e2-06: Likelihood Ratios & Odds-Probability Conversion in Medical Decision-Making

A practical way to calculate the posttest probability of disease is to use the LRs and odds-probability approach.

LRs combine both test sensitivity and specificity into a single measure (a mathematical description of the strength of a diagnostic test), which helps evaluate and interpret a diagnostic test. LRs indicate how many times more (or less) a test result is to be found in persons with disease compared with persons without disease. There are two types of LR—LR positive and LR negative, calculated by the following formulas:

When test results are dichotomized using a single cutoff value to divide "positive" from "negative," every test has two LRs, one corresponding to a positive test (LR+) and one corresponding to a negative test (LR–):

For continuous measures, multiple interval LRs (iLRs) can be defined to correspond to ranges or intervals of test results. The iLR for a test result interval is the probability of a result in that same interval for a disease-positive patient divided by the probability of a result in the same interval for a disease-negative patient. Given the pretest probability of disease and the test result, the iLR is used to calculate the posttest probability (Table e2–6).

LRs can be calculated using the above formulas. They can also be found in some textbooks, journal articles, and online programs (such as www.thennt.com) (see Table e2–7 for sample values). LRs provide an estimation of whether there will be significant change in pretest to posttest probability of a disease given the test result, and thus can be used to make quick estimates of the usefulness of contemplated diagnostic tests in particular situations. An LR of 1 implies that there will be no difference between pretest and posttest probabilities. LRs of more than 10 or less than 0.1 indicate large, often clinically significant differences. LRs between 1 and 2 and between 0.5 and 1 indicate small differences (rarely clinically significant).

The simplest method for calculating posttest probability from pretest probability and LRs is to use a nomogram (Figure e2–7). The clinician places a straightedge through the points that represent the pretest probability and the LR and then reads the posttest probability where the straightedge crosses the posttest probability line.

A more formal way of calculating posttest probabilities uses the LR as follows:

To use this calculation formulation, probabilities must be converted to odds, where the odds of having a disease are expressed as the probability of having the disease divided by the probability of not having the disease. For instance, a probability of 0.75 (75%) is the same as 3:1 odds (Figure e2–8).

Odds is defined as the probability of the event occurring divided by the probability of the event not occurring and is calculated by formula Odds = P/(1 – P), where P is the probability of the event occurring. Probability, on the other hand, is calculated by formula P = Odds/(1 + Odds).

To estimate the potential benefit of a diagnostic test, the clinician first estimates the pretest odds of disease given all available clinical information and then multiplies the pretest odds by the positive and negative LRs. The results are the posttest odds, or the odds that the patient has the disease if the test is positive or negative. To obtain the posttest probability, the odds are converted to a probability (Figure e2–8).

For example, if the clinician believes that the patient has a 60% chance of having a myocardial infarction (pretest odds of 3:2) and the troponin I test is positive (LR+ = 24), then the posttest odds of having a myocardial infarction are

If the troponin I test is negative (LR– = 0.01), then the posttest odds of having a myocardial infarction are

To this point, the impact of only one test on the probability of disease has been discussed, whereas during most diagnostic workups, clinicians obtain clinical information in a sequential fashion. To calculate the posttest odds after three tests, for example, the clinician might estimate the pretest odds and use the appropriate LR for each test:

When using this approach, however, the clinician should be aware of a major assumption: the chosen tests or findings must be conditionally independent. For instance, with liver cell damage, AST and ALT enzymes may be released by the same process and are thus not conditionally independent. If conditionally dependent tests are used in this sequential approach, an inaccurate posttest probability will result.

A key aspect of medical decision making is the selection of a treatment threshold, ie, the probability of disease at which treatment is indicated. The treatment threshold is determined by the relative consequences of different actions: treating when the disease is present; not treating when the disease is absent; treating when the disease is actually absent; or failing to treat when the disease is actually present. Figure e2–9 shows a possible way of identifying a treatment threshold by considering the value (utility) of these four possible outcomes.

Use of a diagnostic test is warranted when its result could shift the probability of disease across the treatment threshold. For example, a clinician might decide to treat with antibiotics if the probability of streptococcal pharyngitis in a patient with a sore throat is greater than 25% (Figure e2–10A).

If, after reviewing evidence from the history and physical examination, the clinician estimates the pretest probability of strep throat to be 15% (ie, clinician's initial assessment), then a diagnostic test such as throat culture (LR+ = 7) would be useful only if a positive test would shift the posttest probability above 25%. Use of the nomogram shown in Figure e2–7 indicates that the posttest probability would be 55% (ie, clinician's new assessment) (Figure e2–10B); thus, ordering the test would be justified since it affects patient management. On the other hand, if the history and physical examination had suggested that the pretest probability of strep throat was 60%, the throat culture (LR– = 0.33) would be indicated only if a negative test would lower the posttest probability below 25%. Using the same nomogram, the posttest probability after a negative test would be 33% (Figure e2–10C). Therefore, ordering the throat culture would not be justified because it does not affect patient management.

Tests are only useful if the result is likely to influence or change patient management. The test-treatment threshold is an important concept in clinical decision making. To apply the threshold model of decision making, it is necessary to have: (1) validated clinical prediction tools so that pretest probability can be accurately determined from available data; (2) evidence-based treatment threshold for the disease; and (3) accuracy of diagnostic tests, in particular LRs of the tests, in consideration.

Up to this point, the discussion of diagnostic testing has focused on test characteristics and methods for using these characteristics to calculate the probability of disease in different clinical situations. Although useful, these methods are limited because they do not incorporate the many outcomes that may occur in clinical medicine or the values that patients and clinicians place on those outcomes. To incorporate outcomes and values with characteristics of tests, decision analysis can be used.

