Question

Sensitivity and specificity essential characteristics of medical tests. Sensitivity is the probability that the test will indicate “disease” given that the individual actually has the disease, and specificity is the probability that the test will indicate “no disease” given that the individual does not have the disease. Answer the following questions for a test with sensitivity 75% and specificity 99%. Let p denote the prevalence of the disease (i.e., proportion of the population with the disease).

(a) For p = 1%, calculate the probability that an individual with a test result indicating “disease” does not actually have the disease and the probability that an individual with a test result indicating “no disease” actually has the disease. What do these values suggest in terms of the accuracy of the test?

(b) How do the probabilities calculated in (a) change as the disease prevalence p changes? Show these relations on a graph and discuss.

Answer 1

Here sensitivity = True Positive Rate = True Positive/ Total positive = 0.75

specificity =True Negative Rate = True Negative/ Total Negative = 0.99

p = Prevalance of the disease

(a) so Here p = 1% = 0.01

We have to calculate

Pr(Not actually having disease l Test positive) = ?

For the first we have to calculate the probability of being tested positive

Pr(Tested Positive) = Sensitivity * Prevalance + (1 - Specificity) * (1 - Prevalance)

= 0.75 * 0.01 + (1 - 0.99) * ( 1 - 0.01)

= 0.0174

Pr(Not actually having disease l Test positive) = (0.01 * 0.99)/ 0.0174 = 0.569

=> Pr(Having disease l Tested Negative) = ?

Pr(Tested Negative) = 0.25 * 0.01 + 0.99 * 0.99 = 0.9826

Pr(Having Disease l Tested Negative) = (0.25 * 0.01)/0.9826 = 0.0025

Here these probabilities suggests that if the test is positive than there is 57% chance is there that the person don't have disease but if the test says negative then there is onnly 0.25% chance that the person have the disease.

(b) Here

first probability when p is the prevalance is

Pr(Not actually having disease l Test positive) = [(1-p) * 0.01)/[0.75p + 0.01 * (1-p)]

= (0.01 -0.01p)/ (0.01 + 0.74p)

Now False positive rate= Pr(Having Disease l Tested Negative) = (0.25p)/[0.25 p + 0.99 * (1-p)]