Chapter adapted and updated, with permission, from Nicoll D et al. Guide to Diagnostic Tests, 7th ed. McGraw-Hill, 2017.
A practical way to calculate the posttest probability of disease is to use the likelihood ratios and odds-probability approach.
Likelihood ratios (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 diseased compared to nondiseased persons. There are two types of LR—LR positive and LR negative, calculated by the following formulas:
When test results are dichotomized using a single cut-off 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 likelihood ratios (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 e2–5).
Table e2–5.Interval likelihood ratios for serum ferritin as a test for iron deficiency anemia. ||Download (.pdf) Table e2–5. Interval likelihood ratios for serum ferritin as a test for iron deficiency anemia.
|Serum Ferritin (mcg/L) ||Interval Likelihood Ratios for Iron Deficiency Anemia |
|≥ 100 ||0.08 |
|45–99 ||0.54 |
|35–44 ||1.83 |
|25–34 ||2.54 |
|15–24 ||8.83 |
|≤ 15 ||51.85 |
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 e2–6 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).
Table e2–6.Examples of likelihood ratios (LR). ||Download (.pdf) Table e2–6. Examples of likelihood ratios (LR).
|Target Disease ||Test ||LR+ ||LR– |
|Abscess ||Abdominal CT scanning ||9.5 ||0.06 |
|Coronary artery disease ||Exercise electrocardiogram (1 mm depression) ||3.5 ||0.45 |
|Lung cancer ||Chest radiograph ||15 ||0.42 |
|Left ventricular hypertrophy ||Echocardiography ||18.4 ||0.08 |
|Myocardial infarction ||Troponin I ||24 ||0.01 |
|Prostate cancer ||Digital rectal examination ||21.3 ||0.37 |
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.
Nomogram for determining posttest probability from pretest probability and likelihood ratios. To figure the posttest probability, place a straightedge between the pretest probability and the likelihood ratio for the particular test. The posttest probability will be where the straightedge crosses the posttest probability line. (Adapted from Fagan TJ. Nomogram for Bayes theorem. [Letter.] N Engl J Med. 1975 Jul 31;293(5):257.)
A more formal way of calculating posttest probabilities uses the LR as follows:
Pretest odds × Likelihood ratio = Posttest odds
To use this 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).
Formulas for converting between probability and odds. (Reproduced, with permission, from Nicoll D et al. Guide to Diagnostic Tests, 7th ed. McGraw-Hill, 2017.)
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