Two Positive Screens for a Rare Condition
A public-health team screens 100,000 people for a rare condition. The prevalence is 0.4%, so about 400 have the condition. The screening test has sensitivity 98% and specificity 95%. A patient receives two positive results from two independent test batches; conditional on the patient's true status, the batches are assumed independent. A clinician tells the patient that because sensitivity is 98%, the chance of having the condition is about 98%. The team also has a budget rule: if the posterior probability after two positives exceeds 90%, the patient is sent directly to confirmatory care; otherwise a third test is ordered. The team wants to know the actual posterior probability after two positives, whether the clinician's statement is defensible, and how the decision changes if the independence assumption is doubtful.
1. Compute the posterior probability of the condition after two positive tests. Show the prior, likelihoods, and Bayes calculation.
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Let D be the condition. Prior P(D)=0.004. Conditional independence gives P(++|D)=0.98^2=0.9604 and P(++|D^c)=0.05^2=0.0025. Bayes: P(D|++)= [0.004·0.9604]/[0.004·0.9604+0.996·0.0025]. Numerator=0.0038416. Denominator=0.0038416+0.00249=0.0063316. Posterior≈0.6067, about 60.7%.
2. Is the clinician's 98% claim correct? Explain what quantity the 98% describes and what quantity the patient needs.
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The 98% figure is sensitivity, P(+|D), not the predictive value P(D|+). The patient needs P(D|++). Even one positive gives only about P(D|+)=[0.004·0.98]/[0.004·0.98+0.996·0.05]≈0.073, or 7.3%. Two positives raise the posterior to about 60.7%, still well below 98%. The clinician's statement confuses the probability of a positive test given disease with the probability of disease given a positive test; the low base rate makes the latter much smaller.
3. Under the budget rule, is a third test ordered? If the two test results are positively correlated conditional on true status, would the posterior after two positives tend to be higher or lower than your answer? Explain.
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The posterior 60.7% is below 90%, so under the stated rule a third test is ordered. If the two batches are positively correlated conditional on true status, the second positive is partly redundant: it does not multiply the odds by the full sensitivity/specificity ratio. The effective likelihood ratio for two positives is smaller than the independent-case ratio, so the posterior after two positives would generally be lower than 60.7%. If correlation is strong, the second result adds little and the posterior may stay near the one-test value.