Interpreting a Two-Way Table of Screening Results
A regional clinic trialled a screening test on 2,000 people. Results: of 200 people who truly had the condition, 170 tested positive. Of the 1,800 who did not have the condition, 180 tested positive. The clinic plans to advertise: 'Our test is 85% accurate.' A journalist argues the test is useless because 'most positive results are wrong'. The clinic wants an honest one-sentence claim for its brochure and must decide whether to recommend mass screening.
1. Build the two-way table and compute the positive predictive value and negative predictive value.
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| Condition | No condition | Total | |
|---|---|---|---|
| Test positive | 170 | 180 | 350 |
| Test negative | 30 | 1620 | 1650 |
| Total | 200 | 1800 | 2000 |
Sensitivity = 170/200 = 0.85 (85%). Specificity = 1620/1800 = 0.90 (90%). Positive predictive value (PPV) = 170/350 ≈ 0.486, i.e. about 48.6% - fewer than half of positives truly have the condition. Negative predictive value = 1620/1650 ≈ 0.982, i.e. about 98.2%.
So the journalist is partly right: most positive results (about 51%) are false positives, because the condition is relatively rare (10% prevalence).
2. Evaluate the clinic's '85% accurate' claim and propose a fairer statement.
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'85% accurate' is ambiguous and misleading. It matches the sensitivity (correctly identifying those with the condition), but it ignores the 180 false positives. Overall correct classification = (170 + 1620)/2000 = 1790/2000 = 89.5%, so even 'accuracy' overall is 89.5%, not 85%.
The honest statement must separate the two directions: 'The test correctly identifies about 85% of people who have the condition and about 90% of people who do not. Of people who test positive, about half will actually have the condition, so a positive result is followed by a confirmatory test.'
The judgement call is that a single 'accuracy' number cannot summarise a screening test; sensitivity, specificity, and PPV answer different questions, and PPV depends on how common the condition is.
3. Would mass screening be justified? Use the predictive values to argue both sides and reach a conclusion.
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Case for: the test catches 170 of 200 true cases (85%), and a negative result is highly reassuring (NPV ~98%), so it efficiently rules out the condition for most people. Case against: 180 people without the condition test positive, so mass screening generates many false positives. Each false positive causes anxiety and follow-up costs; with 350 positives and only 170 true cases, roughly half of all positives need confirmatory testing, which may cost more than it saves.
Conclusion: mass screening is defensible only if a cheap, reliable confirmatory test exists and follow-up is funded. The test is better used to rule out (high NPV) than to diagnose (low PPV). The judgement call is recognising that 'sensitivity 85%' does not settle a screening policy; the PPV and the consequences of false positives do.