# Likelihood ratios; Pre and post test probability

## Diagnosis and Screening: Likelihood ratios; pre and post test probability

What is a likelihood ratio?

The likelihood ratio provides a direct estimate of how much a test result will change the odds of having a disease, and incorporates both the sensitivity and specificity of the test.

The likelihood ratio for a positive result (LR+) tells you how much the odds of the disease increase when a test is positive.

The likelihood ratio of a positive test result (LR+) is sensitivity divided by (1- specificity)

 (LR+) = True positives Sensitivity False positives (1-Specificity)

The likelihood ratio for a negative result (LR-) tells you how much the odds of the disease decrease when a test is negative.

The likelihood ratio of a negative test result (LR-) is (1- sensitivity) divided by specificity.

 (LR-) = False negatives = (1- Sensitivity) True negatives (Specificity)

*Note that the calculation of likelihood ratios is currently not tested as a skill in the UK MFPH Part A examination.

Interpretation of Likelihood Ratios

The further away a likelihood ratio (LR) is from 1, the stronger the evidence for the presence or absence of disease.

• LR >1 indicates that the test result is associated with the presence of the disease.

• LR <0.1 indicates that the test result is associated with the absence of disease.

Pre-test probability (~ prevalence)

This is the proportion of people in the population at risk who have the disease at a specific time or time interval, i.e. the point prevalence or the period prevalence of the disease. In other words, it is the probability −before the diagnostic test is performed − that a patient has the disease. Pre-test probabilities may be estimated from routine data, practice data or clinical judgement.

Post-test probability

This is the proportion of patients testing positive who truly have the disease. It is similar to the positive predictive value but apart from the test performance also includes a patient based probability of having disease

 Post-test probability = Post-test odds (1 + Post-test odds)

Using Pre- and Post-test probability and LR

By comparing the pre- and post-test probabilities, it is possible to determine whether probability of diagnosis has risen (i.e. the post-test probability has increased) or fallen (i.e. post-test probability has decreased). In this way, it is possible to provide comprehensive information about a screening test in order to enable informed choice. In practice assessing post-test probability is commonly done by using a Likelihood Ratio Nomogram (below)

Summary of calculations of key features of screening tests

 Feature Formula Sensitivity a/ (a+c) Specificity d/ (b+d) Positive predictive value a/ (a+b) Negative predictive value d/ (c+d) Accuracy (a+d)/ (a+b+c+d) Likelihood ratio: Positive test Sensitivity/ (1-specificity) Negative test (1-sensitivity)/specificity