Four hundred 400 patients are enrolled in a study to evaluat

Four hundred (400) patients are enrolled in a study to evaluate the accuracy of a test for diabetes disease. 200 of the patients were diagnosed with diabetes. 160 of the patients with diabetes had a positive test as did 10 of the patients without diabetes. What is the probability (i.e., specificity) that a patient tested negative on a test given that he/she does not have diabetes?

Solution

P(That a patient have diabetes) = 200/400= 0.5

Let D denote an event that patient has diabetes then D^c denote that the patient does not have diabetes

Let T denote an event that a patient has a positive test then T^c denote an event that a patient has a negative test

P( patients with diabetes had a positive test) = P(DnT) = 160/400 = 0.4

P( patients without diabetes had a positive test) = P(D^cnT) = 10/400 = 0.025

P(a patient tested negative on a test given that he/she does not have diabetes) = P(T^c|D^c) = P(T^cnD^c) / P(D^c)

P(D^c) = 1 - P(D) = 1-0.5 = 0.5

P(T^cnD^c) = P(TuD)^c = 1- P(TuD) = 1 - [ P(T) + P(D) - P(TnD)]

P(T) = [160 + 10 ] / 400 = 0.425

Therefore, P(T^cnD^c) = P(TuD)^c = 1- P(TuD) = 1 - [ P(T) + P(D) - P(TnD)] = 1 - [ 0.425 + 0.5 - 0.4]

= 1 - [0.525] = 0.475

P(a patient tested negative on a test given that he/she does not have diabetes) = P(T^c|D^c) = P(T^cnD^c) / P(D^c)

= 0.475/0.5 = 0.95

Therefore, probability a patient tested negative on a test given that he/she does not have diabetes is 0.95