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Difficulty: Very hardDiagnostic Testing Performance

A regional health department is evaluating a novel two-tiered screening protocol for a tick-borne pathogen endemic to the area. Epidemiological data indicates that the baseline prevalence of the infection in the target demographic is 10%.

The proposed protocol mandates an initial enzyme immunoassay (EIA). Only individuals who yield a positive result on the EIA will proceed to undergo a confirmatory immunoblot assay. The testing pathway is considered positive only if both assays yield positive results.

The performance characteristics of the individual assays are established as follows:
- Enzyme immunoassay: Sensitivity = 90%, Specificity = 80%
- Immunoblot assay: Sensitivity = 80%, Specificity = 90%

Assuming the two assays are conditionally independent, which of the following best represents the probability that an individual who tests positive on both sequential tests actually harbors the infection?

  1. A
    33%
  2. B
    72%
  3. 80%Answer
  4. D
    97%
  5. E
    98%

Answer

80%
The probability that an individual with positive results on both tests actually has the infection is the Positive Predictive Value (PPV) of the sequential testing strategy. In a population of 1,000 individuals with a 10% prevalence, 100 have the infection and 900 do not. After the first test (90% sensitivity, 80% specificity), there are 90 true positives and 180 false positives. Only these 270 individuals proceed to the second test. After the second test (80% sensitivity, 90% specificity), 72 of the true positives remain positive (90×0.8090 \times 0.80), and 18 of the false positives test positive again (180×0.10180 \times 0.10). The PPV is the number of true positives divided by the total number of positives: 72/(72+18)=72/90=0.8072 / (72 + 18) = 72 / 90 = 0.80, or 80%.

Step-by-Step Solution

1
Calculate the initial number of diseased and healthy individuals
100 diseased and 900 healthy individuals
Establishing a hypothetical cohort of 1,000 individuals based on the 10% prevalence simplifies the sequential probability calculations.
2
Determine the outcomes of the initial enzyme immunoassay (EIA)
90 True Positives (100 * 0.90) and 180 False Positives (900 * 0.20)
Because the protocol is sequential, only these 270 individuals who test positive on the EIA will proceed to the confirmatory test.
3
Determine the outcomes of the confirmatory immunoblot assay on the remaining cohort
72 final True Positives (90 * 0.80) and 18 final False Positives (180 * 0.10)
The confirmatory test is applied only to the 270 individuals who tested positive initially. The false positive rate for this specific step is 1 - 0.90 = 0.10.
4
Calculate the Positive Predictive Value (PPV) of the combined sequential strategy
PPV = 72 / (72 + 18) = 72 / 90 = 0.80 (80%)
The PPV represents the proportion of true positives among all individuals who ultimately test positive on both assays.

Key Concept

Calculating the Positive Predictive Value (PPV) in a sequential testing strategy requires applying the combined false positive and true positive rates to the baseline population prevalence.

Alternative Method

Instead of using a hypothetical cohort, this problem can be rapidly solved using Likelihood Ratios (LR). The pre-test odds are 0.10/0.90=1/90.10 / 0.90 = 1/9. The LR+ for the EIA is Sensitivity/(1−Specificity)=0.90/0.20=4.5Sensitivity / (1 - Specificity) = 0.90 / 0.20 = 4.5. The LR+ for the immunoblot is 0.80/0.10=80.80 / 0.10 = 8. Because the tests are conditionally independent, the combined LR+ is 4.5×8=364.5 \times 8 = 36. The post-test odds after both tests are 1/9×36=41/9 \times 36 = 4. Converting odds back to probability yields 4/(4+1)=80%4 / (4 + 1) = 80\%.
Estimated Time:2m 0s
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Diagnostic Testing Performance

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