A screening test for a rare cancer has 95% sensitivity and 90% specificity. In a population where the cancer prevalence is 0.1%, a patient tests positive. What is the approximate probability that this patient actually has cancer?
AAbout 95%, because the test has 95% sensitivity
BAbout 90%, because the test has 90% specificity
CAbout 1%, because even with a good test, false positives vastly outnumber true positives when prevalence is very low
DAbout 50%, because sensitivity and specificity are both high
In 100,000 people at 0.1% prevalence, 100 have cancer and 99,900 do not. Sensitivity of 95% detects 95 of 100 true cases. Specificity of 90% correctly rules out 89,910 of 99,900 non-cases, but misclassifies 9,990 as positive. Total positives: 95 + 9,990 = 10,085. PPV = 95/10,085 ≈ 0.94%, or about 1%. The overwhelming majority of positive tests are false positives because the non-diseased population is so much larger than the diseased population. This is why screening programs for rare diseases require confirmatory testing.
Question 2 True / False
A test with high sensitivity is most useful for ruling out disease (SnNout: Sensitivity-Negative-rule Out), while a test with high specificity is most useful for ruling in disease (SpPin: Specificity-Positive-rule In).
TTrue
FFalse
Answer: True
A highly sensitive test rarely misses true cases, so a negative result effectively rules out the disease (if the test does not detect it, you almost certainly do not have it). A highly specific test rarely produces false positives, so a positive result effectively rules in the disease (if this stringent test says you have it, you almost certainly do). These mnemonics (SnNout and SpPin) capture the asymmetric clinical utility of sensitivity and specificity and guide the choice of test at different stages of the diagnostic workup.
Question 3 Multiple Choice
A hospital administrator argues that because a test has 99% sensitivity and 99% specificity, it should be used for universal screening of all patients. Why might this be a poor decision for a disease with 0.01% prevalence?
AThe test is too expensive for universal use
BAt 0.01% prevalence, false positives will outnumber true positives by roughly 100:1, generating enormous numbers of unnecessary follow-up procedures
CSensitivity and specificity are unreliable metrics
DUniversal screening requires 100% sensitivity
In 1,000,000 screened patients at 0.01% prevalence, 100 have disease. Sensitivity of 99% detects 99. Specificity of 99% correctly clears 989,901 but produces 9,999 false positives. The PPV is 99/(99 + 9,999) ≈ 0.98%. For every true case found, roughly 101 healthy people receive a false alarm, each requiring expensive and potentially harmful follow-up (biopsies, imaging, anxiety). The cost-benefit calculus of screening depends on prevalence, not just test performance.
Question 4 Short Answer
Explain why predictive values depend on prevalence but sensitivity and specificity do not.
Think about your answer, then reveal below.
Model answer: Sensitivity and specificity are conditional on true disease status — they measure test performance within the diseased and non-diseased groups separately, so they do not change with the proportion of diseased people in the population. Predictive values are conditional on test result — they ask 'given a positive test, what is the probability of disease?' This depends on the ratio of true positives to all positives, which changes with prevalence. As prevalence decreases, the non-diseased group grows, producing more false positives relative to true positives, and PPV drops even if sensitivity and specificity remain constant.
Bayes' theorem formalizes this relationship: PPV = (sensitivity × prevalence) / [(sensitivity × prevalence) + ((1-specificity) × (1-prevalence))]. The denominator includes both true positives and false positives. When prevalence is low, the false positive term dominates because it is multiplied by (1-prevalence), which is close to 1. This is why the same test can have a PPV of 95% in a high-risk clinic and 1% in a general screening program.