Diagnostic tests
Introduced| English | Meaning |
|---|---|
| sensitivity/ˌsensɪˈtɪvɪti/ | sensitivity |
| specificity/ˌspesɪˈfɪsɪti/ | specificity |
| positive predictive value | positive predictive value |
| likelihood ratio | likelihood ratio |
A positive test and a worried patient
- A test that is 90% sensitive and 90% specific comes back positive. Is the chance of disease 90%? In a clinic where 1 in 100 has the disease, it is about 8%.
- Some test numbers belong to the test, and others belong to the population being tested.
The 2×2 table
- Rows: test positive or negative. Columns: diseased or healthy. a = true positives, b = false positives, c = false negatives, d = true negatives.
- Every diagnostic number is read from these four cells.
Match each cell of the 2×2 table to its name.
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Rows are the test; columns are the truth.
Sensitivity and specificity
- Sensitivity 灵敏度 = a/(a + c): of the diseased, the share testing positive. Specificity 特异度 = d/(b + d): of the healthy, the share testing negative.
- They are properties of the test and stay about the same from one clinic to another.
(Synthetic) 50 people have the disease and 45 test positive. What is the sensitivity in %?
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45 ÷ 50 = 90%.
Predictive values
- Positive predictive value 阳性预测值 (PPV) = a/(a + b): of positive results, the share truly diseased. NPV = d/(c + d).
- They depend on prevalence: when disease is rare, false positives from the large healthy group swamp the true positives.
The positive predictive value of a test falls when the disease becomes rarer.
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False positives from the larger healthy group outnumber the true positives.
Likelihood ratios
- Likelihood ratio 似然比: LR+ = sensitivity ÷ (1 − specificity). A positive result multiplies the pre-test odds by LR+.
- Unlike PPV, the likelihood ratio travels with the test to any clinic; you supply the local prevalence.
A test has sensitivity 80% and specificity 95%. What is LR+?
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0.80 ÷ (1 − 0.95) = 0.80 ÷ 0.05 = 16.
Worked example: a natural-frequency tree
- (Synthetic) 1,000 people, prevalence 10%, Se = Sp = 90%: 100 diseased → 90 TP, 10 FN; 900 healthy → 810 TN, 90 FP.
- PPV = 90/180 = 50%; NPV = 810/820 = 98.8%. At 1% prevalence: 9 TP and 99 FP, so PPV = 9/108 = 8.3%.
(Synthetic) 1,000 people, prevalence 2%, Se = Sp = 95%. There are 19 true positives and 49 false positives. What is the PPV in %? Give one decimal place.
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19 ÷ (19 + 49) = 19 ÷ 68 ≈ 27.9%.
Put the steps of a natural-frequency tree in order.
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The last step gives the PPV.
Worked example: odds with LR+
- (Synthetic) LR+ = 0.9/0.1 = 9. Pre-test probability 10% = odds 1:9.
- Post-test odds = 1:9 × 9 = 1:1, a probability of 50%: the same PPV, reached in one line.
Which number can you take from a study to a clinic with a different prevalence?
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LRs depend on sensitivity and specificity only.
Three common slips
- Saying "90% sensitive" means "right 90% of the time". It is right about 90% of the diseased only.
- Using a textbook PPV in a clinic with a different prevalence.
- Mixing the columns. Sensitivity reads down the diseased column; specificity down the healthy column.
Recap
- Se = a/(a + c), Sp = d/(b + d): test properties.
- PPV = a/(a + b), NPV = d/(c + d): they move with prevalence.
- LR+ = Se/(1 − Sp) multiplies the pre-test odds, in any clinic.