18.2 Diagnostic Test Metrics & Screening
Key Takeaways
Sensitivity = TP/(TP+FN) (rule-out / SnNOut when high); specificity = TN/(TN+FP) (rule-in / SpPIn when high); neither depends on prevalence in the 2×2 used to compute them if disease status is known.
PPV = TP/(TP+FP) and NPV = TN/(TN+FN) move with prevalence: at low prevalence, even specific tests yield many false positives and low PPV; high prevalence raises PPV and lowers NPV.
Likelihood ratios: LR+ = sens/(1−spec); LR− = (1−sens)/spec; they reweight pre-test to post-test odds (Bayes) without being prevalence themselves.
ROC curves plot sensitivity vs 1−specificity across cutoffs; AUC summarizes discrimination; lowering the cut-point ↑sensitivity ↓specificity (and vice versa).
Parallel testing increases net sensitivity (fewer missed cases); serial/confirmatory testing increases net specificity (fewer false positives); screening criteria require important disease, acceptable test, effective earlier treatment, and acceptable false-positive burden.
18.2 Diagnostic Test Metrics & Screening
Quick Answer: Sn = TP/(TP+FN); Sp = TN/(TN+FP); PPV = TP/(TP+FP); NPV = TN/(TN+FN). Prevalence changes PPV/NPV, not intrinsic Sn/Sp. LR+ = Sn/(1−Sp); LR− = (1−Sn)/Sp. Parallel tests ↑net Sn; serial tests ↑net Sp. Screening needs important disease, good test, effective early treatment, and tolerable false-positive load.
Diagnostic reasoning on the CBSE is arithmetic plus epidemiology. Fill the 2×2 carefully, label each cell, compute the four core metrics, then ask how prevalence and cutoffs move the answers.
The 2×2 Table
Gold standard disease status defines columns; test result defines rows:
| Disease + | Disease − | Total | |
|---|---|---|---|
| Test + | TP | FP | TP+FP |
| Test − | FN | TN | FN+TN |
| Total | TP+FN | FP+TN | N |
- True positive (TP): disease present, test positive
- False positive (FP): disease absent, test positive
- False negative (FN): disease present, test negative
- True negative (TN): disease absent, test negative
Always verify column totals (people with/without disease) before computing.
Sensitivity, Specificity, PPV, NPV
| Metric | Formula | Question answered | Mnemonic |
|---|---|---|---|
| Sensitivity | TP / (TP + FN) | Of those with disease, who test positive? | SnNOut: high Sensitivity, Negative → rules Out |
| Specificity | TN / (TN + FP) | Of those without disease, who test negative? | SpPIn: high Specificity, Positive → rules In |
| PPV | TP / (TP + FP) | Of positive tests, who truly have disease? | Depends on prevalence |
| NPV | TN / (TN + FN) | Of negative tests, who truly are disease-free? | Depends on prevalence |
Sensitivity and specificity are properties of the test in a defined population/method; predictive values are properties of the test applied at a given prevalence (and spectrum of disease).
Worked example 1 — Full metrics from counts
A new D-dimer–like assay is studied in 1,000 patients with suspected disease; gold standard confirms disease in 200:
| Disease + | Disease − | Total | |
|---|---|---|---|
| Test + | 180 | 160 | 340 |
| Test − | 20 | 640 | 660 |
| Total | 200 | 800 | 1,000 |
- Sensitivity = 180/200 = 0.90 (90%)
- Specificity = 640/800 = 0.80 (80%)
- Prevalence in sample = 200/1000 = 20%
- PPV = 180/340 ≈ 0.529 (53%)
- NPV = 640/660 ≈ 0.970 (97%)
Interpretation: a negative test is reassuring (high NPV here), but a positive test is only a coin-flip–ish for true disease at this prevalence—need confirmatory testing.
Worked example 2 — Same Sn/Sp, lower prevalence (PPV collapse)
Screen 10,000 asymptomatic people; true prevalence 1% (100 diseased, 9,900 healthy). Keep Sn 90%, Sp 80%:
- TP = 0.90 × 100 = 90
- FN = 10
- TN = 0.80 × 9,900 = 7,920
- FP = 9,900 − 7,920 = 1,980
PPV = 90 / (90 + 1980) = 90/2070 ≈ 0.043 (4.3%)
NPV = 7920 / (7920 + 10) ≈ 0.999
Teaching point: At low prevalence, most positive results are false positives. This is why screening tests need high specificity or a two-step strategy, and why PPV from referral clinics cannot be copied to community screening.
Worked example 3 — High prevalence setting
Same Sn 90%, Sp 80%, prevalence 50% in 1,000 people (500 diseased):
- TP = 450, FN = 50, TN = 400, FP = 100
- PPV = 450/550 ≈ 82%
- NPV = 400/450 ≈ 89%
PPV rose and NPV fell relative to the low-prevalence example—prevalence effect.
Likelihood Ratios and Bayes Intuition
Likelihood ratios (LRs) quantify how much a result shifts disease odds:
- LR+ = sensitivity / (1 − specificity) = true-positive rate / false-positive rate
- LR− = (1 − sensitivity) / specificity = false-negative rate / true-negative rate
Bayes update (odds form):
Post-test odds = pre-test odds × LR
Pre-test odds = p / (1 − p); post-test probability = odds / (1 + odds)
Rough clinical rules of thumb:
| LR | Approximate shift |
|---|---|
| LR+ > 10 | Large increase in disease probability |
| LR+ 5–10 | Moderate increase |
| LR+ 2–5 | Small increase |
| LR ≈ 1 | No information |
| LR− 0.2–0.5 | Small decrease |
| LR− 0.1–0.2 | Moderate decrease |
| LR− < 0.1 | Large decrease |
Worked example 4 — LR and probability update
From Example 1: Sn 0.90, Sp 0.80
LR+ = 0.90 / 0.20 = 4.5
LR− = 0.10 / 0.80 = 0.125
Pre-test probability 20% → pre-test odds = 0.20/0.80 = 0.25
Positive test → post odds = 0.25 × 4.5 = 1.125 → post probability = 1.125/2.125 ≈ 53% (matches PPV)
Negative test → post odds = 0.25 × 0.125 = 0.03125 → post probability ≈ 3% (matches 1 − NPV)
LRs separate test discriminatory power from prevalence, then Bayes combines them—high-yield conceptual frame even when the stem only asks for LR+.
