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Test for a Deadly Disease
Posted by urikalish • 6/22/09 • Subscribe to this Discussion [RSS] • Report This Topic
Topics: math
Let’s assume there’s a deadly disease that affects 1 in every 10 people.
Let's say there’s a pretty good test for this disease which is 90% accurate (that means that if a person is sick – the test will say he’s sick in 90% of the cases, and if a person is healthy – the test will say he’s healthy in 90% of the cases).
Now, assuming you took this 90% accurate test, and got a positive result that implies that you are sick - what are the odds that you are really sick?
User Comments
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Anybody...?
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Come on people, don't you like riddles?
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Sounds like math
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90%-
Seems logical, but obviously, it's not the correct answer.
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I don't like deadly disease. -
9%
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No.
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"Now, assuming you took this 90% accurate test, and got a positive result that implies that you are sick - what are the odds that you are really sick?"
Unless this is a trick question revolving around the word "sick" -- sick = ill vs. sick = has the disease -- a 90% accurate test is 90% accurate. Therefore, if it says you have the disease, it is 90% that you have the disease.-
No trick, and no, it's not 90%.
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I'm no statistician, so you'll have to explain how it's not 90%. I thought by definition, a test that is 90% accurate is 90% accurate.
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11%
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No.
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Hint: try using an example of 100 people.
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You made my brain hurt...
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In that case if it's 90% accurate you'd be 100% dead
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in this case if "affects one in ten people" and there is a 90% you are infected (but not necessarily affected) then .9*.1=.09 or a 9% chance that you are both infected and the infection affects you. Again, I would like to see your reasoning for why you disagree, as it is very operational definition dependant, i.e. what do you mean by ill,sick,affected, etc...
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OK, I'll take you half way...
Let's say the population is 100 people.
How many of them are sick?
How many of these sick people will test positive?
How many of them are healthy?
How many of these healthy people will test positive?
Now, assuming you got a positive result - what are the odds that you are really sick?-
10 are sick, 9 will test positive. 90 are healthy, 9 of healthy test positive. 9:10 are the "ODDS" that you are really sick, I don't think you quite understand the terminology involved, sorry to be all smarty pants on you.
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unless your problem was that we were all giving you probabilities and not odds :O
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but then again, if a test is 90% accurate, there is no sense combining that with the statistics of infection rate. they are independent events. If the test is 90% accurate then 9:10 are your odds of being correctly diagnosed. If 1 of 10 are affected then 1:9 are your odds of being affected.
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As I often say, intuition is bad for your health
Let's say the population is 100 people.
How many of them are sick?
10
How many of these sick people will test positive?
9
How many of them are healthy?
90
How many of these healthy people will test positive?
9
So we got a total of 18 people tested positive on this 90% accurate test. You are definitely one of these 18, but what are the chances that you are really sick?-
18 people tested positive.
9 of them are true positives (sick and tested positive).
9 of them are false positives (healthy but tested positive).
Therefore, your chance of being in the first (sick) group is...?
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these 18 people are those who tested positive, 50% of whom are really sick. so in this case your odds are 1:2, but I think that is a dodgy of way of looking at it if you will pardon me. I don't have a PHD in stats, but I definitely took it (and got an A) for my bachellor's:)
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these are independent events, like flipping a coin and rolling dice, thus to say how do they interact? the answer is that they do not. that is my opinion at least.
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using your 100 people analogy, all 100 have to take the test for this to be valid, which they do not necessarily. This would be valid if you took 100 people who took the test and then posed this question.
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We don't really need people to take the test. It's just a simpler way (in my opinion) to calculate the odds without using probabilities and fractions.
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Although the accuracy of this test is 90%, the chances that you are really sick when getting a positive result here is only 50%!
It's very counter-intuitive, but your chances are influenced by how rare the disease is.
As I said before, intuition can be hazardous to your health, or as someone else said it - I think therefore I am wrong.
p.s. Some more counter-intuition examples:
urikalish.blogspot.com/search/label/intuition-
I think you are wrong, therefore reconsider your position based on the above argument:) Someone said that too, me:P Love the riddle though. (I may be wrong as well)
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Let's say the population is 100 people.
How many of them are sick?
100
How many of these sick people will test positive?
90
How many of them are healthy?
