Inter-relationships among the four basic indicators of diagnostic testing : a signal-flow-graph approach

Abstract EN

Ternary problems of conditional probability formulated in conjunction with a normalized version of the two-by-two contingency table might be solved by arithmetic or algebraic techniques to produce useful inter-relationships among the various marginal, conjunctive, and conditional probabilities associated with that table.

In particular only tedious and indirect algebraic methods can be used to inter-relate the four most prominent indicators of diagnostic testing (sensitivity, specificity, positive predictive value (PPV), and negative predictive value (NPV)).

In general, it is theoretically possible to express any one of these four indicators in terms of the other three.

Such a possibility is materialized herein, in a direct and appealing arithmetic fashion, through the use of a (Mason) signal flow graph, which carefully selects for its constituent relations, an appropriate combination of the total probability law, Bayes’ rule and the complementation formula.

Despite the relative simplicity of the four formulas expressing each of the four diagnostic indicators in terms of the other three, they seem not to be well known in the open literature.

We call a set of four values satisfying these formulas (to within permissible round-off errors) a consistent set.

We define two checking functions of these four values that we call the Diagnostic Checking Difference (DCD) and the Diagnostic Checking Ratio (DCR) that are exactly 0 and 1, respectively, for consistent values.

The deviation of the DCD and the DCR from 0 and 1, respectively, is, therefore, a measure of inconsistency for any purported set of the four diagnostic indicators.

An interesting observation is that the unbiased measures of informedness and markedness emerge naturally within these two functions, suggesting that the remaining terms within these checking functions might also serve as unbiased measures.

We made an extensive testing of published sets of the four basic indicators, to check whether these sets are consistent or not.

In a dominant majority of cases, the published sets are consistent, thereby independently attesting to the correctness of our formulas.

However, in a small (albeit significant) number of cases, we came across sets that are dramatically inconsistent.

Even some prominent celebrated medical journals were not inconsistency free.

This is particularly curious, because there is no much room for error when computing the four indicators out from the raw data of true positives, true negatives, false positives, and false negatives.

Since the four indicators constitute only three independent entities (rather than four), an expert is not free to guess the four separately.

Any attempt to estimate the four indicators via expert opinion or some sort of model should be confined to estimating only three of them, with the fourth being deduced numerically from the rest.