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Difficulty: HardDiagnostic and Screening Test Performance

A prospective study evaluates the diagnostic performance of a rapid point-of-care synovial fluid assay for detecting septic arthritis. The study enrolls 1,0001,000 adult patients presenting to the emergency department with acute monoarticular joint effusions. All patients undergo both the rapid assay and a gold-standard synovial fluid culture. The culture confirms septic arthritis in 200200 patients. Among these confirmed cases, the rapid assay is positive in 180180 patients. Across the entire study cohort of 1,0001,000 patients, the rapid assay is positive in a total of 580580 patients. What is the negative likelihood ratio of this rapid assay? (Express your answer as a decimal).

Answer: 0.2

Answer

0.20
To find the negative likelihood ratio (LR−LR-), we must calculate the sensitivity and specificity from the study data. The study gives a gold-standard disease-positive population of 200200. Out of these, 180180 test positive, making the Sensitivity 180/200=0.90180/200 = 0.90. The remaining 800800 patients are disease-negative. Because the total number of positive tests is 580580, the number of false positives (positive test in a disease-negative patient) is 580−180=400580 - 180 = 400. The number of true negatives is therefore 800−400=400800 - 400 = 400. Specificity is 400/800=0.50400 / 800 = 0.50. The formula for LR−LR- is (1−Sensitivity)/Specificity(1 - \text{Sensitivity}) / \text{Specificity}. Substituting the values yields (1−0.90)/0.50=0.10/0.50=0.20(1 - 0.90) / 0.50 = 0.10 / 0.50 = 0.20.

Step-by-Step Solution

1
Determine disease prevalence numbers from the total cohort.
Disease Positive = 200; Disease Negative = 800.
Establishing the absolute numbers for the presence and absence of the disease is required before calculating the true/false positive and negative rates.
2
Calculate the test Sensitivity.
Sensitivity = 180 / 200 = 0.90.
Sensitivity is the proportion of disease-positive individuals who correctly test positive (True Positives / Total Disease Positive).
3
Deduce False Positives (FP) and True Negatives (TN) to calculate Specificity.
FP = 580 - 180 = 400. TN = 800 - 400 = 400. Specificity = 400 / 800 = 0.50.
Because only the total number of positive tests (580) was given, subtracting the true positives (180) isolates the false positives. True negatives are the remainder of the disease-negative group.
4
Calculate the Negative Likelihood Ratio (LR−LR-).
LR−=(1−0.90)/0.50=0.20LR- = (1 - 0.90) / 0.50 = 0.20.
The negative likelihood ratio is calculated as the False Negative Rate divided by the True Negative Rate (Specificity), representing the probability of a negative test in a patient with the disease divided by the probability of a negative test in a patient without the disease.

Key Concept

The negative likelihood ratio (LR−LR-) is calculated as (1−Sensitivity)/Specificity(1 - \text{Sensitivity}) / \text{Specificity}. It requires extracting the true positive and false positive rates from clinical study totals.
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Topics

Diagnostic and Screening Test Performance

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