December 28, 2016

A useful diagnostic test should provide information that helps to discriminate between competing hypotheses. But any practical diagnostic will be imperfect: both false positive and false negative indications are to be expected. So just how useful is a diagnostic test when it is, necessarily, imperfect? In [1], p. 44 shows a static, graphical example of how Bayes's theorem may be used to understand the factors determining the discriminatory power of diagnostic tests. This Demonstration is a dynamic version of that argument. Let ... be the logical truth value (1 or 0) of a proposition about a state variable (e.g., a disease or health risk is present or absent), and let ... be the logical truth value (1 or 0) of a proposition about the outcome of an indicative imperfect diagnostic test (e.g., an X-ray or blood test measurement is either definitely positive or negative for this disease). From a statistical perspective there are three precise numerical inputs that feed into a coherent posterior inference about binary-valued ... after having observed the result of the binary-valued diagnostic signal ... : a sensitivity number, a specificity number, and a base rate number. The first two characterize uncertainty about the outcome of the diagnostic ... as a conditional probability under two different information conditions about the state ... . The sensitivity number ... expresses uncertainty about whether the diagnostic test ... will be positive, that is, ... , assuming that ... is true. The specificity number ... expresses an uncertainty about whether the diagnostic test ... for ... will be negative, that is, ... , assuming that ... is true. The third number, the base rate number, is a marginal or unconditional probability, ... , characterizing uncertainty about the binary state variable ... in the absence of, or prior to knowing, any diagnostic information ... . The discriminatory power of diagnostic information can be measured by the levels and differences between two inverse conditional probability assessments, ... and ... , one for each possible diagnostic test result. This interactive Demonstration creates a graphical depiction of the inverse probabilities ... and ... as functions of the underlying sensitivity, specificity, and base rate inputs. A natural frequency representation of the full joint probability distribution over the random variables ( ... , ... ) is provided in a truth table format above the graph, where the column entries are frequency counts or "cases" in a hypothetical population of a fixed size.