0
How many of these healthy people will test positive?
0
So we got a total of 90 people tested positive on this 90% accurate test. You are definitely one of these 90, but what are the chances that you are really sick? -- 100% -
Let's say the population is 100 people.
How many of them are sick?
90
How many of these sick people will test positive?
81
How many of them are healthy?
10
How many of these healthy people will test positive?
1
So we got a total of 82 people tested positive on this 90% accurate test. You are definitely one of these 82, but what are the chances that you are really sick? -- 81:1 -
I totally agree with funispower. If you take 100 people to take the test, 90 will be negative and 10 will be positive (one being a false positive), period. I don't know where you got that 18 would be positive. If 1 in every 10 people are sick, then your odds of being sick are 1:10. The odds that, being sick, you will have a positive test are 1:9. The odds that having a positive test you are really sick are 9:10 or 90:100.
You are also forgetting that no biological test is done once, precisely because they aren't 100% accurate and that introduces precision to the equation an a little more complicated biostatistic math.
Did I mention that my PhD minor is in Epidemiology and Biostatistics?-
"If you take 100 people to take the test, 90 will be negative and 10 will be positive (one being a false positive)"
- 9 true positives out of the sick population, and another 9 false positives from the healthy population.
"1 in every 10 people are sick, then your odds of being sick are 1:10."
- That is correct, your odds of being sick are 10%, but that wasn't my question. My question was, assuming you already got a positive result, what are the odds that you are really sick. That's an entirely different question and the odds for that are 50%.
"You are also forgetting that no biological test is done once"
- That may be true, but I asked about one test result. -
OK?
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Do my fancy tree and efforts to finally get the correct result mean I may claim my PhD too? From Miskatonic University, perhaps.
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my friend, ok.
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Math.
Bleah. -
This is called conditional probability. -
Let's say the population is 100 people.
How many of them are sick?
50
How many of these sick people will test positive?
45
How many of them are healthy?
50
How many of these healthy people will test positive?
5
So we got a total of 50 people tested positive on this 90% accurate test. You are definitely one of these 50, but what are the chances that you are really sick? -- 11% -
This is all like saying what are the odds you'll be killed crossing the street.
On the other side of the street, either:
You'll be alive.
or
You'll be dead.
Therefore, you've got a 50% chance.
* * *
"That is correct, your odds of being sick are 10%, but that wasn't my question. My question was, assuming you already got a positive result, what are the odds that you are really sick. That's an entirely different question and the odds for that are 50%."
- As I've shown in my three posts above, your 50% only appears if 10% of the population is sick. Any other percentage of sick people completely throws that 50% out the window. And you never said the illness strikes 10% of the population in the original post.
- Therefore, that 50% does not answer the question you asked.
- Still, if a test is 90% accurate, it is 90% accurate.-
You got a 10% of being sick in the first place, and the test may be 90% accurate, and still, after being tested positive, your odds for being really sick are none of these 2 percentages as you can see. With these specific details (10% and 90%), the answer is exactly 50%, but different initial percentages will give you a different answer than 50% as you kindly demonstrated.
p.s. I did say the illness strikes 10% of the population in my first sentence "Let’s assume there’s a deadly disease that affects 1 in every 10 people".
OK?
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"I did say the illness strikes 10% of the population in my first sentence 'Let’s assume there’s a deadly disease that affects 1 in every 10 people'."
-- Ah, so you did. You put it right in the very first sentence of your post. Mea culpa.
But I still think you asked the question wrong for the answer you were looking for.
"Now, assuming you took this 90% accurate test, and got a positive result that implies that you are sick - what are the odds that you are really sick?"
vs.
"Now, assuming you took this 90% accurate test, and got a positive result that implies that you are sick - within the group that tested positive what are the odds that you are really sick?"
The first question is asking about an individual, and the odds are 90%. The second question is asking about a group, and the odds are 50%.-
Sorry, but I don't see why the second sentence is better...
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Look at it this way: What do the odds of the test being correct for you have to do with other people being tested? How do the odds of you actually being sick change because other people are tested?
You get the test, and "score" positive for sickness. What are the odds of you really being sick? The test is 90% accurate.
Now, another person gets the test. How does their health and their test result change the odds of you being really sick? I don't see how their result changes your personal odds.
