How to Calculate A Positve and Negative Likelihood Ratio

A likelihood ratio is a statistical measure used in medical testing to determine how much more (or less) likely a test result is for a person with a particular condition compared to someone without it. This guide explains how to calculate both positive and negative likelihood ratios, their interpretation, and practical applications.

What is a Likelihood Ratio?

The likelihood ratio (LR) is a key concept in diagnostic testing that helps clinicians assess the accuracy of a test result. It compares the probability of a test result occurring in people with a disease to the probability of it occurring in people without the disease.

Likelihood ratios are particularly useful because they:

Provide a standardized way to compare different tests
Help determine whether a positive or negative test result is more meaningful
Can be used to calculate post-test probabilities

Likelihood ratios are different from sensitivity and specificity, which measure the test's ability to correctly identify cases and non-cases, respectively.

Positive vs. Negative Definitions

There are two types of likelihood ratios:

Positive Likelihood Ratio (LR+): Measures how much more likely a positive test result is in people with the disease compared to those without it.
Negative Likelihood Ratio (LR-): Measures how much less likely a negative test result is in people with the disease compared to those without it.

These ratios help determine whether a test result provides useful information for diagnosing a condition.

How to Calculate

The formulas for calculating likelihood ratios are:

Positive Likelihood Ratio (LR+) = Sensitivity / (1 - Specificity)

Negative Likelihood Ratio (LR-) = (1 - Sensitivity) / Specificity

Where:

Sensitivity = True Positive Rate = (True Positives) / (True Positives + False Negatives)
Specificity = True Negative Rate = (True Negatives) / (True Negatives + False Positives)

These formulas show how the likelihood ratios are derived from the test's sensitivity and specificity.

Interpreting Results

The interpretation of likelihood ratios depends on their value:

LR+ > 1: A positive test result increases the likelihood of the disease
LR+ = 1: The test result doesn't change the likelihood of the disease
LR+ < 1: A positive test result decreases the likelihood of the disease
LR- > 1: A negative test result decreases the likelihood of the disease
LR- = 1: The test result doesn't change the likelihood of the disease
LR- < 1: A negative test result increases the likelihood of the disease

Likelihood ratios are often combined with pre-test probabilities to calculate post-test probabilities using Bayes' theorem.

Worked Example

Consider a test for a rare disease with the following characteristics:

Sensitivity (true positive rate): 90% (0.9)
Specificity (true negative rate): 95% (0.95)

Calculating the likelihood ratios:

LR+ = 0.9 / (1 - 0.95) = 0.9 / 0.05 = 18

LR- = (1 - 0.9) / 0.95 = 0.1 / 0.95 ≈ 0.105

Interpretation:

A positive test result makes the disease 18 times more likely
A negative test result makes the disease about 10 times less likely

This example shows how likelihood ratios can help assess the usefulness of a diagnostic test.

Frequently Asked Questions

What is the difference between likelihood ratio and odds ratio?: The likelihood ratio compares probabilities, while the odds ratio compares odds. Likelihood ratios are generally preferred in diagnostic testing because they're more intuitive for interpreting test results.
How do I use likelihood ratios in clinical practice?: Likelihood ratios help clinicians update their assessment of a patient's condition based on test results. They're often combined with pre-test probabilities to calculate post-test probabilities.
What are the limitations of likelihood ratios?: Likelihood ratios assume that test results are independent and don't account for other factors that might affect the diagnosis. They're most useful when used in combination with other clinical information.
Can likelihood ratios be used for screening tests?: Yes, likelihood ratios are particularly valuable for evaluating screening tests where the disease is relatively common and the test has a high false positive rate.
How do I calculate post-test probabilities using likelihood ratios?: You can use Bayes' theorem to combine the likelihood ratio with the pre-test probability to calculate the post-test probability of the disease.