What is the positive likelihood ratio (LR+) of a test with 90% sensitivity and 75% specificity?

Calculating the Positive Likelihood Ratio (LR+)

The positive likelihood ratio (LR+) for a test with 90% sensitivity and 75% specificity is 3.6.

Understanding Likelihood Ratios

Likelihood ratios are valuable diagnostic metrics that help clinicians interpret test results and determine how a particular test result changes the probability of disease.

Formula for Positive Likelihood Ratio

The positive likelihood ratio (LR+) is calculated using the following formula:

LR+ = Sensitivity / (1 - Specificity)

Where:

• Sensitivity = True positive rate
• Specificity = True negative rate
• (1 - Specificity) = False positive rate

Calculation

Given:

• Sensitivity = 90% = 0.90
• Specificity = 75% = 0.75

Therefore:

• LR+ = 0.90 / (1 - 0.75)
• LR+ = 0.90 / 0.25
• LR+ = 3.6

Clinical Interpretation of LR+ = 3.6

An LR+ of 3.6 means that a positive test result is 3.6 times more likely to occur in someone with the disease than in someone without the disease. This represents a moderate shift in pre-test to post-test probability.

Interpretation Guidelines for LR+:

• LR+ > 10: Large, often conclusive increase in likelihood of disease
• LR+ 5-10: Moderate increase in likelihood of disease
• LR+ 2-5: Small increase in likelihood of disease (our result falls here)
• LR+ 1-2: Minimal increase in likelihood of disease
• LR+ = 1: No change in likelihood of disease

Application in Clinical Practice

To apply this LR+ in practice:

1. Determine the pre-test probability of disease based on clinical presentation
2. Convert pre-test probability to pre-test odds: Odds = Probability/(1-Probability)
3. Multiply pre-test odds by the LR+ to get post-test odds: Post-test odds = Pre-test odds × LR+
4. Convert post-test odds back to probability: Probability = Odds/(1+Odds)

Example:

If a patient has a 20% pre-test probability of disease:

• Pre-test odds = 0.20/(1-0.20) = 0.25
• Post-test odds = 0.25 × 3.6 = 0.9
• Post-test probability = 0.9/(1+0.9) = 0.47 or 47%

This demonstrates how a positive test result with LR+ of 3.6 increases the probability of disease from 20% to 47%.

Common Pitfalls to Avoid

1. Confusing likelihood ratios with predictive values (PPV and NPV)
2. Failing to consider the pre-test probability when interpreting test results
3. Misapplying likelihood ratios to tests with multiple result categories (requires multiple LRs)
4. Using likelihood ratios without considering the clinical context

Understanding likelihood ratios helps clinicians make more informed diagnostic decisions by quantifying how test results modify disease probability.