special subjects
Understanding Medical Tests and Test Results
Bayes Theorem
The process of using the pre-test probability of disease and the test characteristics to calculate the post-test probability is referred to as Bayes theorem or Bayesian revision. For routine clinical use, Bayesian methodology typically takes several forms: Odds-likelihood formulation (calculation or nomogram) Tabular approach If the pre-test probability of disease is expressed as its odds and because a test’s LR represents odds, the product of the 2 represents the post-test odds of disease (analogous to multiplying 2 probabilities together to calculate the probability of simultaneous occurrence of 2 events): Pre-test odds×LR=post-test odds Because clinicians typically think in terms of probabilities rather than odds, probability can be converted to odds (and vice versa) with these formulas: Odds=probability/1−probability Probability=odds/odds+1 Consider the example of UTI as given in the table Distribution of Test Results of a Hypothetical Leukocyte Esterase Test in a Cohort of 1000 Women With an Assumed 30% Prevalence of UTI , in which the pre-test probability of UTI is 0.3, and the test being used has an LR 0.13. Thus, a negative test result decreases the pre-test probability from 30% to 13%, again the same result obtained in the NPV calculation in the Distribution of Test Results of a Hypothetical Leukocyte Esterase Test in a Cohort of 1000 Women With an Assumed 30% Prevalence of UTI . Many medical calculator programs that run on handheld devices are available to calculate post-test probability from pre-test probability and LRs. Using a nomogram is particularly convenient because it avoids the need to convert between odds and probabilities or create 2×2 tables. To use the Fagan nomogram , a line is drawn from the pre-test probability through the LR. The post-test probability is the point at which this line intersects the post-test probability line. Sample lines in the figure are drawn using data from the UTI test in the table Although the nomogram appears less precise than calculations, typical values for pre-test probability are often estimates, so the apparent precision of calculations is usually misleading. Illustrative lines are drawn using data from the UTI test in the table Distribution of Test Results of a Hypothetical Leukocyte Esterase Test in a Cohort of 1000 Women With an Assumed 30% Prevalence of UTI. Line A represents a positive test result, drawn from a pre-test probability of 0.3 through an LR+of 4.73 to a post-test value of slightly<0.7, similar to the calculated probability of 0.67. Line B represents a negative test result drawn from pre-test probability of 0.3 through an LR-of 0.34 to a post-test value slightly>0.1, similar to the calculated probability of 13%. LR+=likelihood ratio for a positive result; LR-=LR for a negative result. Adapted from Fagan TJ. Letter: Nomogram for Bayes theorem.New England Journal of Medicine293:257, 1975. Often, LRs of a test are not known, but sensitivity and specificity are known, and pre-test probability can be estimated. In this case, Bayesian methodology can be done using a 2×2 table illustrated in the table Interpretation of a Hypothetical Leukocyte Esterase (LE) Test Result in a Cohort of 1000 Women Assuming a 30% Prevalence of UTI (Pre-Test Probability), Test Sensitivity 71%, and Specificity 85%* using the example from the table 