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Difficulty: MediumDiagnostic and Screening Test Performance

An infectious disease ward at a metropolitan hospital frequently evaluates patients presenting with presumed bacterial meningitis. During the winter season, epidemiological records indicate that 20%20\% of these admitted patients are ultimately confirmed to have a bacterial etiology via standard cerebrospinal fluid (CSF) culture. To expedite initial patient management, the hospital introduces a rapid CSF polymerase chain reaction (PCR) panel. Clinical validation of this rapid PCR panel demonstrates a sensitivity of 80%80\% and a specificity of 95%95\%. If this rapid PCR panel is applied to a representative cohort of 1,0001,000 such admissions, what is the expected positive predictive value (expressed as a percentage) of a positive test result?

Answer: 80 %

Answer

80
To find the positive predictive value (PPV), we must determine the proportion of positive test results that are true positives. In a theoretical population of 1,0001,000 patients with a 20%20\% prevalence, 200200 patients actually have bacterial meningitis and 800800 do not. Using the test's sensitivity of 80%80\%, there are 160160 true positives (200×0.80200 \times 0.80). Using the specificity of 95%95\%, the false positive rate is 5%5\%, yielding 4040 false positives (800×0.05800 \times 0.05). The total number of positive test results is 200200 (160 TP+40 FP160 \text{ TP} + 40 \text{ FP}). The PPV is the ratio of true positives to total positive tests: 160200=0.80\frac{160}{200} = 0.80, which equates to 80%80\%.

Step-by-Step Solution

1
Calculate the actual number of diseased and non-diseased individuals in the cohort.
Disease Present = 200200; Disease Absent = 800800
The prevalence is 20%20\%. In a cohort of 1,0001,000 patients, 1,000×0.20=2001,000 \times 0.20 = 200 have the disease, leaving 800800 without the disease.
2
Calculate the number of True Positives (TP).
TP = 160160
Sensitivity is the proportion of diseased individuals who test positive. 200×0.80=160200 \times 0.80 = 160.
3
Calculate the number of False Positives (FP).
FP = 4040
False positives arise from the non-diseased group. Since specificity is 95%95\%, the false positive rate is 5%5\%. 800×0.05=40800 \times 0.05 = 40.
4
Calculate the Positive Predictive Value (PPV).
PPV = 80%80\%
PPV is the number of True Positives divided by the total number of positive tests. 160160+40=160200=0.80\frac{160}{160 + 40} = \frac{160}{200} = 0.80, which is 80%80\%.

Key Concept

Positive Predictive Value (PPV) depends heavily on the prevalence of the disease in the tested population, in addition to the test's sensitivity and specificity.
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Diagnostic and Screening Test Performance

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