Likelihood Ratios part 2:  Bedside calculation

The terms "odds of disease" and "probability of disease" get thrown around a lot like they’re the same thing, but they’re not. Let’s consider a group of 10 patients, 3 of whom have strep and 7 of whom don’t.  If we randomly choose a patient, the probability that they will have strep is 3/10 or 0.3 or 30%. On the other hand, the odds of having strep in this group are 3 : 7. Here is a table which relates the odds to the probability:

Stated as a mathematical formula (yuck!) this relationship is:

• for an odds of a : b, probability = a / (a + b)
• for a probability of x%, the odds are x : (100-x)

Thus, if the odds are 4:9, the probability is 4 / (4+9) = 4/13 = 0.31 (or 31%). Similarly, if the probability is 15%, then the odds are 15 : (100-15) = 15 : 85.   With a little practice, you can easily convert from probability to odds and back again in your head.

Why should you possibly care about doing this? Well, the likelihood ratio has a very interesting property:

post-test odds of disease = likelihood ratio x pre-test odds of disease

So, for positive and negative tests:

odds of disease for (+) test = odds of disease before testing  x  LR+

odds of disease for (-) test = odds of disease before testing  x  LR-

Now, with a little practice, we can actually estimate the probability of disease given a positive or negative test in our heads! Let’s go through a couple of examples:

You estimate, based on your knowledge of the community, the patient’s age of 10 years, and his symptoms (sore throat, fever, exudate, and adenopathy) that the pre-test probability of strep is approximately 40%. The rapid antigen test for strep is positive; looking at the package insert, you see that it has a sensitivity of 90% and specificity of 90%. The LR+ and LR- are therefore 9 and 0.1. Before proceeding, make sure you understand how we calculated those LR’s using the formulas described above.

First, notice that knowing the sensitivity and specificity doesn’t help you much when it comes to calculating the likelihood of disease in your patient. However, in 3 simple steps, we’ll use the LR’s to do just that:

* It simplifies calculations somewhat to reduce elements to the least common denominator. Thus, 40:60 is the same as 4:6, and is also the same as 2:3. Similarly, 30 : 70 is the same as 3:7.

What if the test is negative? Let’s go through that, using the LR- of 0.1 this time in our calculations:

Now, instead of just knowing that a positive strep test makes disease more likely, and a negative one makes it less likely (or worse yet, thinking that a positive test means the patient has disease and a negative test means they don’t) you can estimate the specific likelihood of disease for your patient. This is truly "patient-centered" medicine, since your interpretation of the laboratory test is specific to your patient’s pre-test probability of disease, which is in turn based on his or her age, symptoms, and signs.

In the above example, a positive test provides pretty convincing evidence of strep (86% probability). On the other hand, many physicians would be uncomfortable not treating a child who had a negative strep test, and therefore still had a 6% chance of having strep. After going through this calculation once, you might decide that in similar patients, you will empirically treat them, since a negative test does not rule out disease. Or, you might decide to get a throat culture inpatients with a negative strep screen, while giving antibiotics to those with a positive strep screen.

Let’s consider another example: an older patient with much less typical symptoms of strep (age 20, sore throat, cough, no adenopathy, and no exudate) and a pre-test probability of disease of 5% by your estimate. If the test is positive (remember, LR+ = 9):

If the test is negative:

In this case, a negative test does rule out disease, and a positive test gives a high enough likelihood of disease that you would probably treat the patient, but remain open to other causes for his or her symptoms. Individualizing treatment in this way is much more powerful than simply doing the same thing for every patient.

Getting the most information from a test

When we order a test result, we’re accustomed to thinking in terms of the results being positive or negative. However, the actual information in the result is often much richer. Consider the diagnosis of iron deficiency anemia (IDA) from the serum ferritin level. Labs generally report a single cutoff for abnormal around 65 mmol/l, with low values suggesting a diagnosis of iron deficiency anemia. Using that value as a "positive" test, the LR+ is 6 and the LR- is 0.12.

But there is more information hidden in these results. You can also calculate a likelihood ratio for each range of ferritin, as shown below:

Doing these calculations is easy. Set up your table as above, with a column showing the percentage of patients with the disease that have a test value in that range, and a second column showing the percentage of patients without disease that have a test value in that range. Then, divide the first column by the second column to calculate the LR for that range. In the table above, for example, 59% / 1.1% = 52.

Once again, likelihood ratios help us provide individualized care, and get the most possible information from a test result.  This is an important advantage of using likelihood ratios!