# Statistical Foundations of Clinical Decisions

8 Statistical Foundations of Clinical Decisions

There is no controversy about the need to improve clinical decision making and maximize the quality of care. Opinions do differ, however regarding the extent to which the tools discussed in this chapter are likely to help in actual clinical decision making. Some individuals and medical centers already use these methods to guide the care of individual patients. Others acknowledge that the tools can help to formulate policy and analyze the cost-effectiveness of medical interventions, such as immunizations,1,2 but they may not use the techniques for making decisions about individual patients. Even when the most highly regarded means are used to procure evidence, such as double-blind, placebo-controlled clinical trials, the applicability of that evidence to an individual patient is uncertain and a matter of judgment.

Regardless of the clinician’s philosophic approach to using these methods for actual clinical care, they can help clinicians to understand the quantitative basis for making clinical decisions in the increasingly complex field of medicine.

# I Bayes Theorem

The numerator of Bayes theorem merely describes cell a (the true-positive results) in Table 7-1. The probability of being in cell a is equal to the prevalence times the sensitivity, where p(D+) is the prevalence (expressed as the probability of being in the diseased column) and where p(T+ | D+) is the sensitivity (the probability of being in the top, test-positive, row, given the fact of being in the diseased column). The denominator of Bayes theorem consists of two terms, the first of which describes cell a (the true-positive results), and the second of which describes cell b (the false-positive results) in Table 7-1. In the second term of the denominator, the probability of the false-positive error rate, or p(T+ | D−), is multiplied by the prevalence of nondiseased persons, or p(D−). As outlined in Chapter 7, the true-positive results (a) divided by the true-positive plus false-positive results (a + b) gives a/(a + b), which is the positive predictive value.

In genetics, a simpler-appearing formula for Bayes theorem is sometimes used. The numerator is the same, but the denominator is merely p(T+). This makes sense because the denominator in a/(a + b) is equal to all those who have positive test results, whether they are true-positive or false-positive results.

## A Community Screening Programs

In a population with a low prevalence of a particular disease, most of the positive results in a screening program for the disease likely would be falsely positive (see Chapter 7). Although this fact does not automatically invalidate a screening program, it raises some concerns about cost-effectiveness, which can be explored using Bayes theorem.

A program employing the tuberculin tine test to screen children for tuberculosis (TB) is discussed as an example (based on actual experience).3 This test uses small amounts of tuberculin antigen on the tips of tiny prongs called tines. The tines pierce the skin on the forearm and leave some antigen behind. The skin is examined 48 hours later, and the presence of an inflammatory reaction in the area where the tines entered is considered a positive result. If the sensitivity and specificity of the test and the prevalence of TB in the community are known, Bayes theorem can be used to predict what proportion of the children with positive test results will have true-positive results (i.e., will actually be infected with Mycobacterium tuberculosis).

Box 8-1 shows how the calculations are made. Suppose a test has a sensitivity of 96% and a specificity of 94%. If the prevalence of TB in the community is 1%, only 13.9% of those with a positive test result would be likely to be infected with TB. Clinicians involved in community health programs can quickly develop a table that lists different levels of test sensitivity, test specificity, and disease prevalence that shows how these levels affect the proportion of positive results that are likely to be true-positive results. Although this calculation is fairly straightforward and extremely useful, it is not used often in the early stages of planning for screening programs. Before a new test is used, particularly for screening a large population, it is best to apply the test’s sensitivity and specificity to the anticipated prevalence of the condition in the population. This helps avoid awkward surprises and is useful in the planning of appropriate follow-up for test-positive individuals. If the primary concern is simply to determine the overall performance of a test, however, likelihood ratios, which are independent of prevalence, are recommended (see Chapter 7).

Box 8-1 Use of Bayes Theorem or 2 × 2 Table to Determine Positive Predictive Value of Hypothetical Tuberculin Screening Program

# Part 3 Use of 2 × 2 Table, with Numbers Based on Study of 10,000 Persons

## B Individual Patient Care

Suppose a clinician is uncertain about a patient’s diagnosis, obtains a test result for a certain disease, and the test is positive. Even if the clinician knows the sensitivity and specificity of the test, this does not solve the problem, because to calculate the positive predictive value, whether using Bayes theorem or a 2 × 2 table (e.g., Table 7-1), it is necessary to know the prevalence of the disease. In a clinical setting, the prevalence can be considered the expected prevalence in the population of which the patient is part. The actual prevalence is usually unknown, but often a reasonable estimate can be made.

For example, a clinician in a general medical clinic sees a male patient who complains of easy fatigability and has a history of kidney stones, but no other symptoms or signs of parathyroid disease on physical examination. The clinician considers the probability of hyperparathyroidism and decides that it is low, perhaps 2% (reflecting that in 100 similar patients, probably only 2 of them would have the disease). This probability is called the prior probability, reflecting that it is estimated before the performance of laboratory tests and is based on the estimated prevalence of a particular disease among patients with similar signs and symptoms. Although the clinician believes that the probability of hyperparathyroidism is low, he or she orders a serum calcium test to “rule out” the diagnosis. To the clinician’s surprise, the results of the test come back positive, with an elevated level of 12.2 mg/dL. The clinician could order more tests for parathyroid disease, but even here, some test results might come back positive and some negative.

Under the circumstances, Bayes theorem could be used to help interpret the positive test. A second estimate of disease probability in this patient could be calculated. It is called the posterior probability, reflecting that it is made after the test results are known. Calculation of the posterior probability is based on the sensitivity and specificity of the test that was performed and on the prior probability of disease before the test was performed, which in this case was 2%. Suppose the serum calcium test had 90% sensitivity and 95% specificity (which implies it had a false-positive error rate of 5%; specificity + false-positive error rate = 100%). When this information is used in the Bayes equation, as shown in Box 8-2, the result is a posterior probability of 27%. This means that the patient is now in a group of patients with a substantial possibility, but still far from certainty, of parathyroid disease. In Box 8-2, the result is the same (i.e., 27%) when a 2 × 2 table is used. This is true because, as discussed previously, the probability based on the Bayes theorem is identical to the positive predictive value.

Box 8-2 Use of Bayes Theorem or 2 × 2 Table to Determine Posterior Probability and Positive Predictive Value in Clinical Setting (Hypothetical Data)

# Part 3 Use of a 2 × 2 Table to Calculate First Positive Predictive Value

In light of the 27% posterior probability, the clinician decides to order a serum parathyroid hormone concentration test with simultaneous measurement of serum calcium, even though this test is expensive. If the parathyroid hormone test had a sensitivity of 95% and a specificity of 98%, and the results turned out to be positive, the Bayes theorem could be used again to calculate the probability of parathyroid disease in this patient. This time, however, the posterior probability for the first test (27%) would be used as the prior probability for the second test. The result of the calculation, as shown in Box 8-3, is a new probability of 94%. The patient likely does have hyperparathyroidism, although lack of true, numerical certainty even at this stage is noteworthy.

Box 8-3 Use of Bayes Theorem or 2 × 2 Table to Determine Second Posterior Probability and Second Positive Predictive Value in Clinical Setting