# The Discriminatory Power of Diagnostic Information from Discrete Medical Tests

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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.

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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.

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Contributed by: John Fountain and Philip Gunby (March 2011)
Open content licensed under CC BY-NC-SA

## Details

A richer explanation of Bayes's theorem in the 2×2 case is available in the Demonstration Bayes's Theorem and Inverse Probability. A comprehensive explanation of the natural frequency method of representing probability distributions in the 2×2 discrete case is available in the Demonstration Comparing Ambiguous Inferences when Probabilities Are Imprecise.

Snapshot 1: a reduced base rate (20% down to 5%)

Snapshot 2: an improved specificity (70% up to 95%)

Snapshot 3: an improved sensitivity (80% up to 95%)

[1] J. M. Bernardo and A. F. M. Smith, Bayesian Theory, New York: John Wiley & Sons, 1994.

## Permanent Citation

John Fountain and Philip Gunby

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