## Clinical Context A positive screening test in a patient requires interpretation using Bayes' theorem and calculation of positive predictive value (PPV). ## PPV Calculation **Key Point:** PPV tells us the probability of disease given a positive test result — this is the clinically relevant question when a screening test is positive. Using Bayes' theorem: $$PPV = \frac{\text{Sensitivity} \times \text{Prevalence}}{\text{Sensitivity} \times \text{Prevalence} + (1 - \text{Specificity}) \times (1 - \text{Prevalence})}$$ $$PPV = \frac{0.90 \times 0.02}{(0.90 \times 0.02) + (0.15 \times 0.98)}$$ $$PPV = \frac{0.018}{0.018 + 0.147} = \frac{0.018}{0.165} \approx 10.9\%$$ ## Interpretation & Next Step | Finding | Value | Meaning | |---------|-------|----------| | PPV | ~11% | Only 11% of positive tests represent true disease | | False positive rate | ~89% | 89% of positive tests are false positives | | Post-test probability | 11% | Despite positive test, cancer probability is still low | **High-Yield:** Even with a positive screening test, when prevalence is low (2%), the PPV remains modest (~11%). However, a positive FOBT **must be investigated** with the gold standard test (colonoscopy) to exclude cancer, because: 1. The test is positive (not negative) 2. Cancer is a serious condition that cannot be missed 3. Colonoscopy is the definitive diagnostic test **Clinical Pearl:** Do not confuse "low PPV" with "no action needed." A positive screening test always warrants confirmation with the gold standard, even if PPV is modest. The next step is diagnostic colonoscopy. ## Why Colonoscopy Is Correct Colonoscopy is the gold standard for colorectal cancer diagnosis. A positive FOBT, regardless of PPV, requires endoscopic evaluation to rule out malignancy.
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