WOLFRAM|DEMONSTRATIONS PROJECT

The Discriminatory Power of Diagnostic Information from Discrete Medical Tests

total count

100

1000

10000

your selection of parameters - bolder

base rate

sensitivity

specificity

benchmark (BM) choices - lighter

BM base rate

0.2

BM sensitivity

0.8

BM specificity

0.7

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

P(D=1|S=1)

expresses uncertainty about whether the diagnostic test

will be positive, that is,

D=1

, assuming that

S=1

is true. The specificity number

P(D=0|S=0)

expresses an uncertainty about whether the diagnostic test

for

S=1

will be negative, that is,

D=0

, assuming that

S=0

is true. The third number, the base rate number, is a marginal or unconditional probability,

P(S=1)

, 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,

P(S|D=1)

and

P(S|D=0)

, one for each possible diagnostic test result. This interactive Demonstration creates a graphical depiction of the inverse probabilities

P(S|D=1)

and

P(S|D=0)

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.