98 more people are tested. How does their health and their test results change the odds of you being really sick? I don't see how their results change your personal odds.
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Basically, if you are sick theres a 90% chance you will get a positive result
But that doesnt mean theres a 90% chance that a positive result means you're sick.
the problem, is the answer would be 90% if half the population were sick. (ie, if you contrasted 10 results of sick people, to 10 results of healthy people)-
simpler:
10 groups of 10's.
one of these groups are sick, 9 positive results.
9 of these groups are healthy, another 9 positive results.
so out of 18 positive results, half are accurate.
and you could have any one of those 18 results, you don't know.
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I think the answer is even easier to find than that.
You have two options, you are either sick, or you're not sick. What is the chance that you are sick? 50%.
No matter how many people test, no matter how reliable the test is, you will always have a 50% chance of being sick because you either are or you're not. (granted, if you are in an area where the sickness is).
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I did statistics, and spent a few months being told nothing is simple. It's ground in, I hated it lol, but I enjoyed doing this.
But to my viewpoint, if I was told this test is not very accurate, as far as I am concerned, my worries would be are I, arent I. So it would be 50/50 in my eyes.
But if the sickness was more prevalent, more people had it, it would become more likely that your positive result is true. -
That ignores prior probabilities. I used to think your answer is always 50% too until I started studying probability.
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I was being a little bit facetious
If you take a test, and the test is positive, what is the likelyhood you are sick? 50%. Kinda like when you take a pregnancy test, and the test comes out positive, what is the lieklyhood you are pregnant? 50%
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You mean, there's a 50% possibility that I'm pregnant?
I better call my folks. -
Ya, I'm with Anok. You go girl!! LOL
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^ Yeah it's not that simple, because there's the interaction between the independent events. I've taken many courses in statistics and this problem should be solved in the same way as a lie - detecter problem. You should point out the frequency of the false positives, but I'll just assume that it's 10% (100 - 90%) for the sake of this explanation.
Chances of being sick = 0.10
Chances of not being sick = 0.90
Chances of test saying you're sick = 0.90
Chances of test saying you're not sick = 0.10
Then we get the following
Chances of the test saying you are sick and being sick
(0.10 * 0.90) / (0.10 * 0.90 + 0.10 * 0.90) = 0.5
So that brings us to 50%
This is reasonable because a false positive rate of 10% is very large.
This is the same methodology as for a lie detector's actual accuracy.-
You guys are way over complicating this
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You're disturbing a sleeping mathematical dragons sleep.
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Do not meddle in the affairs of dragons, for you are crunchy, and good with ketchup.
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Your question does not make sense. You ask us 'You take a test that is 90% accurate. It's positive, so that means there is a 90% chance you are infected and the test is right.
All the other statistics don't mean shit in this question. 1 out of 10 people get infected? Sure that's fine but it doesn't affect the accuracy of the test, which remains 90% even if 5 out of 10 people get infected. Even if 10 out of 10 people get infected the accuracy of the test will always remain 90%.
So there's your answer, 90%.-
its 90% accurate, in that sick people will get a positive result.
But that doesn't mean a positive result is 90% likely to be right.
Its like if I shoot a kitten, I'm 99.9% likely to kill it.
But If I found a dead kitten, the chances it was killed by someone shooting it are less than 1%
But if I found a dead kitten, in a country where the locals shoot kittens for fun, it is more likely that the dead kitten was killed by shooting. So the likelyhood it was shot is higher. -
The actual probability is of course affected by the accuracy of the test, but also by the rareness of the disease, which is very counter-intuitive.
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The amount of sick people do affect the chances of getting false positives. The more sick people there are, the less your chances are of getting a false positive. The less sick people there are, the greater your chances of getting a false positive.
This is a different probability from the accuracy of the test.
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The odds are 1 in 10 :-)-
The odds are 50%
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I'm still not seeing how that works.
You're among 18 people who have tested positive for the disease. You're saying it's a 50% chance that you're actually sick.
Say one person in the group dies in an accident. Do your odds of being actually sick change?
Say everyone (except you) dies in a bus accident. Do your odds of being actually sick change? -
Tree diagrams?
I was about to upload photos of murdered kittens -
@Bullgrit,
No, it stays 50%, even if the other 17 people die in a weird accident. Using 100 people is just an easier way of describing the odds instead of using formulas and fractions.
