Likelihood Ratios: Cirrhosis

At a recent noon conference, we reviewed likelihood ratios (LRs).  I drew some examples from an article in JAMA's series The Rational Clinical Examination on assessing the probability of cirrhosis in patients with liver disease.

The mathematical definition of likelihood ratios is a little off-putting, but their significance is easy to understand if you just remember that the positive likelihood ratio is true positives divided by false positive, and the negative likelihood ratio is true negatives over false negatives.  If you remember this, you can figure out the math if you need to so long as you also remember the definitions of sensitivity and specificity.

From this definition, you can see that what likelihood ratios tell you is how much more likely a given clinical sign or test result is to occur in somebody with the disease than it is to occur in somebody without the disease.  Thus, for palmar erythema, the clinical sign shown below, the likelihood ratio of 5.0 means that palmar erythema is 5.0 more likely to occur in the presence of cirrhosis than it its absence.


This is still not a totally intuitive way to think about things, but the real value of likelihood ratios is that you can use them to translate directly from pre-test probability to post-test probability.  Suppose that the pre-test probability of cirrhosis in a given patient (which you would get from epidemiological studies of people like your patient) is 10%.  Given that, you can use a nomogram like the one shown below (reproduced from Sackett et al.) to get the post-test probability as shown.




The powerful thing about this method is that the post-test probability you've just calculated is the pre-test probability for the next sign you observe - so long as the two signs have independent mechanisms.  If they had the same mechanism (say, peripheral edema and pleural effusion in CHF) you would just be proving the same thing over and over; but if they have different mechanisms you can "chain" the likelihood ratios.  So, to get back to this patient, if his probability of cirrhosis is increased to 30% by the finding of palmar erythema, and he also has ascites (LR = 7.2), then you can use the nomogram again as follows:



The red line represents the change in probability conferred by the finding of palmar erythema.  The green line represents the finding of ascites on top of palmar erythema, which is why it starts where the red one ends.

This leads us to an important observation about LRs, which is that their meaning completely depends on the pre-test probability of the thing you're looking for.  If the probability of the condition is high, a sign associated with a modest LR may be virtually diagnostic; however, if it's wildly improbable, even very high LRs may not help you much.