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Zorluk: OrtaDiagnostic Testing Performance

A 32-year-old man comes to an urgent care clinic in mid-July with a 2-day history of a mild cough and a low-grade fever. The physician considers testing him for a novel respiratory virus. The manufacturer's data for a rapid point-of-care antigen test indicates a sensitivity of 90%90\% and a specificity of 90%90\%. Based on current local epidemiological data for patients presenting with similar symptoms during the summer, the estimated prevalence (pretest probability) of this virus is 1%1\%. The patient undergoes the rapid test, and the result is positive. Which of the following is closest to the probability that this patient actually has the viral infection?

  1. 8%8\%Cevap
  2. B
    10%10\%
  3. C
    81%81\%
  4. D
    90%90\%
  5. E
    99%99\%

Cevap

8%8\%
The probability that a patient with a positive test result actually has the disease is defined as the Positive Predictive Value (PPV). PPV depends heavily on the pretest probability, which represents the disease prevalence in this specific clinical context. Using a hypothetical cohort of 10,000 patients, a 1%1\% prevalence means 100 individuals have the virus and 9,900 do not. A 90%90\% sensitivity applied to the 100 infected patients yields 90 true positives. A 90%90\% specificity means a 10%10\% false positive rate applied to the 9,900 uninfected patients, yielding 990 false positives (9,900×0.109,900 \times 0.10). The total number of positive tests is 90+990=108090 + 990 = 1080. The PPV is 90/1080≈0.083390 / 1080 \approx 0.0833, which rounds to 8%8\%. Because the disease is rare, the massive absolute number of false positives overwhelmingly dilutes the true positives, leading to a surprisingly low probability of actual infection despite the positive test.

Adım Adım Çözüm

1
Set up a hypothetical population of 10,000 patients and calculate the base number of true disease cases.
With a 1%1\% prevalence, 100 patients truly have the virus (10,000×0.0110,000 \times 0.01) and 9,900 do not (10,000−10010,000 - 100).
Establishing absolute numbers based on the pretest probability (base rate) is a necessary first step to accurately calculate predictive values.
2
Calculate the number of true positive test results.
100 diseased patients ×0.90\times 0.90 sensitivity = 90 true positives.
Sensitivity defines the proportion of diseased individuals who will correctly test positive.
3
Calculate the number of false positive test results.
9,900 healthy patients ×0.10\times 0.10 false positive rate (1−0.901 - 0.90 specificity) = 990 false positives.
Specificity defines the proportion of healthy individuals who test negative; the remaining healthy patients will incorrectly test positive.
4
Calculate the Positive Predictive Value (PPV).
PPV=TruePositivesTruePositives+FalsePositives=9090+990=901080≈0.0833PPV = \frac{True Positives}{True Positives + False Positives} = \frac{90}{90 + 990} = \frac{90}{1080} \approx 0.0833, or 8.3%8.3\%.
PPV defines the probability that a patient with a positive test result actually has the target disease.

Anahtar Kavram

Positive Predictive Value (PPV) Dependence on Prevalence

Alternatif Yöntem

An alternative approach uses Likelihood Ratios (LR). The Positive Likelihood Ratio (LR+) is calculated as Sensitivity/(1−Specificity)=0.90/0.10=9Sensitivity / (1 - Specificity) = 0.90 / 0.10 = 9. The pretest odds are calculated from the prevalence: Odds=Probability/(1−Probability)=0.01/0.99≈0.0101Odds = Probability / (1 - Probability) = 0.01 / 0.99 \approx 0.0101. The post-test odds equal PretestOdds×LR+=0.0101×9=0.0909Pretest Odds \times LR+ = 0.0101 \times 9 = 0.0909. Converting back to probability: Probability=Odds/(1+Odds)=0.0909/1.0909≈0.0833Probability = Odds / (1 + Odds) = 0.0909 / 1.0909 \approx 0.0833 or 8.3%8.3\%, which is closest to 8%8\%.
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Diagnostic Testing Performance

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