Likelihood ratios

Last revised by Stefan Tigges on 6 Dec 2023

Likelihood ratios (LR) are an alternative to positive and negative predictive values for estimating the likelihood of disease after diagnostic testing. The general formula for a likelihood ratio is the probability (P) that someone with a disease will have a particular test result divided by the probability that someone without a disease will have that same result:

  • LR = P(test result with disease) / P(test result without disease)

There are 2 types of likelihood ratios, positive and negative:

A positive likelihood ratio (LR+) is the probability that someone with a disease will have a positive test result divided by the probability that someone without a disease will have a positive result.

A negative likelihood ratio (LR-) is the probability that someone with a disease will have a negative test result divided by the probability that someone without a disease will have a negative result. The formula for a positive likelihood ratio is:

  • LR+ = P(positive test result with disease) / P(positive test result without disease) or

  • LR+ = P(test+|disease+) / P(test+|disease-) or

  • LR+ = true positive rate / false positive rate or

  • LR+ = sensitivity/ (1-specificity)

The formula for a negative likelihood ratio is:

  • LR- = P(negative test result with disease) / P(negative test result without disease) or

  • LR- = P(test-|disease+) / P(test-|disease-) or

  • LR- = false negative rate / true negative rate

  • LR- = (1-sensitivity) / specificity

The higher the positive likelihood ratio (LR+), the more likely that someone testing positive has the disease in question. The lower the negative likelihood ratio (LR-), the more likely that someone testing negative does not have the disease in question. An LR+ or an LR- of one means that people with and without disease are equally likely to test positive or negative respectively. Likelihood ratios of one mean that a test cannot discriminate between those who do and those who do not have the disease. As a general rule, an LR+ greater than 10 and an LR- less than 0.1 show that a test reliably discriminates between people who do and do not have disease. If we know the pre-test probability of disease, we can use LR+ to calculate the post-test probability of disease among those who test positive using the following formula: Post-test odds = pre-test odds x LR+.

Example

If a cancer screening test has 90% sensitivity and 80% specificity, then LR+ = sensitivity / (1-specificity) = 0.9 / (1-0.8) = 0.9 / 0.2 = 4.5: this means that someone with the disease is 4.5 times as likely to test positive as someone without the disease. The LR- =( 1-sensitivity) / specificity = (1-0.9) / 0.8 = 0.1 / 0.8 = 1.125: this means that someone with the disease is 0.125 times as likely to test negative as someone without the disease.

If we know the prevalence of cancer in our population (pre-test disease probability), we can calculate the post-test probability of disease: the process is not difficult but is awkward because of the need to convert probabilities to odds and odds into probabilities. For our example, we’ll assume a prevalence of 10%. First, we calculate the pre-test odds of disease:

  • pre-test odds = pre-test probability / (1-pre-test probability) = 0.1 / (1-0.9) = 0.1 / 0.9 = 0.111

Next, we plug our values for pre-test odds and LR+ into our formula:

  • post-test odds = pre-test odds x LR+ = 0.111 x 4.5 = 0.5

Last, we convert the post-test odds into a probability:

  • post-test probability = post-test odds / (1+ post-test odds) = 0.5 / (1+0.5) = 0.5 / 1.5 = 0.33

This means that the probability of having the disease has increased from 10% to 33% for a person testing positive.

Nomograms are also available to derive post-test probability, obviating the need for calculations.

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