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Difficulty: HardDiagnostic and Screening Test Performance

A biotechnology firm has developed a novel non-invasive stool DNA assay for colorectal cancer. To validate the assay's performance, a study is conducted in a high-risk gastrointestinal clinic among patients presenting with hematochezia. In this validation cohort, the prevalence of colorectal cancer is 50%50\%. The assay demonstrates a sensitivity of 95%95\% and a specificity of 95%95\%.

Following regulatory approval, a large health maintenance organization (HMO) decides to implement this stool DNA assay as a primary screening tool for all asymptomatic, average-risk members aged 50 to 75. Epidemiological data indicates that the point prevalence of colorectal cancer in this average-risk HMO population is 5%5\%.

Assuming the test's intrinsic performance characteristics remain constant across populations, what is the approximate probability that an average-risk HMO member with a positive stool DNA assay result actually has colorectal cancer?

  1. A
    5%5\%
  2. B
    10%10\%
  3. 50%50\%Answer
  4. D
    95%95\%
  5. E
    99%99\%

Answer

The probability is 50%50\%, which corresponds to the Positive Predictive Value (PPV) in the new screening population.
The question asks for the probability that a patient with a positive test actually has colorectal cancer, which is the definition of Positive Predictive Value (PPV). Unlike sensitivity and specificity, which are intrinsic characteristics of the diagnostic test, PPV varies with disease prevalence. To solve this, we can calculate the outcomes in a hypothetical population of 10,000 average-risk HMO members. In this group, a 5%5\% prevalence means 500 patients have colorectal cancer and 9,500 do not. Using the test's 95%95\% sensitivity, there will be 475 true positives (500×0.95500 \times 0.95). Using the test's 95%95\% specificity, the false positive rate is 5%5\% (1−0.951 - 0.95). Applying this false positive rate to the 9,500 healthy individuals yields 475 false positives (9,500×0.059,500 \times 0.05). The total number of positive test results is 950 (475 true positives+475 false positives475 \text{ true positives} + 475 \text{ false positives}). The PPV is the ratio of true positives to total positives: 475950=0.50\frac{475}{950} = 0.50, or 50%50\%.

Step-by-Step Solution

1
Identify the relevant test parameters and the new population prevalence.
Sensitivity = 95%95\%, Specificity = 95%95\%, New Prevalence = 5%5\%.
Sensitivity and specificity are intrinsic to the test and do not change across populations, but PPV must be recalculated using the new population's prevalence.
2
Set up a hypothetical screening population (e.g., 10,000 individuals) and calculate the number of true positives.
5%5\% of 10,000 = 500 individuals with disease. True Positives (TP) = 500×0.95=475500 \times 0.95 = 475.
This determines the numerator for the positive predictive value calculation.
3
Calculate the number of false positives in the disease-free portion of the population.
95%95\% of 10,000 = 9,500 individuals without disease. False Positives (FP) = 9,500×(1−0.95)=4759,500 \times (1 - 0.95) = 475.
False positives must be combined with true positives to find the total number of positive test results in the population.
4
Calculate the positive predictive value (PPV).
PPV=TPTP+FP=475475+475=475950=0.50PPV = \frac{TP}{TP + FP} = \frac{475}{475 + 475} = \frac{475}{950} = 0.50, or 50%50\%.
PPV represents the probability that a positive test result is a true positive.

Key Concept

Positive Predictive Value (PPV) is highly dependent on the prevalence of the disease in the tested population. Applying a test with high sensitivity and specificity to a low-prevalence population significantly lowers its PPV due to an absolute increase in false positives.
Estimated Time:2m 0s
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Topics

Diagnostic and Screening Test Performance

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