The prevalence of previously undetected diabetes in a population to be screened is approximately 1.5% and it is assumed that 10,000 persons will be screened. The screening test will measure blood serum sugar content. A value of 180 mg percent or higher is considered positive. The sensitivity and the specificity associated with this screening are 22.9% and 99.8% respectively.
a. Set up a two by two table with the appropriate numbers in each cell of the table. Round to the nearest whole number, but only after you have completed all the calculations down through and including item f). (1 pt. per box including totals)
+ - Total
Total
Prevalence = 1.5%
Population = 10,000
Sensitivity = 22.9%
Specificity = 99.8%
Calculate the following values: (2 pts. each)
b. The percentage of false positives among all those without disease (the type I error rate).
c. The percentage of false negatives among all those with disease (the type II error rate).
d. The predictive value of a positive test.
e. The predictive value of a negative test.
f. Based on the calculations above, how many false positives and negatives will occur if 100,000 people are screened?
To set up the two by two table, we will use the given information and calculations.
Let's start with the table:
```
+ - Total
+
-
Total
```
Now, let's calculate the values and fill in the table:
Step a: Prevalence = 1.5% of 10,000 people = 0.015 * 10,000 = 150 people with diabetes in the population
```
+ - Total
+ ? ? 150
- ? ? ?
Total ? ? 10,000
```
Step b: The sensitivity of the test is 22.9%, which means that 22.9% of people with diabetes will test positive:
22.9% of 150 = 0.229 * 150 = 34.35 ≈ 34 people
```
+ - Total
+ 34 ? 150
- ? ? ?
Total ? ? 10,000
```
Step c: The specificity of the test is 99.8%, which means that 99.8% of people without diabetes will test negative:
99.8% of (10,000 - 150) = 0.998 * 9850 = 9838.30 ≈ 9,838 people
```
+ - Total
+ 34 ? 150
- ? 9838 ?
Total ? ? 10,000
```
Step d: The percentage of false positives among all those without disease (type I error rate) can be calculated using the specificity:
100% - Specificity = 100% - 99.8% = 0.2%
0.2% of 9850 = 0.002 * 9850 = 19.70 ≈ 20 people
```
+ - Total
+ 34 ? 150
- ? 9838 ?
Total ? 20 10,000
```
Step e: The percentage of false negatives among all those with the disease (type II error rate) can be calculated using the sensitivity:
100% - Sensitivity = 100% - 22.9% = 77.1%
77.1% of 150 = 0.771 * 150 = 115.65 ≈ 116 people
```
+ - Total
+ 34 ? 150
- ? 9838 116
Total ? 20 10,000
```
Step f: Based on the calculations above, let's calculate the number of false positives and negatives if 100,000 people are screened:
False Positives = Type I error rate * (Total - prevalence) = 0.002 * (100,000 - 150) = 197.70 ≈ 198 people
False Negatives = Type II error rate * Prevalence = 0.771 * 150 = 115.65 ≈ 116 people
```
+ - Total
+ 34 ? 150
- ? 9838 116
Total 198 20 100,000
```
So, if 100,000 people are screened, we can expect approximately 198 false positives and 116 false negatives.