Question

The prostate surface antigen test (PSA) is commonly used to detect early signs and symptoms of prostate cancer in men. However, the PSA test has come under quite a bit of public scrutiny for its relatively high false positive rate around 25%. Assume the prevalence of prostate cancer in men above the age over 65 is 47%. Investigators looking at the performance of the PSA test sample a group of 10,000 men over the age of 65, and come up with 5,500 positive results.What percent of men taking the test with prostate cancer will get a positive result?What is the predictive value positive for the PSA test in this example?.

Answer 1

probabilty of having cancer P(C)=0.47

and not having cancer =P(C^c) =1--0.47=0.53

let probabilty of true +ve =P(TP) and false +ve =P(FP) =0.25

hence probabilty of +ve result =P(C)*P(TP|C)*P(C^c)*P(FP|C^c)

5500/10000 =0.47*P(TP|C)+0.53*0.25

P(TP|C) =0.888298

hence 88.83% of men taking the test with prostate cancer will get a positive result

	diseased	not diseased
test positive	4175	1325	5500
test negative	525	3975	4500
	4700	5300	10000

.What percent of men taking the test with prostate cancer will get a positive result?

= 4175/4700

=0.8882978

What is the predictive value positive for the PSA test in this example?.

= 4175/5500

= 0.759090909