Epidemiological Measures

Sensitivity, Specificity, and Predictive Values
Eric Delmelle
March2026

Chapter Overview

• Diagnostic Test Accuracy: What makes a test reliable?
• Sensitivity: Detecting true cases — avoiding false negatives
• Specificity: Excluding non-cases — avoiding false positives
• Predictive Values: PPV and NPV — interpreting results in context
• The Trade-off: Balancing sensitivity and specificity in practice
• Real-World Applications: COVID-19, cancer screening, and more

1 Why Diagnostic Accuracy Matters

• Accurately diagnosing and classifying diseases is critical for effective treatment and disease tracking.
• Errors in classification — whether due to poor coding, faulty tests, or reporting mistakes — can significantly impact health outcomes and policy decisions.
• Every diagnostic test has imperfections; understanding them helps us use tests wisely.

The 2×2 Contingency Table

• All diagnostic test outcomes can be organized into a 2×2 table comparing test results against true disease status.

	Disease Present	Disease Absent
Test Positive	True Positive (TP)	False Positive (FP)
Test Negative	False Negative (FN)	True Negative (TN)

• True Positive (TP): Sick person correctly identified as sick
• True Negative (TN): Healthy person correctly identified as healthy
• False Positive (FP): Healthy person incorrectly flagged as sick
• False Negative (FN): Sick person incorrectly cleared as healthy

2 Sensitivity and Specificity

• Sensitivity and Specificity are the two fundamental measures of a diagnostic test’s performance.
• They are intrinsic properties of a test — they do not depend on disease prevalence in the population.
• Choosing the right balance between them depends on the consequences of false negatives vs. false positives.

Sensitivity: Avoiding False Negatives

• Sensitivity measures how well a test identifies people who actually have a disease.
• A highly sensitive test minimizes false negatives, meaning fewer sick people go undiagnosed.
• If a test has low sensitivity, it will fail to detect many cases, allowing diseases to spread unnoticed.

\[\text{Sensitivity} = \frac{TP}{TP + FN} \times 100\%\]

“Of all people who ARE sick, what proportion does the test correctly identify?”

Example: COVID-19 Testing Sensitivity Issues

• Early PCR tests for COVID-19 had sensitivity rates of 70–80%.
• Up to 30% of infected individuals received false-negative results.
• False negatives led to infected individuals unknowingly spreading the virus, worsening the pandemic.

🔑 When the cost of missing a case is high (e.g., infectious disease, cancer), prioritize high sensitivity.

Specificity: Avoiding False Positives

• Specificity measures how well a test excludes people who do not have the disease.
• A highly specific test minimizes false positives, meaning fewer healthy people are mistakenly diagnosed as sick.
• If a test has low specificity, people might receive unnecessary treatments for diseases they don’t actually have.

\[\text{Specificity} = \frac{TN}{TN + FP} \times 100\%\]

“Of all people who are NOT sick, what proportion does the test correctly clear?”

Example: False Positives in Cancer Diagnosis

• Low specificity cancer screenings sometimes identify benign tumors as malignant.
• This leads to unnecessary biopsies and emotional distress.
• False positives in COVID-19 antibody tests led some people to believe they were immune when they had never actually been infected.

🔑 When the cost of a false alarm is high (e.g., invasive treatment, stigma), prioritize high specificity.

Why Sensitivity and Specificity Matter

• False negatives can delay treatment and increase transmission of infectious diseases.
• False positives can lead to unnecessary medical interventions and public panic.
• Balancing both sensitivity and specificity is crucial in public health decisions.

Situation	Priority
Screening for infectious disease	High Sensitivity
Confirmatory testing before treatment	High Specificity
Rare disease with costly treatment	High Specificity
Cancer early detection programs	High Sensitivity

3 The Sensitivity–Specificity Trade-off

• There is an inherent trade-off: increasing sensitivity typically decreases specificity and vice versa.
• The cut-off threshold for a positive test determines where on this trade-off curve a test sits.
• ROC curves (Receiver Operating Characteristic) visualize this trade-off across all possible thresholds.

