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Difficulty: MediumDiagnostic and Screening Test Performance

An internal medicine resident is reviewing the literature on a new rapid immunoassay for detecting *Legionella pneumophila* antigen in urine. The manufacturer's validation study compared the new immunoassay against the gold-standard respiratory culture. The study included 250 patients with culture-confirmed *Legionella* pneumonia and 500 patients with pneumonia caused by other pathogens (culture-negative for *Legionella*). The rapid immunoassay returned a positive result for 230 of the patients with confirmed *Legionella* pneumonia. Additionally, the assay returned a positive result for 20 of the patients without *Legionella* pneumonia.

Based on these study results, what is the positive likelihood ratio of this rapid immunoassay?

Answer: 23

Answer

23
The positive likelihood ratio (LR+LR+) is defined as the probability of a positive test result in a patient with the disease divided by the probability of a positive test result in a patient without the disease. It is calculated using the formula: LR+=Sensitivity1−SpecificityLR+ = \frac{Sensitivity}{1 - Specificity}. First, we determine the sensitivity (true positive rate) by dividing the number of true positives by the total number of patients with the disease (230/250=0.92230 / 250 = 0.92). Next, we determine the false positive rate (1−Specificity1 - Specificity) by dividing the number of false positives by the total number of patients without the disease (20/500=0.0420 / 500 = 0.04). Finally, dividing the sensitivity by the false positive rate (0.92/0.040.92 / 0.04) yields an LR+LR+ of 23. This means a positive immunoassay result is 23 times more likely to occur in a patient with *Legionella* pneumonia than in a patient with pneumonia from another cause.

Step-by-Step Solution

1
Extract the clinical data from the vignette to determine true positives and false positives.
True Positives (TP) = 230; Total Disease Positive = 250. False Positives (FP) = 20; Total Disease Negative = 500.
These variables are required to calculate the test's sensitivity and the false positive rate.
2
Calculate the sensitivity of the immunoassay.
Sensitivity=230250=0.92Sensitivity = \frac{230}{250} = 0.92
Sensitivity measures the proportion of actual positives that are correctly identified by the test.
3
Calculate the false positive rate, which is equivalent to 1−Specificity1 - Specificity.
False Positive Rate=20500=0.04\text{False Positive Rate} = \frac{20}{500} = 0.04
The positive likelihood ratio formula requires the false positive rate in its denominator.
4
Calculate the positive likelihood ratio (LR+).
LR+=0.920.04=23LR+ = \frac{0.92}{0.04} = 23
The LR+ quantifies how much more likely a positive test result is to occur in a patient with the disease compared to a patient without the disease.

Key Concept

Positive Likelihood Ratio (LR+) Calculation
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Topics

Diagnostic and Screening Test Performance

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