Likelihood ratios

When we decide to order a diagnostic test, we want to know which test (or tests) will best help us rule-in or rule-out disease in our patient.  In the language of clinical epidemiology, we take our initial assessment of the likelihood of disease (“pre-test probability”), do a test to help us shift our suspicion one way or the other, and then determine a final assessment of the likelihood of disease (“post-test probability”).  Take a look at the diagram below, which graphically illustrates this process of “revising the probability of disease”.

Likelihood ratios tell us how much we should shift our suspicion for a particular test result. Because tests can be positive or negative, there are at least two likelihood ratios for each test.  The “positive likelihood ratio” (LR+) tells us how much to increase the probability of disease if the test is positive, while the “negative likelihood ratio” (LR-) tells us how much to decrease it if the test is negative.  The formula for calculating the likelihood ratio is:

    probability of an individual with the condition having the test result   
      LR  =       probability of an individual without the condition having the test result

Thus, the positive and likelihood ratio are:

   probability of an individual with the condition having a positive test
     LR+  =     probability of an individual without the condition having a positive test

               probability of an individual with the condition having a negative test
     LR-  =      probability of an individual without the condition having a negative test

You can also define the LR+ and LR- in terms of sensitivity and specificity:

LR+ = sensitivity / (1-specificity)

LR- = (1-sensitivity) / specificity

Let’s consider an example:

In a study of the ability of rapid antigen tests to diagnose strep pharyngitis, 90% of patients with strep pharyngitis have a positive rapid antigen test, while only 5% of those without strep pharyngitis have a positive test.  The LR+ for the ability of rapid antigen tests to diagnose strep pharyngitis is:

            LR+ = 90% / (100%-95%) = 90% / 5% = 18

Don't get too caught up in the calculations. the important thing is to understand the meaning of a likelihood ratio. They have unique properties that make them particularly relevant to clinicians:

• The LR+ corresponds to the clinical concept of "ruling-in disease"

• The LR- corresponds to the clinical concept of "ruling-out disease"

• The LR+ and LR- don't change as the underlying probability of disease changes (predictive values do change, as you just learned)

• LR's using multiple "levels" of positive (i.e. not just a simple yes/no or positive/negative result) provide much richer, more useful information to you as a clinician.

Interpreting likelihood ratios: general guidelines

The first thing to realize about LR’s is that an LR > 1 indicates an increased probability that the target disorder is present, and an LR < 1 indicates a decreased probability that the target disorder is present.  The following are general guidelines, which must be correlated with the clinical scenario:

Take a look at a collection of likelihood ratios for common tests, provided by the folks at the Cochrane Centre in Oxford, England.

The decision to order a test is also based on our initial assessment of the likelihood of the target disorder, and how important it is to rule-in or rule-out disease. For example, a chest x-ray might have a good likelihood ratio for pneumonia. But if you believe a patient has a simple cold, this test, no matter how good the LR, probably shouldn’t be ordered.  It is sometimes helpful to be able to calculate the exact probability of disease given a positive or negative test.  We saw that this is next to impossible using sensitivity and specificity at the bedside (unless you can do Bayes’ Theorem in your head!).  Next, we’ll learn how we can use likelihood ratios to quickly estimate the probability of disease in our patients.