Question

The prevalence of undetected diabetes in a population to be screened is 1.5%, and it is assumed that 25,000 persons will be screened. The screening test will measure blood serum glucose content. A value of 180 mg or higher is considered positive. The sensitivity and specificity associated with this screening test are 22.9% and 99.8%, respectively.

a. How many patients tested positive for diabetes according to the gold standard?

b. How many patients are true positives?

c. How many patients are false negatives?

d. How many patients tested negative for diabetes according to the gold standard?

e. How many patients are true negatives?

f. How many patients are false positives?

g. How many patients have blood glucose levels of less than or equal to 180 mg%?

h. How many patients have blood glucose levels of <180 mg%?

i. What is the predictive value of a positive test?

j. What is the predictive value of a negative test?

Answer 1

Before calculations we shall see what type 1 and type 2 errors are

Type1: Rejecting null hypo even though it we shouldn't (false positive)

In our case, positive denotes that person has diabetes.

We use specificity to calculate this error by multiplying '(1-specificity rate)* Exp no. of healthy patients'

Interpretation: Positive false results of patients ( People are healthy but results are positive)

Type 2:Not rejecting null hypo even if the evidence(test) is against(false negative)

In our case

We use use sensitivity to calculate this error by multipling '(1-sensitivity rate) * Exp no. of ill patients'

Interpretation: Negative false results of patients (People are ill but results are negative)

Expected no. of people having diabetes(total no. of unhealthy) = 25000 * 1.5%

Expected no. of people having diabetes = 375

We make attributes table to calculate other values and understand better

	Positive	Negative	Total
No. of ill (unhealthy)	375*22.9% = 86	375(1-22.9%)=289	375
No. of healthy	24625(1-99.8%)=49	24625*99.8%=24576	25000-375=24625
Total	135	24865	25000

I have rounded off the numbers to the nearest integer. As you can see that some values are found using formula and the rest are simply calculated by subtracting and adding.

	Positive	Negative	Total
No. of ill (unhealthy)	86	289	375
No. of healthy	49	24576	24625
Total	135	24865	25000

a. How many patients tested positive for diabetes according to the gold standard? = 135 (total ill )

b. How many patients are true positives? = 86  (ill and positive)

c. How many patients are false negatives? = 289 (ill and negative)

d. How many patients tested negative for diabetes according to the gold standard? = 24865 (total negative)

e. How many patients are true negatives? = 24576 (negative and healthy)

f. How many patients are false positives? = 49 (positive and healthy)

g. How many patients have blood glucose levels of less than or equal to 180 mg%? = 24625 (total healthy)

h. How many patients have blood glucose levels of <180 mg%? 24625

i. What is the predictive value of a positive test? = 86/135= 63.7% (positive and actually have it)

j. What is the predictive value of a negative test? = 24576/24865 = 98.8% (negative and actually are healthy)