Decision analysis is a quantitative evaluation of the outcomes that result from a set of choices in a specific clinical situation. Although it is infrequently used in routine clinical practice, the decision analysis approach can be helpful to address questions relating to clinical decisions that cannot easily be answered through clinical trials.

The basic idea of decision analysis is to model the options in a medical decision, assign probabilities to the alternative actions, assign values (utilities) (eg, survival rates, quality-adjusted life-years, or costs) to the various outcomes, and then calculate which decision gives the greatest expected value (expected utility). To complete a decision analysis, the clinician would proceed as follows: (1) draw a decision tree showing the elements of the medical decision; (2) assign probabilities to the various branches; (3) assign values (utilities) to the outcomes; (4) determine the expected value (expected utility) (the product of probability and value [utility]) of each branch; and (5) select the decision with the highest expected value (expected utility). The results obtained from a decision analysis depend on the accuracy of the data used to estimate the probabilities and values of outcomes.

Figure e2–11 shows a decision tree in which the decision to be made is whether to treat without testing, perform a test and then treat based on the test result, or perform no tests and give no treatment. The clinician begins the analysis by building a decision tree showing the important elements of the decision. Once the tree is built, the clinician assigns probabilities to all the branches. In this case, all the branch probabilities can be calculated from: (1) the probability of disease before the test (pretest probability), (2) the chance of a positive test result if the disease is present (sensitivity), and (3) the chance of a negative test result if the disease is absent (specificity). Next, the clinician assigns value (utility) to each of the outcomes.

After the expected value (expected utility) is calculated for each branch of the decision tree, by multiplying the value (utility) of the outcome by the probability of the outcome, the clinician can identify the alternative with the highest expected value (expected utility). When costs are included, it is possible to determine the cost per unit of health gained for one approach compared with an alternative (cost-effectiveness analysis). This information can help evaluate the efficiency of different testing or treatment strategies.

Although time-consuming, decision analysis can help structure complex clinical problems, assist in difficult clinical decision-making by providers and patients, and improve the quality of clinical decisions. It is particularly useful in situations where diagnostic tests are esoteric or expensive or both (eg, paraneoplastic antibody panel, DNA sequencing–based gene mutation panel for hereditary cancers, etc).

Vast resources are being spent on diagnostic testing. New laboratory tests are being developed and marketed all the time, including companion diagnostic tests (tests that are designed to be paired with a specific drug) and DNA sequencing–based mutation analyses. Direct to consumer tests are gaining popularity among patients and health care providers. Clinicians need to know how to appraise published studies on new diagnostic tests (eg, performance characteristics) and how to determine whether a new test is superior to existing test(s) in terms of clinical utility and cost-effectiveness. Careful examination of the best available evidence is essential.

Evidence-based medicine is the care of patients using the best available research evidence to guide clinical decision making. It relies on the identification of methodologically sound evidence, critical appraisal of research studies for both internal validity (freedom from bias) and external validity (applicability and generalizability), and the dissemination of accurate and useful summaries of evidence to inform clinical decision making. Systematic reviews can be used to summarize evidence for dissemination, as can evidence-based synopses of current research. Systematic reviews often use meta-analysis: statistical techniques to combine evidence from different studies to produce a more precise estimate of the effect of an intervention or the accuracy of a test. The concept of evidence-based medicine can be readily applied to diagnostic testing since appropriate use of a diagnostic test is part of the decision-making process.

The Choosing Wisely campaign, launched by the American Board of Internal Medicine in 2012, encourages clinicians and their patients to examine the usefulness of certain laboratory tests and medical procedures, and promotes choosing care that is necessary, evidence-based, and not harmful. More than 80 specialty societies have joined the campaign as partners, and identified tests and procedures commonly used in their respective field that either are considered unnecessary or that should be questioned. Those unnecessary tests and procedures are available online at the Choosing Wisely website (www.choosingwisely.org).

Clinical practice guidelines are systematically developed statements intended to assist practitioners in making decisions about health care. Clinical algorithms and practice guidelines are now ubiquitous in medicine, developed by various professional specialty societies or independent expert panels. Diagnostic testing is an integral part of such algorithms and guidelines (eg, high-sensitivity D-dimer test in the guidelines for diagnosis and management of venous thromboembolism). Their utility and validity depend on the quality of the evidence that shaped the recommendations, on their being kept current, and on their acceptance and appropriate application by clinicians. Although some clinicians are concerned about the effect of guidelines on professional autonomy and individual decision making, many organizations are trying to use compliance with practice guidelines as a measure of quality of care. It is important to note, however, that evidence-based guidelines are used to complement, not replace, clinical judgment and shared decision making tailored to individual patients. Furthermore, personalized treatment (ie, using advanced diagnostic or prognostic tools and incorporating patient preferences to guide an individual patient's treatment) is increasingly recommended as part of practice guidelines. The approach is in keeping with the rapidly evolving field of personalized medicine, in which diagnostic testing plays an important role in selecting the best possible therapies that are tailored to individual characteristics of each patient.

Computerized and mobile information technologies provide clinicians with information from laboratory, imaging, physiologic monitoring systems, and many other sources. Computerized clinical decision support systems, along with test order communication systems, have been increasingly used to develop, implement, and refine computerized protocols for specific processes of care derived from evidence-based practice guidelines. Artificial intelligence–based algorithms could help diagnose and manage complicated diseases faster and more efficiently. It is important that clinicians use established information technology to deliver standard medical care in their practice.