98% accurate = 98% of people with cancer test positive
98% positive with cancer
98% negative without cancer
I asked what chance you have of >>actually<< having cancer if your test is positive.
You can't look at your body and say "Hey, I see lung cancer!". YOU don't know anything of the state of your having cancer until you take the test.
You = fail hard. How can you miss this every time I phrase it?
mugs
Lifer
- Apr 29, 2003
- 48,920
- 46
- 91
You're not very good at this. You changed the question. In the OP you stated that 98% accurate means that 98% of people who test positive have cancer. You didn't say anything about false negatives. Given that false positives and false negatives occur for different reasons, and thus are unlikely to occur at the same rate, the reader should not assume that the false negative rate is 2%.
Be explicit.
Edit: I think I mixed up fall positive and false negative, but my brain hurts and I don't feel like fixing it
Testing positive because you have cancer is different than having cancer because you tested positive.
I said
You say I said, which I didn't
One states the actual cancer status of a person relative to the test result, the other states the test result relative to the actual cancer status.
If A = "Have Cancer" and B = "Positive" then
98% of A test B
you're saying I say
98% of B then A.
He was explicit. He said that 98% of people who have cancer test positive.
However, the OP didn't give us anything about false positives. If you don't have cancer, what is the chance of testing positive? It doesn't have to be 2%. The OP fails.
e: This is why some people were saying 1.0000. Since the OP didn't give us a false positive rate, we can only assume that there are no false positives (only false negatives at a 2% rate)
Exactly. This is where Positive Predictive Power/Negative Predictive Power, specificity, and sensitivity come into play.
http://en.wikipedia.org/wiki/False_positive_paradox
You don't need to give a false positive rate if I already tell you that it's accurate X% of the time. There are 4 possibilities, since cancer is a binary condition
1. Test shows you're positive, you're positive (Accurate, 98% chance)
2. Test shows you're positive, you're negative (inaccurate, 2%)
3. Test shows you're negative, you're positive. (inaccurate, 2%)
4. Test shows you're negative, you're negative (Accurate 98%)
You don't need to give a false positive rate if I already tell you that it's accurate X% of the time. There are 4 possibilities, since cancer is a binary condition
1. Test shows you're positive, you're positive (Accurate, 98% chance)
2. Test shows you're positive, you're negative (inaccurate, 2%)
3. Test shows you're negative, you're positive. (inaccurate, 2%)
4. Test shows you're negative, you're negative (Accurate 98%)
Wrong!
Example:
A test has a "90% accuracy" rating...that is, 90% of people who take the test get the correct result.
Let's say the affliction occurs at 50%. If there is a 0% incidence of false-positives, and a 20% incidence of false-negatives, we have:
50% of the population does not have the affliction, and gets a negative result.
40% of the population has the affliction, and gets a positive result.
10% of the population has the affliction, and gets a false negative.
end result = 90% of the population gets an "accurate" result.
However, if false-positives and false-negatives are each at 10%,
45% of the population gets a true negative,
45% of the population gets a true positive,
5% of the population gets a false negative, and
5% of the population gets a false positive.
In BOTH cases, the "accuracy" is 90%, but the ways in which the test failed is different.
The example gets even better if the affliction isn't at 50%, like they never are.
In real life, the false-positive rate is almost never the same as the false-negative rate.
DrPizza
Administrator Elite Member Goat Whisperer
In addition to what mugs said about us not assuming that false positives and false negatives occur at the same rate, you also mentioned: "you receive a positive result" - You don't seriously think they give you your result based on one trial, considering those running the test are completely aware of the answer to the question you intended to pose. 
A Casual Fitz
Diamond Member
- May 16, 2005
- 4,649
- 1,018
- 136
Okay I know I chopped this up but it was just to clear things up for me. I'm reading this just like Numenorean. You take a 98% accurate test and get a positive result. You are 98% likely to have cancer, right?
