Fagan Nomogram

The Fagan nomogram [1] is a nomogram [2] to computes the probability of the presence of a given condition based on an imperfect test and varying pre-test probabilities. It is a very handy tool to understand Bayes’s Theorem “physically”. People typically have a sense that the less powerful the test, the less likely it is that a detection (a positive result on the test) means the presence of the condition. But people are much less likely to grasp the role of the pre-test probability [3].

As an illustration, consider the example illustrated in the default setting of the nomogram below. If only 10% of the population exhibit a particular kind of condition, then even if a test gives a ratio of true positives to false positives at 10 to 1, only 50% of the people tested positive will actually exhibit the condition.

Similarly, if only 10% of the population exhibit the condition, then in order to be 90% sure that a positive result indicates the presence of the condition, then the test can give a false-positive result only 1 every 100 times it gives true positive results. Intuitively, what’s going on is that the base-10 logarithm of the “likelihood ratio” (10 and 100 respectively in the examples above) is the “number of nines added to the baseline probability”, interpreting a probability of 0.1 as having “negative 1 nine”.

The original nomogram was meant to be used with a physical ruler to do the calculations. Here, you can move the circles around to change the settings.

References

  1. Fagan TJ. Letter: Nomogram for Bayes theorem. NEJM, July 1975.

  2. Wikipedia article on Nomograms.

  3. Casscells W, Schoenberger A, Graboys TB. Interpretation by physicians of clinical laboratory results. NEJM 299(18):999-1001, Nov. 1978.