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Question:

A pediatrician examines a population of 1,000 children with a new test to diagnose asthma.  The prevalence of asthma in this population is 10%.  The sensitivity of the pediatrician's diagnostic test is 95%, and the specificity is 80%.  Which of the following is the likelihood that a child from this population with a positive test result really has asthma?

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Explanation:

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The positive predictive value (PPV) of a diagnostic test is the probability (ie, likelihood) that an individual truly has the disease given a positive test.  PPV is equal to the number of individuals who have the disease and who test positive (ie, true positives [TP]) divided by the total number of individuals with a positive test result (TP + false positives [FP]):

PPV = TP / (TP + FP)

To determine PPV in this example, it helps to construct a 2 × 2 table with the expected data for this population according to the properties of the diagnostic test.  In this example, the pediatrician examined 1,000 children in a population with a 10% prevalence of asthma (ie, 100 children have asthma, 900 do not).  The sensitivity, which is equal to TP / (TP + false negatives [FN]), and the specificity, which is equal to true negatives (TN) / (TN + FP), are 95% and 80% respectively.  Therefore, the number of TP is 95 (ie, 100 × 0.95), the number of FN is 5 (ie, 100 − 95), the number of TN is 720 (ie, 900 × 0.80), and the number of FP is 180 (ie, 900 − 720).

The PPV can be calculated as follows:

PPV = TP / (TP + FP) = 95 / (95 + 180)

Predictive values depend on the prevalence of the disease in the study population; as the disease prevalence increases, PPV increases and NPV decreases, and vice versa.

(Choice A)  5 / (5 + 95) is the false negative rate (FNR).  FNR is equal to FN / (FN + TP) and describes the proportion of individuals with a known positive condition for which the test result is negative.  It is also known as the miss rate.

(Choice C)  95 / (95 + 5) is the test's sensitivity.  Sensitivity is equal to TP / (TP + FN) and describes the proportion of individuals with a known positive condition for which the test result is positive.  It is an intrinsic measure of the test's ability to correctly identify individuals with the disease, but by itself, it does not provide enough information to interpret a positive test result in a particular individual.

(Choice D)  720 / (720 + 180) is the test's specificity.  Specificity is equal to TN / (TN + FP) and describes the proportion of individuals with a known negative condition for which the test result is negative.

(Choice E)  720 / (720 + 5) is the test's negative predictive value (NPV).  NPV is equal to TN / (TN + FN) and describes the proportion of individuals with a negative test result who really do not have the disease.

Educational objective:
Positive predictive value is the probability that an individual has a disease given a positive test.  It is calculated as follows: true positives / (true positives + false positives).