The screening test headline
A regional health survey estimates that 2% of adults have a particular condition. A screening test has sensitivity 95%, meaning it is positive for 95% of people who have the condition, and specificity 90%, meaning it is negative for 90% of people who do not have the condition. A local newspaper reports: 'The test is 95% accurate, so a positive result means a 95% chance of having the condition.' A mathematics class is asked to check this claim and to advise a patient who has just tested positive. The class is also told that a second, independent test using the same sensitivity and specificity is available. The clinician warns, however, that the two tests use the same biomarker and that their errors may be positively correlated. The class must distinguish between the sensitivity of the test and the probability of disease given a positive result, and must explain how the second test and the independence assumption change the advice.
1. Compute the probability that a randomly selected adult who tests positive actually has the condition. Explain why it is not 95%.
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Use natural frequencies for 10,000 adults. With prevalence 2%, about 200 have the condition and 9,800 do not. Of the 200 diseased, sensitivity 95% gives 190 true positives and 10 false negatives. Of the 9,800 healthy, specificity 90% gives 8,820 true negatives and 980 false positives. Total positives =190+980=1,170. Therefore
P(disease | positive)=190/1170=19/117≈0.162, about 16.2%.
The newspaper's 95% is the sensitivity P(positive | disease), not the predictive value P(disease | positive). Because the condition is rare, most positives are false positives. The base rate matters.
2. The patient takes a second independent test with the same sensitivity and specificity and again tests positive. Assuming conditional independence given disease status, compute the new probability of disease.
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Assuming the two tests are conditionally independent given disease status, the probability of two positives is 0.95^2=0.9025 for a diseased person and 0.10^2=0.01 for a healthy person. By Bayes:
P(D | ++)=0.02(0.9025)/[0.02(0.9025)+0.98(0.01)] =0.01805/(0.01805+0.0098) =0.01805/0.02785≈0.648.
So a second positive raises the probability to about 64.8%, much higher than 16.2% but still not 95%. The independence assumption is doing real work here.
3. The clinician says the two tests are not truly independent because they use the same biomarker. Explain how positive correlation of errors affects the 64.8% figure.
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If the two tests use the same biomarker, their errors are likely positively correlated: a person without the disease who produces one false positive is more likely to produce another. That means P(++ | no disease) is greater than 0.01, the value used under independence. In the Bayes calculation, a larger false-positive repeat probability increases the denominator without changing the numerator, so the posterior P(D | ++) is lower than 64.8%. In the extreme case of perfectly correlated errors, the second result adds no information and the posterior stays at about 16.2%. The clinician should therefore treat 64.8% as an optimistic upper estimate and prefer a truly independent confirmatory test.