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The only way these kind of calculations make sense to me is to draw them in a tree-diagram. Each node of the tree has to add up to 100% and I have to annotate each branch carefully or I end up worrying about a deadly disease for nothing.
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I want another math/probability problem like this. My brain needs the exercise. Serious. -
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I tried this with 1000 people:
Out of 1000 people, 100 of those have have the disease. The test is 90% accurate, so 90 of those 100 test positive. From the original 1000, 900 will not have the disease. Of those 900, 90 will be false positives (right? 10% of them test positive even though they don't have the disease).
This makes the total number of people who tested positive 990. Out of all of those 990 who tested positive, 10% or 99 have the disease. This means 99/990 or 0,1 or 10% is your chance of having the disease.
What am I missing?-
You added the wrong numbers togheter
The number of people who tested positive is 180 out of a 1000, not 990,
you added 900 and 90 by accident.
Its 90 true positives, plus 90 false positives. -
Thanks, I see what you're saying.
Thanks for the help!
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OK the tree diagram helps:
The probability of having the disease is equal to the amount of people who have the disease and test positive, divided by the total number of people who test positive.
The total number of people who have the disease here is 0,1 %, or 10 out of the 100. The total number of people who test positive is 0,1 x 0,9 for those who have the disease (or 9 out of the original 100), and 0,9 x 0,1 of those who do not have cancer (also 9 out of the original 100). This means the total amount of people who test positive in the original population of 100 is 18.
Finally, your answer is 9 who test positive and do have the disease divided by 18 who test positive in the total population, or 0,5 ie 50%.-
Tree diagrams are genius
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Go Poodle!!!
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Took me a while, mostly because I was too lazy to draw out the tree diagram. I'm actually more visual than abstract minded, so the tree helps me to add the right branches.
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OMG FlamingPoodle you are a hoot. A diagram!! LOLOLOLOL Love it. How long did it take you to make this?
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All this is based on an acceptance of Bayesian theory, though, which is really only of a limited practical use. Remember it is an "interpretation of the concept of probability." Echoing what Deray said above 3 days ago, this model is absolutely useless for epidemiological purposes, in fact it is kind of silly. This is because you are thinking of this situation in a vacuum.
If this was actually a real-life scenario you were describing, even if the test was only run once, you would likely see a probability of far greater than 50/50 that the positive test results are in face true positives and not false positives. This is because in reality there will be numerous factors that you haven't listed, like the fact that the infect population will be much more likely to feel symptoms that prompt them to get testing in the first place. The fact of the matter is you cannot gauge a real probability here without multiple trials.
This is what we might call a trick of math, or as has been said elsewhere above, a riddle. But when the PhD in biostatistics says you are wrong, you ought to accept it!-
Bayesian analysis has LOADS of practical use. If you have anti-spam, chances are it's using a Bayesian filter.
But when the PhD in biostatistics says you are wrong, you ought to accept it.
That's called appeal to authority. It's a logical fallacy. According to yudkowsky.net/rational/bayes , most people get Bayesian riddles wrong. Even if they are PhDs. This is because these riddles are so counter-intuitive.
Besides, I made a tree. That has to count for something. -
@diabolicomix
Sorry, but in this case, she is wrong.
p.s. I'm a computer science graduate, but also studied biology and chemistry at the university for about 2 years. Knowing personally both types of students, I estimate that about 90% of computer science students will solve this riddle correctly, but only about 10% of the biology students. Maybe you should do the math yourself and not trust your doctor when assessing your survival chances... -
Yes, and I made a tree!
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You miss the point, it isnt abou the math, and that is why the computer science students aren't in charge of public health.
You are neglecting extrinsic factors, plain and simple. This problem treats it as if the entire population is getting tested, when it is far more likely in a medical that people are only going to be coming to the doctor for a test when they exhibit symptoms or have potentially been exposed.
But even that isn't the end of it. Nobody know what other factors might be out there, that is why you need scientific tests to determine what is actually going on. Sure when you limit consideration to a few arbitrary factors you can get a bayesian solution, but what good does this do a doctor, or a patient?
Thus if you have a test that is 90% accurate, you arent going to see 50% of patients getting false positives.
