Question

[Positive test] A patient is tested for a virus that is believed to be present in 5% of the population. If the test result is positive, compute the chance the patient actually has the virus… a. If the false-positive rate is 4% and the false-negative rate is 1% b. If the false-positive rate is 2% and the false-negative rate is 0.3% c. If the false-positive rate is 0.1% and the test never gives a false negative. d. If the test never gives a false-positive or a false-negative.

Answer 1

a. For this case the confusion matrix will be :

		Predicted
		Positive	Negative	Total
Actual	Positive	0.04	0.01	0.05
	Negative	0.04	0.91	0.95
	Total	0.08	0.92	1

The chance the patient actually has the virus given the test result is positive =

= P(Predicted positive / Actual Positive) = P(Predicted Positive and Actual Positive)/ P(Actual Positive)

= 0.04 / 0.05 = 0.80

b. For this case the confusion matrix will be:

		Predicted
		Positive	Negative	Total
Actual	Positive	0.047	0.003	0.05
	Negative	0.02	0.93	0.95
	Total	0.067	0.933	1

The chance the patient actually has the virus given the test result is positive =

= P(Predicted positive / Actual Positive) = P(Predicted Positive and Actual Positive)/ P(Actual Positive)

= 0.047 / 0.05 = 0.94

c. For this case the confusion matrix will be:

		Predicted
		Positive	Negative	Total
Actual	Positive	0.05	0	0.05
	Negative	0.001	0.949	0.95
	Total	0.051	0.949	1

The chance the patient actually has the virus given the test result is positive =

= P(Predicted positive / Actual Positive) = P(Predicted Positive and Actual Positive)/ P(Actual Positive)

= 0.05 / 0.05 = 1

d. For this case the confusion matrix will be:

		Predicted
		Positive	Negative	Total
Actual	Positive	0.05	0	0.05
	Negative	0	0.95	0.95
	Total	0.05	0.95	1

The chance the patient actually has the virus given the test result is positive =

= P(Predicted positive / Actual Positive) = P(Predicted Positive and Actual Positive)/ P(Actual Positive)

= 0.05 / 0.05 = 1