False Positive Paradox Calculator
Calculate positive predictive value and understand the false positive paradox using Bayes' theorem
Calculate False Positive Paradox
Proportion of positive cases correctly identified (true positive rate)
Proportion of negative cases correctly identified (true negative rate)
Proportion of the population with the condition
Total number of people being tested (for demonstration)
Predictive Values
Formula used: PPV = (Sensitivity × Base Rate) / [Sensitivity × Base Rate + (1 - Specificity) × (1 - Base Rate)]
This is Bayes' theorem applied to medical testing
HIV Test Example
Medical Test Scenario
Test: HIV screening test
Sensitivity: 99% (detects 99% of HIV cases)
Specificity: 99% (correctly identifies 99% of non-HIV cases)
Base Rate: 0.1% (1 in 1000 people have HIV)
Surprising Result
PPV = (0.99 × 0.001) / [(0.99 × 0.001) + (0.01 × 0.999)]
PPV = 0.00099 / (0.00099 + 0.00999)
PPV ≈ 9%
Only 9% of positive tests are true positives!
Key Concepts
Sensitivity
True Positive Rate - correctly identifying positive cases
Specificity
True Negative Rate - correctly identifying negative cases
Base Rate
Prevalence - proportion of population with the condition
PPV
Positive Predictive Value - probability condition exists given positive test
Overcoming the Paradox
Increase test specificity to reduce false positives
Test high-risk populations (higher base rate)
Use confirmatory testing for positive results
Consider clinical context and symptoms
Understanding the False Positive Paradox
What is the False Positive Paradox?
The false positive paradox occurs when a high proportion of positive test results are actually false positives, even when the test has high sensitivity and specificity. This counterintuitive phenomenon happens when the condition being tested for has a low base rate (prevalence) in the population.
Why Does This Happen?
- •Low base rate means few people actually have the condition
- •Even small false positive rates affect many healthy people
- •True positives are outnumbered by false positives
- •People ignore base rates and focus on test accuracy
Bayes' Theorem Formula
PPV = (SE × BR) / [SE × BR + (1-SP) × (1-BR)]
- PPV: Positive Predictive Value
- SE: Sensitivity (true positive rate)
- SP: Specificity (true negative rate)
- BR: Base Rate (prevalence)
Key Insight: Base rate has enormous impact on PPV. Low base rates dramatically reduce the probability that a positive test is correct, regardless of test accuracy.
Common Misconceptions
- • "99% accurate = 99% chance positive is correct"
- • "Larger sample sizes fix the problem"
- • "Higher sensitivity eliminates false positives"
Real-World Examples
- • COVID-19 testing in low-prevalence areas
- • Cancer screening programs
- • Drug testing in general population
Practical Solutions
- • Target high-risk populations
- • Use confirmatory testing
- • Improve test specificity