Purpose: We apply the information theory concept of “channel capacity” to quantify the performance of a diagnostic test as a function of test sensitivity (Sn) and test specificity (Sp).
Method: The expected value of the amount of information a diagnostic test provides is equal to the “mutual information” between the test result (R) and the disease state (D). For the case in which only two test results and two disease states are considered, mutual information, I(D;R), is a function of Sn, Sp, and the pretest probability of disease (P). The channel capacity (C) of the test is the maximal value of I(D;R) for a given Sn and Sp. After deriving the expression for I(D;R) in terms of Sn, Sp, and P, we solve for the value of P that maximizes I(D;R). C is obtained by using this value of P to calculate I(D;R).
Result: Let J = (Sp)ln(Sp)-(Sn)ln(Sn)+(1-Sp)ln(1-Sp)-(1-Sn)ln(1-Sn), M = 1-Sn-Sp, and F = 1+eJ/M. Then the value of P that optimizes I(D;R) for a given Sn and Sp is (1-SpF)/MF.
Using this value for P,
C = (1-P)[(Sp)log2(Sp)+(1-Sp)log2(1-Sp)]
+(P)[(1-Sn)log2(1-Sn)+(Sn)log2(Sn)]
- [(1-P)(Sp)+(P)(1-Sn)]log2[(1-P)(Sp)+(P)(1-Sn)]
- [(1-P)(1-Sp)+(P)(Sn)]log2[(1-P)(1-Sp)+(P)(Sn)].
The following graph expresses C as a function of Sn when Sn = Sp.
Conclusion: C is a ratio scale measure of diagnostic test performance that is equal to the upper limit of the number of bits of information that a test can be expected to provide. For the case in which there are only two possible test results and only two possible disease states, C can be expressed as a function of Sn and Sp. It has a range from 0 bits (when Sn = Sp = 0.5) to 1 bit (when Sn = Sp = 1 or Sn = Sp = 0). As a consequence of the nonlinear relationship between C and both Sn and Sp, when Sn = Sp these two parameters must be at least 0.89 (or less than 0.11) for C to be only 0.5 bits. C provides a convenient and meaningful single parameter measure of diagnostic test performance.
See more of: The 32nd Annual Meeting of the Society for Medical Decision Making