ROC Curves and Cut-Point Tradeoffs
Many tests are continuous (troponin, glucose, PSA). Choosing a cutoff defines positive vs negative:
- Lower cutoff → more positives → ↑sensitivity, ↓specificity (fewer FN, more FP)
- Higher cutoff → fewer positives → ↓sensitivity, ↑specificity (more FN, fewer FP)
ROC curve: y-axis sensitivity; x-axis 1 − specificity (false-positive rate) as cutoff sweeps. AUC (area under ROC) summarizes discrimination: 0.5 = chance; 1.0 = perfect. Comparing AUCs compares tests independent of a single cutoff, but clinical use still needs a chosen threshold based on harms of FN vs FP.
Worked example 5 — Cutoff choice
Glucose cutoff for "abnormal":
| Cutoff | Sensitivity | Specificity |
|---|---|---|
| 100 mg/dL | 95% | 70% |
| 126 mg/dL | 78% | 92% |
Screening when missing disease is costly may favor the lower cutoff (higher Sn); confirmatory diagnostic contexts may favor higher Sp. There is no universal "best" number without utilities.
Parallel vs Serial Testing
| Strategy | Definition | Net effect |
|---|---|---|
| Parallel | Multiple tests at once; positive if any positive | ↑ net sensitivity, ↓ net specificity |
| Serial (sequential) | Second test only if first positive; positive if all positive | ↑ net specificity, ↓ net sensitivity |
Worked example 6 — Independent tests (approximation)
Two tests each Sn 80%, Sp 90%, used in parallel for a patient who is positive if either is positive:
- Net sensitivity ≈ 1 − (1−0.8)(1−0.8) = 1 − 0.04 = 96%
- Net specificity ≈ 0.9 × 0.9 = 81% (both must be negative to call negative)
Serial: call disease only if both positive:
- Net sensitivity ≈ 0.8 × 0.8 = 64%
- Net specificity ≈ 1 − (1−0.9)(1−0.9) = 99%
Exams use this for "screen with sensitive test, confirm with specific test."
Screening Criteria (Wilson–Jungner Style Teaching)
Population screening of asymptomatic people is justified only when a package of conditions holds (classic public-health teaching, not a single law):
- Important health problem (severity and/or frequency)
- Accepted treatment for recognized disease
- Facilities for diagnosis and treatment available
- Latent or early symptomatic stage exists
- Suitable test or examination (accurate, acceptable, safe)
- Test acceptable to the population
- Natural history adequately understood
- Agreed policy on whom to treat
- Cost balanced against expenditure on medical care as a whole
- Case finding should be a continuous process, not a one-off
CBSE framing: if early treatment does not improve outcomes, screening only creates labels, anxiety, and procedures (overdiagnosis pathway). Combine with lead-time/length-time concepts from Section 18.1.
False-Positive Burden
Even excellent tests harm at scale.
Worked example 7 — Population false positives
Screen 100,000 people, prevalence 0.2% (200 true cases), Sn 99%, Sp 99%:
- TP = 0.99 × 200 ≈ 198
- FN ≈ 2
- FP = 0.01 × 99,800 ≈ 998
- PPV ≈ 198/(198+998) ≈ 16.5%
About five false positives per true positive. Each FP may trigger invasive biopsy, radiation, or labeling—false-positive burden is a core screening harm, not a footnote.
Integration for CBSE Vignettes
- Write the 2×2; identify which margin is fixed (disease totals vs test totals).
- Sn/Sp use disease columns; PPV/NPV use test rows.
- If prevalence falls and Sn/Sp fixed, expect PPV ↓, NPV ↑.
- "Best test to rule out" → maximize sensitivity (or favorable LR−).
- "Best to confirm" → maximize specificity (or favorable LR+).
- Continuous biomarker + threshold debate → ROC/cutoff tradeoff.
- Two-step algorithm → serial testing for specificity.
- Community screening pitch with unchanged mortality → question overdiagnosis/lead time/false positives, not only Sn.
Master the arithmetic and the prevalence story; most diagnostic-test items are variants of Examples 1–3 and 7.
A test has sensitivity 90% and specificity 95%. In a clinic where disease prevalence is 1%, what is the approximate positive predictive value among those who test positive?
About 15%
About 90%
About 95%
About 1%
A biomarker has sensitivity 0.80 and specificity 0.90. What is LR+ and how should a positive result be used?
LR+ = 0.89; essentially no information
LR+ = 8.0; moderately increases post-test odds of disease (pre-test odds × 8)
LR+ = 0.22; strongly decreases disease probability
LR+ = 1.14; only valid in case-control studies
A hospital uses Test A and Test B simultaneously and treats a patient as positive if either test is positive. Compared with using either test alone, net diagnostic performance most likely shows:
Higher sensitivity and higher specificity
Lower sensitivity and higher specificity
Higher sensitivity and lower specificity
Unchanged sensitivity with doubled PPV regardless of prevalence
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