Question

For every 100 men who have PSA blood test:

10 will have an elevated PSA

90 will have a normal PSA

Of the 10 with elevatged PSA:

3 are true positive

7 are false negative

Of the 90 normal:

1 is false positive

89 are true negative

(a)P(cancer)=?

(b) How "sensitive" is PSA test? P(+|cancer)=?

(c) How "specific" is the PSA test? P(-|no cancer)=?

(d) If a man gets an elevated PSA test (+), what is the probability that he has prostate cancer? PPV= P(cancer|+)=?

Answer 1

(a)

Since out of 100, 10 will have an elevated PSA so

P(cancer) = 10 /100 = 0.10

(b)

P(+ and cancer) = 3/100

P(+|cancer) = P(+ and cancer) / P(cancer) = (3/100) / (10/100) = 0.30

(c)

P(- and no cancer) = 89 / 100

P(no cancer) = 90 /100

P(-|no cancer) = P(- and no cancer) / P(no cancer) = (89/100) / (90/100) = 0.989

(d)

P(+) = 4 /100

P(cancer | +) = P(cancer and +) / P(+) = (3/100) / (4/100) = 0.75