Praxis1452
Platinum Member
- Jan 31, 2006
- 2,197
- 0
- 0
Mo0o
Lifer
- Jul 31, 2001
- 24,227
- 3
- 76
I dont think this can be solved since you dont know the specificity of the test, only the sensitivity. If you could provide the PPV or NPV then it's doable. But as it is, it's impossible.
Edit: I just read your answer post. Judging from your OP, I simply took the word "accurate" to imply "sensitivity" since you said "98% of those who have cancer test positive" which is a 98% sensitivity. Since you explicitly said this, I took it to mean there is no information on the specificity of the test. Although by reading your answer post you somehow assumed we would know that the specificity is also 98%
Edit: I just read your answer post. Judging from your OP, I simply took the word "accurate" to imply "sensitivity" since you said "98% of those who have cancer test positive" which is a 98% sensitivity. Since you explicitly said this, I took it to mean there is no information on the specificity of the test. Although by reading your answer post you somehow assumed we would know that the specificity is also 98%
Its amazes me ever time that on a board that tries to act so educated so many people can get such a simple probability question wrong...
Not to mention the fact that its a classic problem many people should probably have heard before.
Not to mention the fact that its a classic problem many people should probably have heard before.
As jagec pointed out, this isn't necessarily always the case. A test can be XX% "accurate" overall, but still have different levels of sensitivity and specificity.
And again you're changing your mind??
Okay...this has already been answered. You said the test is 98% accurate. If your test comes back as positive, then you have a 98% chance of having cancer. If that's incorrect, then the test isn't 98% accurate as you state.
Your response to my statement that you didn't say anything about having cancer is just you highlighting my point. You were asking about your chances of having cancer, not saying that you do have cancer which is what I was pointing out because later in the post you said this:
So now you are asking the probability of a positive test result if you DO have cancer - not what the probability is that you have cancer if you do test positive. Can you see the difference?
You = FAIL
Fayd
Diamond Member
Mo0o is right... the question should have been framed properly before people can be expected to answer it properly.
There is a problem in your solution you assumed something which we don't know to be true.
P(T|C') != 1 - P(T|C) .
100%. It says "98% who HAVE cancer test positive". "You" tested positive, so it's 100% that you have cancer.
You are also part of the 98% who have cancer that test positive, instead of being in the 2% who have cancer, but DON'T test positive.
You are also part of the 98% who have cancer that test positive, instead of being in the 2% who have cancer, but DON'T test positive.
well, for one thing I wasn't responding to Moo, I didn't even make it through the whole thread when I posted that, but I guess it seems pretty clear for me. Although I did recognize it immediately as a famous paradox. Although now ya'll are just getting me into another pet peeve of mine hanging on some tiny semantic anomaly to try to change the original question (ala. plane on a treadmill). Its supposed to be a lesson on misinterpreting statistical data similar to some of the somewhat more disturbing fallacies.
I think it would more cool though to actually discuss the implications of these paradoxes instead of just looking at the math. It has some serious impacts on real life where people often look at very high probabilities without considering the associated conditional probabilities. For example the OPs question where a person might decide out ofthe blue to have a test for cancer which is 98% accurate, and test positive and be freaking out, but the odds are TINY that they actually have cancer (which seems paradoxical at first). Also, for example if you find a fingerprint at a crime scene and can only do a test against a felony database and find a best match which is 99.9999% accurate. This might seem good at first, but if the database is a million people large then your chances of having the right guy are relatively small (with no additional evidence), especially when you consider the 300 other people in this country who would have an even closer fingerprint test. Or take a person whos husband ded from poison he had only a 1/1000 chance of being killed by naturally, and the same persons ex-husband died 5 years ago from the same thing, also 1/1000. This might seem like a CLEAR case of murder, but that is only a one in a million, and if you have a million women who have been married twice you would EXPECT that one unlucky woman would have both husbands die in such peculiar ways, so without any ADDITIONAL evidence you cannot assume guilt (though many would, and people have actually been convicted in several famous cases based purely on a poor interpretation of probability.)
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