False positives

#1
Hi everyone,

This is my first post and had a basic (I think) question. Are false positives by definition a conditional probability?

I was having a debate with a classmate about the following statement: "10% of all mammograms are false positives".

To me this translated into P(+ | No Breast Cancer) where the first event is result of the mammogram and second event here is whether the patient has breast cancer or not.

A classmate claims that the statement should be written as the joint probability P(+ and No Breast Cancer) because the statement refers to ALL mammograms.

Any clarification is appreciated, thank you.
 

Dragan

Super Moderator
#2
Hi everyone,

Are false positives by definition a conditional probability?

...A classmate claims that the statement should be written as the joint probability P(+ and No Breast Cancer) because the statement refers to ALL mammograms.Any clarification is appreciated, thank you.
It's a conditional probability - not a joint probability. What you are describing is what is referred to as a Type I error. As such, it can be described as what is the probability of committing a Type I error given (or conditioned on) that the null hypothesis is true - hence a conditional probability. Note that sampling distributions associated with a statistic are derived assuming that the null hypothesis is true.
 
#3
Thanks for the clarification on this. The last part you mention sampling distributions are derived assuming null hypothesis is true - I thought sampling distributions of point estimates are typically normally distributed (assuming independence and large enough sample size) and are derived from multiple samples. How is this assuming the null is true?
 

rogojel

TS Contributor
#4
I was having a debate with a classmate about the following statement: "10% of all mammograms are false positives".

.
Hi,
actually the statement is wrong and with the wrong statement your friend is right :). You would never get a statement like 10% of all mammograms are false positives - the statement of interest is like 10% of all POSITIVE mammograms are false positives. In the first case it would be a joint probability, in the second case a conditional probability.

regards
 
#5
Hi,
actually the statement is wrong and with the wrong statement your friend is right :). You would never get a statement like 10% of all mammograms are false positives - the statement of interest is like 10% of all POSITIVE mammograms are false positives. In the first case it would be a joint probability, in the second case a conditional probability.

regards
Thanks for your reply rogojel. I think I see your point - though a little confused. This is an actual statement from an article: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1360940/

Also in my online course this statement, which in its entirety in my lecture reads "An article published in 2003 suggests that up to 10% of all mammograms are false positive."
is taught to us as a conditional probability...

This is a little concerning if the lecture is incorrect.
 
#6
@rogojel the statement JFish is referring to is in the first paragraph after the conclusion section. Im curious as well. Maybe it's implied that it is referring to positive test results? The only way it can be a false positive is if the test read positive to begin with?
 
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rogojel

TS Contributor
#7
hi,
I would not think that the lecture is incorrect, possibly this is only a bad formulation.As given, this information is not very helpful if we do not know the percentage of all positive tests, false or not.If we have that number then we would calculate the conditional probability of false positives.
 

rogojel

TS Contributor
#8
The only way it can be a false positive is if the test read positive to begin with?
hi,
you can have four groups true positives, true negatives, false positives and false negatives. The statement, literally, says that group 3 is 10% of the total.This info is not helping in understanding the quality of the test, this is why I meant that is incorrect. It is not incorrect as a mathematical statement, of course.
 
#9
I think the article is just looking at all false positives across all patients, and making it a point to try and reduce that number, "while maintaining high sensitivity" as they say. Otherwise just lowering the sensitivity of the test could lower the false positive rate. So I think you're right, this sounds like a joint probability in context.
 

rogojel

TS Contributor
#11
I think the article is just looking at all false positives across all patients, and making it a point to try and reduce that number, "while maintaining high sensitivity" as they say. Otherwise just lowering the sensitivity of the test could lower the false positive rate. So I think you're right, this sounds like a joint probability in context.

The 10% false positives is definitely wrong as a number:

False Positives with Additional Testing and Anxiety.

Magnitude of Effect: On average, 10% of women will be recalled from each screening examination for further testing, and only 5 of the 100 women recalled will have cancer.[10]

http://www.cancer.gov/types/breast/hp/breast-screening-pdq
 
#12
I guess this means only half a percent of women will be false positive? The actual number here I'm not trying to dispute, just more the interpretation of the statement as conditional or joint probability.
 

rogojel

TS Contributor
#13
I guess this means only half a percent of women will be false positive? The actual number here I'm not trying to dispute, just more the interpretation of the statement as conditional or joint probability.
I would say 10% positive overall and 5% of the positives a true positive.So, a half percent true positive. This would match the prevalence a bit better I guess.

regards