Questions: Positive and Negative Predictive Values

Question 1

A test for a rare disease has 99% sensitivity and 99% specificity. In a population where 1 in 1,000 people have the disease, a patient receives a positive result. Approximately what is the probability the patient actually has the disease?

• A. About 99% — the test is 99% accurate, so a positive result almost certainly indicates disease
• B. About 50% — since sensitivity and specificity are equal, positive and negative results are equally informative
• C. About 9% — at this prevalence, false positives vastly outnumber true positives
• D. About 1% — the result provides no information beyond the base rate

Answer: (C)

PPV = (sensitivity × prevalence) / [(sensitivity × prevalence) + (1−specificity) × (1−prevalence)]. With prevalence = 0.001, sensitivity = 0.99, specificity = 0.99: PPV = (0.99 × 0.001) / [(0.99 × 0.001) + (0.01 × 0.999)] = 0.00099 / (0.00099 + 0.00999) ≈ 9%. There are roughly 10 false positives for every true positive at this prevalence, because even a 1% false-positive rate generates many false alarms when applied to 999 truly disease-free individuals. The test hasn't changed — only the population has — and PPV falls dramatically. Option 0 is the classic misconception: 'accuracy' conflates sensitivity/specificity with clinical predictive value.

Question 2

A highly effective screening program detects early-stage cancer in a low-prevalence population. Many patients with positive screens turn out to be disease-free after confirmatory testing. The medical team proposes abandoning the screening program because of its low positive predictive value. The best response is:

• A. The program should be abandoned — a low PPV means the test is not working correctly
• B. The PPV should be interpreted in context: low PPV is expected and acceptable in low-prevalence screening when the cost of missing a case exceeds the cost of confirmatory testing
• C. The sensitivity should be increased to raise PPV, even if specificity falls
• D. The program should switch to a test with higher specificity to directly raise PPV without needing prevalence data

Answer: (B)

Low PPV in population screening is mathematically inevitable and clinically acceptable. Screening applies a test to a low-prevalence population to catch disease before symptoms, accepting false positives as the cost of not missing true cases. The standard response is two-stage design: positive screens advance to a more specific confirmatory test applied to a population now enriched for disease (high effective prevalence), yielding high PPV at the confirmation stage. Abandoning the program because of low first-stage PPV misunderstands the design intent of screening.

Question 3

A test with 99% sensitivity and 99% specificity will generally have a PPV above 90% in any clinical setting where it is applied.

• T. True
• F. False

Answer: False

PPV depends on prevalence, not just test characteristics. As shown by the Bayesian calculation, a 99%/99% test applied to a population with 1-in-1,000 prevalence has PPV ≈ 9%. The test's technical performance is excellent, but when false positives from 999 healthy people outnumber true positives from 1 diseased person, most positive results are false. PPV can range from near 0% (very low prevalence) to near 100% (very high prevalence) for the same test. Treating sensitivity and specificity as 'accuracy' is the most dangerous misconception in diagnostic test interpretation.

Question 4

When a positive-screen population advances to confirmatory testing, the effective prevalence seen by the confirmatory test is much higher than the base population's prevalence, which substantially raises the confirmatory test's PPV.

• T. True
• F. False

Answer: True

This is the mathematical rationale for two-stage screening designs. Suppose 1% of the general population has a disease. After a first-stage test (sensitivity 95%, specificity 90%), roughly 95 of 100 true cases and 990 of 9,900 healthy individuals test positive — about 1,085 positives total. Among these, 95/1085 ≈ 8.8% have disease: the effective prevalence for the confirmatory test is now ~9%, not 1%. Applying a highly specific confirmatory test to this enriched group yields much higher PPV. Each stage of testing reshapes the effective prevalence seen by the next.

Question 5

A physician tells a patient: 'This test is 99% accurate, so your positive result means there is a 99% chance you have the disease.' Under what conditions would this reasoning be seriously wrong, and why?

Model answer: The reasoning is seriously wrong when disease prevalence in the tested population is low. 'Accuracy' (sensitivity/specificity) describes how the test performs on people known to have or not have disease. PPV — the probability of disease given a positive result — depends critically on how common the disease is in the population being tested. In a low-prevalence setting, even a 99% accurate test produces many false positives relative to true positives, so PPV can be far below 99%. The physician should apply Bayes' theorem: PPV = (sensitivity × prevalence) / [(sensitivity × prevalence) + (1−specificity)(1−prevalence)]. The relevant question is not 'how good is the test?' but 'given this patient's prior probability of disease, what does this positive result mean?'

This error is pervasive in clinical practice and patient communication. The fix requires explicitly stating the prevalence (or pre-test probability) being assumed. A positive rapid strep test in a child with sore throat and fever (high pre-test probability) has very different post-test meaning than the same positive in a routine screening of healthy adults (low pre-test probability). Sensitivity and specificity are fixed properties of the test; PPV is a property of the test applied to a specific clinical context.