Stats >> Define >> Fagan nomogram

Dear Professor Mean, I have a picture of a Fagan nomogram. This is supposed to help me estimate the impact of a diagnostic test, but to me it just looks like a piece of abstract art. Can you explain how this thing works. --Restless Ron

Dear Restless,

I know that it looks like junk now, but if you wait a few years it will be worth more than a Jackson Pollock.

The Fagan nomogram is a graphical tool for estimating how much the result on a diagnostic test changes the probability that a patient has a disease. A picture of the Fagan nomogram appears below.

To use this tool, you need to provide your best estimate of the probability of the disease prior to testing. This is usually related to the prevalance of the disease, though this may be modified up or down on the basis of certain risk factors that are present in your patient pool or possibly in this particular patient. You also need to know the likelihood ratio for the diagnostic test.

With this information, draw a line connecting the pre-test probability and the likelihood ratio. Extend this line until it intersects with the post-test probability. The point of intersection is the new estimate of the probability that your patient has this disease.

More details

Here are details on how the graph works and how you could construct a similar graph yourself. The principle is very much similar to a slide rule.

First, the computations involved use odds rather than ratios. If you multiply the pre-test odds by the likelihood ratio, you will get the post-test odds. And since multiplication of two numbers is equivalent to adding their logarithms, we use a log scaling for both the odds and the likelihood ratio.

The official formula is:

and the Fagan nomogram uses the equivalent formula

So although the labels on the left and right are written in terms of probability, the tick marks are spaced at the log odds. For technical reasons.we have to set the scaling of the log likelihood ratio to 1/2 that of the log odds. We also have to invert the scale for the log pre-test odds.

So if you wanted to construct this graph yourself, simply plot a range of log odds at x=+1. Plot an inverted range of log odds at x=+1. Write labels in terms of probabilities rather than odds. Then plot 1/2 of the log likelihood ratio values at x=0.

Example

Here are a couple examples of how to use the Fagan nomogram.

Boere-Boonekamp describes an early test for developmetnal dysplasia of the hip. The likelihood ratio for a postive result from this test is 7 for boys (5 for girls). The prevalance of this condition is 1.5% in boys (6% for girls). Suppose one of our patients is a boy with no special risk factors. The diagnostic test is positive. What can we say about the chances that this boy will develop hip dysplasia?

The post-test probability is a bit below 10%.

Suppose this boy had a family history of hip dysplasia that would increase our pre-test probability to 25%. How much would our assessment change if we.had a negative test result?

The likelihood ratio for a negative result is 0.09 for boys (0.2 for girls) So we would draw a line connecting the pre-test probability of slightly more than 20% to a likelihood ratio about just a bit below .1

This leads to a post-test probability around 3%.

Summary

Restless Ron wants to understand how to use a Fagan nomogram to calculate disease probabilities. Professor Mean explains that you draw a line connecting the pre-test probability of disease and the likelihood ratio. When you extend this line to the right, it intersects at the post-test probability of disease.

Further reading

The example of hip dysplasia comes from the Boere-Boonekamp reference. Fagan originally published the nomogram in 1975. The CEBM web site has an interactive Fagan nomogram, but you need to have the Shockwave plug-in to use it.

NHS Research and Development Centre for Evidence Based Medicine.
Bedenoch D. (Accessed July 3, 2001)
http://cebm.jr2.ox.ac.uk/

Early Detection of Developmental Dysplasia of the Hip in the Netherlands: The Validity of a Standardized Assessment Protocol in Infants
Boere-Boonekamp MM, Kerkhoff THM, Schuil PB, Zielhuis GA.
American Journal of Public Health 1998;88(2):285-288.

"Nomogram for Bayes theorem" Fagan TJ (1975) New England Journal of Medicine; 293: 257.

Summary

Restless Ron wants to understand how to use a Fagan nomogram to calculate disease probabilities. Professor Mean explains that you draw a line connecting the pre-test probability of disease and the likelihood ratio. When you extend this line to the right, it intersects at the post-test probability of disease.

Stats >> Define >> Fagan nomogram

This page was last modified on 07/27/10 . Send feedback to ssimon@cmh.edu.