I'm really not questioning the math here, and I don't find the solution to this problem counter intuitive. But it really just doesn't tell us anything useful.
Please look into the distinction between bayesian and frequentist interpretations of probability and consider this further.
My appeal to authority here is not fallacious, because she actually made her point very fluently above irrespective of her education. In any case, PHd or not, she is right.
Also, love the tree, FP. Riddles are a nice pastime but its best to leave the epidemiology to the professionals and not the bayesian statisticians. -
We were actually discussing the chances of having the disease given the above scenario and not about making public health recommendations. In this case, it's about the maths and not about the practical issues. This is just a riddle.
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Yes but by his commentary uri does not seem to realize this.
To wit, uri said: "Maybe you should do the math yourself and not trust your doctor when assessing your survival chances..."
I think I'll trust my doctor, thanks. -
Medical doctors generally do not have PhDs in mathematics or statistics.
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As FP mentioned, this is just a riddle, and an entertaining one... so in that vein I've got a riddle of my own. It's called the Monty Hall problem,
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who doesn't know what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?-
I'm going to say that it totally depends on the reason the Host Opened the Door in the first place.
If I picked the car - and the host wants me to think I picked a goat - then he wants me to second guess myself - if I switch - then I switch to the goat.
If I don't switch - then I get the car.
It's a probability issue - but I would say that it statistically makes more sense to switch --- unless you see the goat piddle - the goat that is already shown is bleating at the next door at the goat in the door you shouldn't be choosing...
But technically - the probability issue goes from 1/3 to 1/2...
Yet - the answer lies within the reason the host opens the door completely. -
Bizarrely that double posted.
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The probability of goats being on game shows is too low to answer.
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@diabolicomix,
You are wrong.
People, doctors included, really suck at conditional probabilities. I read in the link Poodle posted here that "only around 15% of doctors get it right" (and it doesn't surprise me at all since it IS counter-intuitive). You may wanna think again before trusting your doctor with your chances of actually being sick. If I were you, I would get from my doctor the numbers for overall rareness of the disease in the general population, and the accuracy of the test, but I'll do the rest of the math myself thank you very much. On second thought, just tell me what you think I've got - I'll Google the rest...-
[I would get from my doctor the numbers for overall rareness of the disease in the general population, and the accuracy of the test, but I'll do the rest of the math myself thank you very much]
Is this a joke? -
Seriously though, don't do that, for the sake of your own health.
Bayesian statistical theory isn't as hard to grasp as you seem to think it is. What may be harder to grasp is its lack of practical application. If you go to the doctor and test positive for a disease, the tidy little equation you propose isn't going to be enough to tell if its a false positive. Perhaps if you could get all the variables and probabilities, and get them accurately, you could do the math you suggest and achieve some meaningful results, but there is a lot more going on than you can fit into a math problem-- real life is far more nuanced. This is why we have the scientific method and professional doctors who are trained to properly diagnose diseases.
Riddles are one this, but this sort of pronouncement that a computer science student is somehow better equipped diagnose disease than a trained physician is reckless indeed. -
Read the link by Poodle. I didn't say don't trust your doctors at all. Just don't trust them with calculating your conditional probabilities.
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Fortunately that really doesn't matter when it comes to diagnosing an illness.
Doctors don't deal in subjective probabilities, they deal in objective probabilities.
BTW I read that link before I even made my first post to this thread, it doesn't change anything I have said thus far and it doesn't make you right. What do you want, congratulations that you understand an arcane branch of highly theoretical statistics? Congrats. Now if you get sick go to a doctor!
PS Have you noticed how you weaken your position every time you reply? -
There's nothing subjective about this, and doctors still get it wrong 85% of the time.
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Nothing subjective about Bayesian probability? Maybe you should study up a little more...
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Maybe I should, but what's subjective about 10% and 90%?
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All probability is inherently subjective, we are talking of the chances of an outcome relative to a subject. (In fact, according to wikipedia, Bayesian is synonymous with Subjective) Doctors don't deal in this kind of relational probability. The 10% probability of infection is of little consolation to the person infected, since there is a 100% chance that he is infected. Likewise the 90% chance of accuracy means nothing to the subject who gets a *false* positive.
