Interpreting a Diagnostic Test in a Low-Prevalence Herd
A dairy herd of 500 cows is being screened for bovine tuberculosis (bTB) using the caudal fold tuberculin (CFT) test as part of a state eradication program. The herd has no history of bTB and is in a low-prevalence area (estimated herd prevalence 0.1%). The CFT test has a sensitivity of 80% and specificity of 99% at the individual animal level. The state veterinarian must decide whether to depopulate the entire herd if any positive reactors are found, or to conduct confirmatory testing. The owner is concerned about economic losses and asks for the probability that a positive test truly indicates infection. The veterinarian also considers using the comparative cervical test (CCT) as a confirmatory test, which has a sensitivity of 70% and specificity of 99.5%.
1. Calculate the positive predictive value (PPV) of the CFT test in this herd, assuming individual animal prevalence is 0.1%. Show your calculations and interpret the result.
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Prevalence (P) = 0.001. Sensitivity (Se) = 0.80. Specificity (Sp) = 0.99. PPV = (Se × P) / [(Se × P) + ((1 - Sp) × (1 - P))]. Numerator = 0.80 × 0.001 = 0.0008. Denominator = 0.0008 + (0.01 × 0.999) = 0.0008 + 0.00999 = 0.01079. PPV = 0.0008 / 0.01079 ≈ 0.074, or 7.4%. Interpretation: Only about 7.4% of positive CFT results are true positives; over 92% are false positives. This low PPV is due to the low prevalence. Therefore, depopulating based on a single CFT positive would be unjustified; confirmatory testing is essential.
2. If the CCT is used as a confirmatory test on CFT-positive animals, what is the probability that an animal testing positive to both tests is truly infected? Assume the tests are independent conditional on infection status.
Read the worked answer
Sequential testing: PPV of the series = (Se1 × Se2 × P) / [(Se1 × Se2 × P) + ((1 - Sp1) × (1 - Sp2) × (1 - P))]. Se1=0.80, Se2=0.70, Sp1=0.99, Sp2=0.995, P=0.001. Numerator = 0.80 × 0.70 × 0.001 = 0.00056. Denominator = 0.00056 + (0.01 × 0.005 × 0.999) = 0.00056 + 0.00004995 = 0.00060995. PPV = 0.00056 / 0.00060995 ≈ 0.918, or 91.8%. Interpretation: Using the CCT as a confirmatory test greatly increases the PPV to about 92%. However, the overall sensitivity of the series is 0.80 × 0.70 = 0.56, so many infected animals would be missed. In a low-prevalence setting, this strategy minimizes false positives but may delay detection. The veterinarian should consider the goals: if the aim is to prevent herd depopulation based on false positives, confirmatory testing is prudent; if the aim is to detect all infected animals, the series sensitivity is low. The independence assumption may not hold; if tests are correlated, the PPV may be lower.