Application of Logistic Regression to the Analysis of Diagnostic Data: Exact Modeling of a Probability Tree of Multiple Binary Variables

In the analysis and presentation of diagnostic relationships by means of conventional multiple logistic regression, the following limitations occur: 1) the model starts not from the prior disease odds but from the posterior disease odds for all test variables having a zero value; 2) apart from the odds ratio, other test characteristics cannot be read from the model; 3) the sequence of entering of terms is guided by pure statistical criteria and not primarily by the criterion gain in certainty; 4) interactions are not very comprehensibly represented and are difficult to interpret. A method dealing with these limitations with respect to the analysis of data on the relationships between binary tests and disease outcome is described. Essential is the transformation of any test variable x to (x - x0), where x0 is that specific (virtual) value of x so that: posterior disease odds = prior disease odds, and consequently LRx0 = 1. Moreover, a simple branching structure is introduced while the terms are entered in order of decreasing gain in certainty. Examples are given for one-, two-, and three-test situations with and without dependency and interaction of tests, and general formulas are presented. For situations with the same prior probability, and the same overall discrimination of the separate test variables, all equations clearly have a common basis. Inclusion of new variables does not affect coefficients previously included in the model, and terms without a significant contribution can be skipped without affecting other coefficients. Standard errors and confidence intervals of test characteristics can be computed using the BMDP LR program. Further study needs to be done on the inclusion of continuous variables and cost-benefit aspects, and comparison with the performance of the CART program.