*7.3. Conclusions*

Information statistics have a useful role to play in the evaluation and comparison of diagnostic tests. In some cases, information measures may complement useful concepts such as test sensitivity, test specificity, and predictive values. In other situations, information measures may replace more limited statistics. Mutual information, for example, may be better suited as a single parameter index of diagnostic test performance than alternative statistics. Furthermore, information theory has the potential to help us learn about and teach about the diagnostic process. Examples include concepts illustrated above, including the importance of pretest probability as a determinant of diagnostic information, the amount of information lost by dichotomizing test results, the limited potential of some diagnostic tests to reduce diagnostic uncertainty, and the ways in which diagnostic tests can interact to provide diagnostic information. These are concepts that can all be effectively communicated graphically.

It is hoped that this review will help to motivate new applications of information theory to clinical diagnostic testing, especially as data from newer diagnostic technologies becomes available. The challenge will be to develop systems that accurately diagnosis and treat patients by integrating increasingly large amounts of personalized data [39,40]. A potential role for information theory functions in this process is suggested by their applicability to multidimensional data.

**Supplementary Materials:** R code for calculations and figures are available online at http://www.mdpi.com/1099- 4300/22/1/97/s1.

**Funding:** This research received no external funding.

**Acknowledgments:** The author is very grateful for multiple helpful suggestions by reviewers and the academic editor.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **Appendix A Modified Relative Entropy Is not a Distance Metric**

We first show that the modified relative entropy function (Expression (8)) satisfies the triangle equality. Let the modified relative entropy from probability distribution *b* to probability distribution *a* when the true probability distribution is *c* be expressed as

$$d\_{\mathcal{E}}(a,b) = \sum\_{i} c\_i \log \frac{a\_i}{b\_i}.$$

Then,

$$d\_c(\mathbf{x}, \mathbf{y}) + d\_c(\mathbf{y}, \mathbf{z}) = \sum\_i c\_i \log \frac{\mathbf{x}\_i}{y\_i} + \sum\_i c\_i \log \frac{y\_i}{z\_i} = \sum\_i c\_i \log \frac{\mathbf{x}\_i}{z\_i} = d\_c(\mathbf{x}, \mathbf{z}).$$

Despite satisfying the triangle inequality, *dc*(*a*, *b*) does not meet the criteria for a distance metric [41] (p. 117) because it can be negative. If we try to circumvent this problem by defining the measure as the absolute value of *dc*(*a*, *b*), then the triangle inequality is still satisfied:

$$\left|d\_{\mathfrak{c}}(\mathfrak{x},\mathfrak{y})\right| + \left|d\_{\mathfrak{c}}(\mathfrak{y},\mathfrak{z})\right| \ge \left|d\_{\mathfrak{c}}(\mathfrak{x},\mathfrak{y}) + d\_{\mathfrak{c}}(\mathfrak{y},\mathfrak{z})\right| = \left|d\_{\mathfrak{c}}(\mathfrak{x},\mathfrak{z})\right|.$$

Nevertheless, *dc*(*a*, *<sup>b</sup>*) still fails to qualify as a distance metric because it is not necessarily the case that *dc*(*a*, *<sup>b</sup>*) <sup>=</sup> 0 implies that *<sup>a</sup>* <sup>=</sup> *<sup>b</sup>*. For example, if *<sup>a</sup>*<sup>1</sup> <sup>=</sup> *<sup>b</sup>*<sup>2</sup> <sup>=</sup> 1/4 , *<sup>a</sup>*<sup>2</sup> <sup>=</sup> *<sup>b</sup>*<sup>1</sup> <sup>=</sup> 3/4, and *<sup>c</sup>*<sup>1</sup> <sup>=</sup> *<sup>c</sup>*<sup>2</sup> <sup>=</sup> 1/2, then *dc*(*a*, *<sup>b</sup>*) <sup>=</sup> 0 but *<sup>a</sup> b*.

#### **References**


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