the disease is present. The prior probability of P(T-│D+) is 1 - P(T-│D-).
Secondly, the other method of constructing the decision tree commences with branches of disease status. The subsequent branching states the test outcomes. The primary branching probability shows the likelihood of outcomes for the patient within the specific population, in terms of presence or absence of disease. In each branch, the incidence or prevalence within the specific population are indicated. The probabilities are denoted by P(D+) and P(D-). The test outcomes must be shown in the right hand branch of the
disease status. The subsequent branching probabilities are the likelihood of tests in each outcome. The probabilities are denoted by P(T+│D+), P(T+│D-), P(T-│D+), and P(T-│D-).
In particular, subsequent pathways for the probabilities of sensitivity and specificity are indicated. These probabilities are indicated by the test results, which are routinely calculated and reported, as shown in Figure 3.4.
Figure 3.4 Test approaches to decision problems with disease status
The four possible outcomes associated with this decision tree are than calculated:
True Positive = P(D+) * P(T+│D+) False Negative = P(D+) * P(T-│D+) False Positive = P(D-) * P(T+│D-) True Negative = P(D-) * P(T-│D-)
The actual chorological method used in the second method of constructing the decision tree, whereby the initial chance node separates the population according to the disease status involves the same information as the first method, and it can be illustrated that the probabilities associated with each pathway are equivalent in the two methods.
True positive from the first method =
P(T+)*P(D+ T+)
=
P(T+ D+)*P(D+)
P(T+)*
P(T+)
=
P(D+ T+)*P(D+)
Test Positive T+ D+ True Positive (TP) P(T+│D+)
Disease P(D+)
Test Negative T- D+ False Negative (FN)
Disease P(T-│D+)
Test Positive T+ D- False Positive (FP) P(T+│D+)
No Disease P(D-)
Test Negative T- D- True Negative (TN) (P(T-│D-)
True positive from the second method =
P(D+ T+)*P(D+)
True positive from first method = true positive from second method
Whilst it is not possible to indentify the actual disease status for an individual patient consulting with the doctor, using prevalence and incident data it is possible to calculate the expected numbers with and without disease for a given population. Hence, when modelling a health policy decision affecting a population, the second approach can be used, enabling the direct application of conditional probabilities to the following branches.
Given that most medical information that is provided is conditional upon the disease present and absent, the second method provides an easier method by which to examine a population based decision problem. In light of this, applications of decision analysis to economic evaluations of health care commonly employ the second method.