• Feb 24th 2013, 08:44 PM
jacksun
The following is a homework question that I'm having a lot of trouble with. Despite even possessing the answer to the question, I still can't work it out for myself. If someone could kindly explain and work out the answer for me, I would very much appreciate it. Thank you.

QUESTION:
Approximately 1 in 14 men over the age of 50 has prostate cancer. The level of 'prostate specific antigen' (PSA) is used as a preliminary screening test for prostate cancer.

7% of men with prostate cancer do not have a high level of PSA. These results are known as 'false
negatives'.

75% of those men with a high level of PSA do not have cancer. These results are known as 'false
positives'.

If a man over 50 has a normal level of PSA, what are the chances that he has prostate cancer?
• Feb 24th 2013, 11:46 PM
chiro
Hey jacksun.

To get you started I'm going to ask if you can convert the statements into probabilities. For example P(High PSA|No Cancer) and P(Low PSA|Cancer) [Hint: Relate them to false positives and false negatives].
• Feb 25th 2013, 05:29 AM
jacksun
Quote:

Originally Posted by chiro
Hey jacksun.

To get you started I'm going to ask if you can convert the statements into probabilities. For example P(High PSA|No Cancer) and P(Low PSA|Cancer) [Hint: Relate them to false positives and false negatives].

Is this what you mean:

The probability that a man over the age of 50 has prostate cancer is 1/14

The probability that a man with a high PSA does not have cancer (false negative) is 7/100

The probability that a man with a low PSA does have cancer (false positive) is 75/100 = 3/4
• Feb 25th 2013, 05:26 PM
chiro
Try writing them in probability form.

It will be easier since you are trying to find P(Has Cancer|Normal PSA) in terms of the actual probability.
• Feb 25th 2013, 07:42 PM
jacksun
Quote:

Originally Posted by chiro
Try writing them in probability form.

It will be easier since you are trying to find P(Has Cancer|Normal PSA) in terms of the actual probability.

I'm sorry, but I'm afraid I'm just not getting this right now; I'm very confused.

P(Man over age 50 has cancer) = 1/14 = .07

P(No High PSA|Cancer) = .07

P(High PSA|No Cancer) = .75
• Mar 2nd 2013, 06:57 AM
jacksun
• Mar 2nd 2013, 04:02 PM
chiro
• Mar 2nd 2013, 04:27 PM
HallsofIvy
I think a simple way to handle problems like this is to imagine a specific number of cases (chosen to avoid fractions!). Here, imagine 1400 men over 50. 1400/14= 100 have prostate cancer. 7% of those, 7, will not have a "high level of PSA" which means that 100- 7= 93 men with prostate cancer will have a high level of PSA. Since "75% of those men with a high level of PSA do not have cancer", 25% will have prostate cancer: so 7 is 25%= 1/4 of all men who have a high level of PSA. That, in turn, means that there are a total of 4(7)= 28 men, of our original 1400, have a high level of PSA and, therefore, 1400- 28= 1372 of them have a "normal" level of PSA. And we have already seen that 7 men who have prostate cancer do NOT have a high level of PSA.
• Mar 2nd 2013, 04:58 PM
jacksun
Quote:

Originally Posted by HallsofIvy
so 7 is 25%= 1/4 of all men who have a high level of PSA.

But how can that be - I thought the 7, as stated earlier, represented the men who will NOT have a high level of PSA.
• Mar 2nd 2013, 05:01 PM
jacksun
By the way, I know that the final answer to this question is 0.7%
• Mar 2nd 2013, 05:47 PM
Following on Hallsofivy's suggestion, pick a population size that is easy to work with, 1400. I will be describing the different possibilities by using the letter c and p, upper case C represents having cancer, and upper case P represents having high levels of PSA. For example, Cp, would refer to a person who has cancer but low levels of PSA.

1 in 14 means there are 100 people with cancer - so Cp = 7/100, therefore CP = 93/100.

75% of people with high PSA levels do not have cancer, so 25% are CP, therefore the total high PSA numbers is 93/0.25=372, hence cP=279.

So the probability that Cp occurs is 7/(1400-372), which is approximately 0.0068.

Cheers
• Mar 2nd 2013, 06:01 PM
jacksun