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Difficulty: Very hardDiagnostic and Screening Test Performance

Emergency medicine researchers are investigating a novel rapid point-of-care ultrasound algorithm for deep vein thrombosis (DVT). In their study cohort of symptomatic patients with a high pre-test probability, the true prevalence of DVT is 40%40\%.

During the initial evaluation using standard diagnostic criteria, the algorithm yields a sensitivity of 85%85\% and a specificity of 85%85\%.

To minimize missed diagnoses, the researchers broaden the criteria for a positive scan. Statistical analysis of the cohort reveals that this protocol change reduces the absolute number of false-negative results by exactly 50%50\% compared to the standard criteria. However, this adjustment simultaneously causes the absolute number of false-positive results to increase by exactly 100%100\% compared to the standard criteria.

Calculate the negative predictive value (NPV) of the ultrasound algorithm using the broadened criteria. Express your answer as a percentage rounded to the nearest whole number.

Answer: 93 %

Answer

93
To calculate the new NPV, one must first determine the absolute number of patients in each category of a 2x2 contingency table. Assuming a cohort of 1,0001,000 patients with a 40%40\% prevalence, 400400 patients have DVT and 600600 do not. At the baseline 85%85\% sensitivity and specificity, there are 340340 true positives, 6060 false negatives (400−340400 - 340), 510510 true negatives, and 9090 false positives (600−510600 - 510). The broadened criteria reduce the false negatives by 50%50\% (from 6060 to 3030) and increase the false positives by 100%100\% (from 9090 to 180180). Because the total number of patients without DVT is fixed at 600600, an increase in false positives to 180180 means the true negatives must decrease to 420420 (600−180600 - 180). The new Negative Predictive Value is the number of true negatives divided by the total number of negative test results: NPV=TNTN+FN=420420+30=420450≈0.9333\text{NPV} = \frac{\text{TN}}{\text{TN} + \text{FN}} = \frac{420}{420 + 30} = \frac{420}{450} \approx 0.9333. Expressed as a percentage and rounded to the nearest whole number, this is 93%93\%.

Step-by-Step Solution

1
Determine the number of patients with and without the disease in a hypothetical cohort.
In a 1,0001,000 patient cohort with 40%40\% prevalence: 400400 have DVT (disease positive) and 600600 do not have DVT (disease negative).
Creating a 2x2 contingency table with absolute numbers simplifies predictive value calculations.
2
Calculate the baseline number of false negatives (FN) and false positives (FP).
Baseline FN = 400−(0.85×400)=60400 - (0.85 \times 400) = 60. Baseline FP = 600−(0.85×600)=90600 - (0.85 \times 600) = 90.
The protocol modifications are given as percentage changes relative to the baseline absolute error numbers.
3
Apply the protocol changes to find the new absolute numbers of FN and FP.
New FN = 60−(0.50×60)=3060 - (0.50 \times 60) = 30. New FP = 90+(1.00×90)=18090 + (1.00 \times 90) = 180.
Broadening criteria decreases missed cases (FN) by 50%50\% but doubles the false alarms (FP).
4
Calculate the new number of true negatives (TN).
New TN = (Total Disease Negative) - (New FP) = 600−180=420600 - 180 = 420.
The denominator for NPV requires the TN, which must be derived from the fixed disease-negative population.
5
Calculate the Negative Predictive Value (NPV).
NPV=TNTN+FN=420420+30=420450≈0.9333\text{NPV} = \frac{\text{TN}}{\text{TN} + \text{FN}} = \frac{420}{420 + 30} = \frac{420}{450} \approx 0.9333 (93.33%93.33\%).
NPV represents the probability that a patient with a negative test result truly does not have the disease.

Key Concept

Recalculating predictive values after a diagnostic threshold shift by tracking absolute changes in a 2x2 contingency table.
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Topics

Diagnostic and Screening Test Performance

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