The prevalence of asthma in the population you serve has been estimated to be 10 percent. You have read about a new test for asthma that has a sensitivity of 90% and a specificity of 99%. Given this information, if one of your patients has a positive test which of the following represents the probability of a true diagnosis of asthma?
A. 9%
B. 99%
C. 10%
D. 91%
E. 90%
Correct answer:D – 91%
Key Point: Positive predictive value (PPV) depends on the prevalence of disease and can be calculated with the sensitivity and specificity of a given test.
Discussion:The answer to this question can be calculated in two ways. Both utilize the information given to the reader in the question stem:
Method 1: Calculate the likelihood ratio from the sensitivity and specificity given by the question stem.
+LR = sensitivity/(1-specificity)
Understand that the prevalence in this case is equal to the pretest probability because we are given no additional information about the patient being tested. This means the posttest probability is the same as the positive predictive value.
Convert the pretest probability to pretest odds by the following formula,
Pretest odds = pretest probability/(1 – pretest probability)
Calculate the pretest odds to posttest odds by multiplying by the +LR,
Pretest odds x +LR = Posttest odds
Convert the posttest odds to posttest probability which we said earlier is the same as the PPV using the following formula,
PPV = Posttest odds/(1+posttest odds)
Using the numbers provided by the question stem for prevalence, sensitivity and specificity this series of calculations gives the value for PPV as 91%.
Method 2: Construct a 2 x 2 risk table to calculate the PPV by assuming a total of 1000 patients in the population (or any other easy number).
Since the prevalence is 10%, and we know the sensitivity is 90% and being defined as true positives/all positive patients we get the following risk table:
|
Asthma |
No asthma |
Positive test |
90 |
? |
Negative test |
10 |
? |
Totals |
100 |
? |
We can use the assumption of 1000 patients and the specificity to complete the remainder of the risk table:
|
Asthma |
No asthma |
Positive test |
90 |
9 |
Negative test |
10 |
891 |
Totals |
100 |
900 |
The PPV can then be calculated with the definition of true positives/all positive tests, or 90/99, or 91%.