Shifting the Threshold

• Lowering the threshold → More positives → ↑ Sensitivity, ↓ Specificity
  • Good for initial screening where missing cases is costly
• Raising the threshold → Fewer positives → ↓ Sensitivity, ↑ Specificity
  • Good for confirmatory tests where false alarms are costly

📌 Mnemonic — SnNout / SpPin: A highly Sensitive test, when Negative, rules OUT disease. A highly Specific test, when Positive, rules IN disease.

4 Predictive Values

• Sensitivity and specificity describe test performance; predictive values tell us what a result means for the patient.
• Unlike sensitivity/specificity, predictive values depend on disease prevalence.
• Critical for interpreting test results in clinical and public health contexts.

Positive Predictive Value (PPV)

• PPV = the probability that a person with a positive test actually has the disease.
• PPV is higher when prevalence is high — positive tests are more likely to be real.

\[\text{PPV} = \frac{TP}{TP + FP} \times 100\%\]

“Of all people who test positive, what proportion are truly sick?”

Negative Predictive Value (NPV)

• NPV = the probability that a person with a negative test actually does not have the disease.
• NPV is higher when prevalence is low — negative tests are more likely to be true negatives.

\[\text{NPV} = \frac{TN}{TN + FN} \times 100\%\]

“Of all people who test negative, what proportion are truly healthy?”

How Prevalence Affects Predictive Values

• Same test, different populations → different PPV/NPV!
• A test with 95% sensitivity and 95% specificity:

Prevalence	PPV	NPV
1% (low-risk population)	~16%	~99.9%
10% (moderate risk)	~68%	~99.4%
50% (high-risk population)	~95%	~95%

• Testing low-prevalence populations with even a specific test yields many false positives.

Example: COVID-19 Antibody Testing

• In early 2020, antibody tests had ~96% specificity — seemingly excellent.
• In a population with 1% true prevalence, PPV was only ~20%.
• 4 out of 5 positive results were false positives — people incorrectly believing they had immunity.
• This example illustrates why prevalence context is essential when interpreting any test.

5 Worked Example

• A new test for tuberculosis (TB) is evaluated on 1,000 individuals.
• True disease prevalence: 10% (100 people have TB, 900 do not).
• Test sensitivity: 90% | Test specificity: 85%

Step 1: Build the 2×2 Table

	TB Present (n=100)	TB Absent (n=900)	Total
Test Positive	90 (TP)	135 (FP)	225
Test Negative	10 (FN)	765 (TN)	775
Total	100	900	1,000

• TP = 100 × 0.90 = 90
• FN = 100 × 0.10 = 10
• TN = 900 × 0.85 = 765
• FP = 900 × 0.15 = 135

Step 2: Calculate All Four Measures

Sensitivity = 90 / (90 + 10) = 90% ✓

Specificity = 765 / (765 + 135) = 85% ✓

PPV = 90 / (90 + 135) = 40% ← Only 40% of positives are real!

NPV = 765 / (765 + 10) = 98.7% ← Negative test is very reassuring

Interpreting the Results

• Despite good sensitivity and specificity, PPV is only 40% at 10% prevalence.
• A positive result should trigger a confirmatory test before treatment.
• A negative result is very reliable — NPV of 98.7%.
• If prevalence were 50%, PPV would rise to ~75%.

🎯 Key takeaway: Test performance must always be interpreted in the context of who is being tested.

6 Summary

Measure	Formula	Depends on Prevalence?	Key Question
Sensitivity	TP / (TP+FN)	No	Did we catch all sick people?
Specificity	TN / (TN+FP)	No	Did we clear all healthy people?
PPV	TP / (TP+FP)	Yes	Is this positive result real?
NPV	TN / (TN+FN)	Yes	Is this negative result trustworthy?

Key Takeaways

• Sensitivity and Specificity are fixed properties of a test; use them to choose the right test.
• PPV and NPV depend on prevalence; use them to interpret results for patients.
• No test is perfect — understanding the trade-offs is essential for good clinical and public health decisions.
• Always ask: “What population is being tested, and what are the consequences of errors?”