4.4: Likelihood ratios

As we discussed earlier, 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”).  Here is another threshold diagram that illustrates the diagnostic process for a patient with sore throat:

Threshold model strep

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. On the other hand, the “negative likelihood ratio” (LR-) tells us how much to decrease it if the test is negative. The general formula for calculating the likelihood ratio is:

    probability that an individual with disease has the test result   
      LR  =       probability that an individual without disease has the test result

Thus, the positive and likelihood ratio are:

                         probability that an individual with disease has a positive test
     Positive LR = LR+  =     probability that an individual without disease has a positive test

                                       probability that an individual with disease has a negative test
     Negative LR = LR-  =      probability that an individual without disease has a negative test

You can also define the LR+ and LR- in terms of sensitivity and specificity. Notice that the numerator for LR+ is the definition for sensitivity (probably that an individual with disease has a positive test), and the denominator is the converse of specificity. For LR-, the numerator is the converse of sensitivity and the denominator is specificity. So:

LR+ = sensitivity / (1-specificity)

 

LR- = (1-sensitivity) / specificity 

Let’s consider an example. In a study of the ability of rapid antigen tests ("strep screens") to diagnose strep pharyngitis, 80% of patients with strep pharyngitis have a positive rapid antigen test, while 95% of those without strep pharyngitis have a negative test. Thus, the sensitivity is 80% and the specificity is 95%. The LR+ for the ability of rapid antigen tests to diagnose strep pharyngitis is:

            LR+ = 80% / (100%-95%) = 80% / 5% = 18

The negative likelihood ratio is:

            LR- = (100%-80%)/95% = 20/95 = 0.21

Likelihood ratios have unique properties that make them particularly useful for clinicians and healthcare decision-makers, which we'll discuss shortly. Perhaps most important is that:

What do we mean by ruling-in and ruling-out disease? Remember our discussion of the threshold model of diagnosis? Well, a disease is "ruled out" when the probability is below the test threshold, and is "ruled in" when the probability of disease is above the treatment threshold. Here are two other important properties that make likelihood ratios very useful: So, LR+ and LR- are stable properties of a test, like sensitivity and specificity, but are easier to interpret and apply, like predictive values. By "multiple cutpoints" we mean that instead of just being positive or negative, a test can provide results like "Very low risk", "Low risk", "Moderate risk" and "High risk". Each result has its own likelihood ratio.


Interpreting likelihood ratios: general guidelines

The first thing to realize about LR’s is that a LR greater than 1 increases the probability that the target disorder is present, and a LR less than 1 decreases the probability that the target disorder is present.  The following are general guidelines, which must be correlated with the clinical scenario:

LR

Interpretation

> 10

Large and often conclusive increase in the likelihood of disease

5 - 10

Moderate increase in the likelihood of disease

2 - 5

Small increase in the likelihood of disease

1 - 2

Minimal increase in the likelihood of disease

1

No change in the likelihood of disease

0.5 - 1.0

Minimal decrease in the likelihood of disease

0.2 - 0.5

Small decrease in the likelihood of disease

0.1 - 0.2

Moderate decrease in the likelihood of disease

< 0.1

Large and often conclusive decrease in the likelihood of disease

Here is a collection of likelihood ratios for the diagnosis of appendicitis, from the Essential Evidence database:


Sensitivity Specificity LR+ LR-
Adults



CT scan 94 95 18.8 0.06
Ultrasound 86 81 4.5 0.17
C-reactive protein > 1.0 mg/dl 64 72 2.3 0.5
Children



Ultrasound 86 95 17.2 0.15
CT scan (non-contrast) 97 93 13.9 0.03
Ultrasound followed by CT if indeterminate (non-contrast) 99 89 9.0 0.01
C-reactive protein > 1.0 mg/dl 64 82 3.6 0.44

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 CT scan might have a good likelihood ratio for ruling in (or out) appendicitis in a child with abdominal pain. But if you believe a patient has a simple gastroenteritis, and that appendicitis is very unlikely, CT shouldn’t be ordered given the cost, radiation exposure, and fact that a positive scan is likely a false positive. So, beginning with ultrasound and only getting CT if the results remain unclear or the patient doesn't improve might be the best option. Clearly, there is more to the evaluation of diagnostic tests than a simple assessment of accuracy. More on that later.

Not all tests are dichotmous (yes/no, positive/negative). In the next section we will learn about tests that are "polytomous".

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