Likelihood Ratios

A Likelihood Ratio (LR) incorporates. both a diagnostic test’s sensitivity and specificity into one single value by relating the two measurements in the form of a ratio.

Recall that the Sensitivity of a test is the probability (measured as a percentage) that a test will correctly identify those individuals who truly have (true positives) the health condition for which the test is designed to detect. The Specificity of a test is the probability (measured as a percentage) that a test will correctly identify those individuals who truly don’t have (true negatives) the health condition for which the test is designed to detect. For a more detailed exploration of Sensitivity and Specificity, click here.

There are two different types of LR’s that can be calculated for any given test: Positive Likelihood Ratios (LR+) and Negative Likelihood Ratios (LR-).

• Ratio is always ≥ 1
• LR+ is only relevant with a positive test result
• Calculated by dividing the probability of a positive test result in persons with the health condition (i.e. sensitivity) by the probability of a positive test finding in persons without the condition
  • LR+ = Sensitivity / (1-Specificity)
• The larger the numerical value of LR+, the more confidently one can rule in a condition

Rule of Thumb: a test with an LR+ ≥ 10 is a good test to rule in a suspected health condition when implementation of the test yields a positive test result

• Ratio is always ≤ 1
• LR- is only relevant with a negative test result
• Calculated by dividing the probability of a negative test result in persons with the health condition by the probability of a negative test result in persons without the condition (i.e. specificity)
  • LR- = (1- Sensitivity) / Specificity
• The smaller the numerical value of LR- the more confidently one can rule out a condition

Rule of Thumb: a test with an LR- ≤ 0.10 is a good test to rule out a suspected health condition when implementation of the test yields a negative test result

It is rare that a single test will be good at both ruling in and out the same condition; therefore, when choosing which diagnostic tests to use in a clinical scenario, it is important to understand how diagnostic a test result actually is. To do this, one must use LR’s in conjunction with pre and post-test probabilities.

Pre-test probability is a scientifically derived measurement (as a percentage) of the likelihood that an individual, given their specific medical history and current symptoms, has a suspected health condition. Relating pre-test probability to a diagnostic test’s LR’s helps put that diagnostic test into context.

Example: the prevalence of appendicitis in the general population is about 1%; however, individuals reporting to the ED with right lower quadrant (RLQ) abdominal pain have a 26-30% probability of having appendicitis. This means that the same test designed to detect appendicitis must be weighted differently depending on the setting in which it is preformed (random person off the street without any abdominal symptoms vs. ED patient with RLQ abdominal pain). Said another way, one’s level of clinical suspicion of appendicitis before an appendicitis test is preformed affects how much diagnostic value the clinician should assign to the test result they get when they preform their appendicitis test.

As implied by its name, post-test probability is the probability of the individual having a given condition after taking into account the results of a given test. You must use pre-test probability and a test’s LR’s to determine post-test probability. Though this can be done mathematically, an easier way for most people to approach this is visually, through the use of a nomogram.

Let us continue with our RLQ abdominal pain patient in a hospital setting. We first take the prevalence of appendicitis for this patient demographic (roughly 30%) then we use our diagnostic test’s LR+ (8) for RLQ pain to determine the post-test probability that a person has appendicitis if the test is positive.

After taking into account our positive test result, we can see that post-test probability (~80%) has significantly increased in relation to the pre-test probability(~30%), making us much more confident in our suspected diagnosis of appendicitis. The same method could be used to calculate Post-test Probability if the test result were negative; however, LR- would be used in place of LR+. This would lead to a lower post-test-probability than pre-test probability and, if the post-test probability were low enough, might help us rule out the suspected condition.