BAYESIAN APPROACH TO PROBABILITY

The commonsense meaning of the term "probability" can readily become confusing within the mathematical morass of formal statistics. In practice, we commonly use and interchange two definitions for probability. In the frequentist approach, we view the probability of an occurrence as the fraction of occasions that the selected event would occur if the trial were repeated many times. So we say that the probability of a fair coin coming up heads is .5 because if we toss it many times, we would get about 50% heads.

However, in many real-life situations we cannot repeat the trials many times to get a frequency of occurrence for an event. In the subjectivist approach, we assign a probability to an event based on our best guess or opinion. Such subjective assessments are very similar to what we do in medicine when we make an interpretation based on opinion and experience. This approach is often termed bayesian statistics because of a mathematical

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The powerful practical insight that comes from bayesian statistics lies in the requirement that we start statistical calculations with an initial probability of an event. This very much matches the real-life situation, in which any test or prediction depends on the patient under study. It is generally medically unwise to consider a diagnosis for a patient unless we take the patient's own characteristics into account. It is obvious to any clinician that an abnormal cardiogram suggesting myocardial ischemia would not mean the same thing in a healthy 30-year-old as it would in his obese hypertensive grandfather. In a similar argument, a negative exercise stress test does not give us great confidence that a very high-risk patient is free of cardiac disease (because the chance of disease is so high and stress tests are notorious for missing disease in some patients).

In medical situations, bayesian statistics are mostly used for the interpretation of diagnostic or predictive data. The term "bayesian" is often an indication that the patients' or population's a priori chances of disease should be considered when applying a predictive test.

When using diagnostic or predictive tests, the concepts of sensitivity and specificity are very useful because they are not dependent on the population. Sensitivity is, roughly, the ability of the test to pick up the disease, and specificity indicates the ability of the test to avoid false alarms. To use these terms we define four possible situations:

True positive (TP): The patient has disease and the test is positive.

False positive (FP): The patient does not have disease and the test is positive.

True negative (TN): The patient does not have disease and the test is negative.

False negative (FN): The patient has disease and the test is negative.

By definition,

Sensitivity = TP/(TP + FN)

Specificity = TN/(TN + FP)

The ideal test has a sensitivity of 1, which indicates that the test picks up all the patients with the disease, and a specificity of 1, which indicates that the test never falsely claims that a healthy patient is diseased.

It is also sometimes useful to define the following:

Positive predictive value (PPV) = TP/(TP + FP)

Negative predictive value (NPV) = TN/(TN + FN)

Remember that although the sensitivity and specificity of a test do not depend on the characteristics of the population, they certainly do depend on the quality of the test. However, PPV and NPV are dependent on the tested population—this situation is where our comment about "bayesian" is relevant. (Although it is not the goal of this chapter to prove these claims, the reader may be convinced by considering what happens to sensitivity, specificity, PPV, and NPV in situations in which the test is applied to a population in whom the disease is absent so that only TN and FP are nonzero: PPV is vanishingly small and NPV approaches 1. Similarly, for a population with almost all patients having disease, PPV goes to 1 and NPV vanishes.)

The bayesian insight can now be stated that we apply diagnostic or predictive tests only to populations of subjects who are somewhat likely to test positive, but not "excessively" so. If we test subjects who are very likely to test positive, a positive test is confirmatory and a negative may be more likely to be a false-negative than a true-negative result. Similarly, if the tested population is very unlikely to have the specified disease, a negative test is only confirmatory and a positive test is more likely to be a false-positive than a true result.