Question

Community health workers were conducting a screening test for diabetes. People who tested positive on the screening test were referred for follow-up testing to confirm if they had diabetes. There were 300 people who tested positive during the screening in the community. Thirty of these people were later found to not have diabetes. There were 500 people who tested negative during the screening in the community. The community health workers later found out that 20 of the people who tested negative during the screening in the community did actually have diabetes.

a. How many false positives were there on the screening test?

b. How many true negatives were there on the screening test?

c. Calculate and interpret the sensitivity of the test. (show your work)

d. Calculate and interpret the specificity of the test. (show your work)

e. Calculate and interpret the positive predictive value of the test. (show your work)

f. Calculate and interpret the negative predictive value of the test. (show your work)

Answer 1

Let's first identify the meaning of the  terms used in the question,

Sensitivity - The ability of a test to identify all positive samples as genuinely positive. For example, a test detecting all positive cases from a pool of samples has got high sensitivity. It is also termed as true positive rate. In other words, the screening test will be positive for those with disease.

Specificity - The ability of a test to identify all negative samples as genuinely negative. For example, if a test is negative for all healthy individuals in a pool of samples, the test is highly specific. It is also termed as true negative rate. In other words, the screening test will be negative for those without disease.

Positive predictive value - It is the probability that all the subjects tested positive in the test is truly having the disease. In other words, the odds of having a disease when the test is positive.

Negative predictive value - It is the probability that all subjects tested negative in the test truly don't have the disease. In other words, the odds of not having a disease when the test is negative.

Answer to the question

a. Number of false positives - 30

b. Number of true negatives - 480

c. sensitivity of the test -

the formula is number of true positives

number of true positives + number of false negatives

that is number of true positives

total number of individuals with illness

substituting the values here,   270

270+20

270

290

= 0.93 = 93% sensitive

d. specificity of the test

the formula is number of true negatives

number of true negatives + number of false positives  

that is number of true negatives

total number of individuals without the illness

substituting the values here, 480

480+30

480

510

= 0.94 = 94% specific

e. positive predictive value of the test

the formula is number of true positives

number of true positives + number of false positives

that is number of true positives

number of samples that tested positive

substituting the values 270

270+30

= 270

300

= 0.9 = 90%

Hence, if a test is positive there is 90% chance it is correct

f. Negative predictive value of the test

the formula is number of true negatives

number of true negatives + number of false negatives

that is number of negatives

number of samples that tested negative

Substituting the values 480

480+20

= 480

500

= 0.96 = 96%

Hence, if a test is negative, there is a 96% chance it is correct