The solution to the riddle has no bearing on medicine, so what does it matter that doctors don't understand the arcane underlying probabilistic mechanisms? Conversely, there is apparently at least one computer science student who doesn't understand the way epidemiological statistics works, but I bet it's not going to affect this student's use of bayesian statistics to make a reasonably effective spam filter. Likewise a doctor is not going to be helped by the artificial 50/50 outcome from your riddle, because in reality a positive result on a highly accurate test is going to be a good bet that the patient has the disease, not even money.
I know statistics can be fun and everything, but you really oughtn't go blundering around talking about how bad doctors are at predicting the likelihood a patient has an illness when you don't really know what you're talking about. -
You are of course wrong, but I will post a comment to your attack only in about 4 hours, since my kids wanna go to a chocolate restaurant...
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Maybe do yourself a favor and do a bit of research first. For example:
califmedicineman.blogspot.com/2005/05/probability-and-medicine.html
Some highlights:
"The dynamics of a clinical situation will determine the probability of a given patient developing a specific disease. A smoker has a higher probability of getting lung cancer than a nonsmoker, but an individual will either get it or not period. This sounds straight forward but a lot of people have problems with it....This becomes very important as physicians increasingly embrace evidence-based medicine (EBM). In the desire to cite statistics of medical outcomes (such as the chance of developing a certain disease or the likelihood that a certain treatment will work) it is very important to recognize that every patient is different. The study population of a particular study will surely have a cross-section of many different types of participants. The patient's observed probability will be closer to patients more like himself -- maybe closer in ways that weren't even imagined or assessed by the researchers."
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Here's a real simple answer for those who really need to know statistics.
FLIP A COIN.
50%
Why?
Because either you have it or you don't.
And if you are 10% of 10 (as the numbers groan down to about...) then you are half way possible of being ill.
So flip your coin - and you'll know.
And don't forget best out of 3...
Then 5.
LOL! -
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let positive be the case that you test positive for the disease and negative for case test healthy
P(sick) = 10% ; P(healthy) = 90%;
P(positive|healthy) = 10%; P(negative|healthy) = 90%;
P(positive|sick) = 90%; P(negative|sick) = 10%;
P(healthy & positive) = P(positive|healthy) * P(healthy) = 10% * 90% = 9%
P(sick & positive) = P(positive|sick) * P(sick) = 90% * 10% = 9%
P(positive) = P(healthy & positive) + P(sick & positive) = 9% + 9% = 18%
P(sick|positive) = P(sick & positive) / P(positive)
= 9% / 18% = 50%-
Well done!
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well.. I did take a prob/stat course
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Yes, but did you draw a tree? I don't think so..
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@diabolicomix
Let’s assume research shows that it is more likely for a smoker to have this fatal disease when it’s less likely for a non-smoker. If we know our patient is a smoker, then of course we’ll use the relevant odds as our basic probability; there’s no reason to use the general population statistics when there’s a more accurate probability for his specific group.
You may be 100% healthy, or a 100% sick, but you can’t tell which one is it without a test. Like many tests, it probably generates some false positives and false negatives, which is unrelated to the smoking habits of the patient.
So, knowing more on the patient may provide us with a better basic probability to begin with, but the test’s accuracy will not change. Let’s assume that this disease affects smokers at a higher rate of 1 in 5. The accuracy of the test is still 90%. A smoker takes the test and the result comes back positive. By Poodle’s article, most doctors will tell this smoker that he’s got 90% chance of dying (instead of the real 69% in this case).
Getting back to our initial example using data for the general population… If a doctor tells you that there’s a 90% chance you will die in a few weeks, when there’s actually 50% chance that you’re not even sick and still have a few decades ahead of you – that’s really bad. If the article is right, then 85% of the doctors will do exactly that! So, as you can see, this riddle is very relevant to modern medicine and you can start spreading it to every doctor on planet Earth…
p.s. I see you think I’m not very good at probabilities or biology. Maybe you’re right, but I sure fooled the system (I got 99 out of 100 in my last university statistics class and 97 out of 100 in my overall biology and chemistry degree). After about 2 years, I had to quit biology since it really kept me away from my wife and kids, so I only have one degree (in computer science). If you’re right, then that was a good decision since the world is a little safer place without me in the system spreading false medical probabilities -
I've just begun to teach myself probability, but the way I understand it is that that's the advantage of Bayesians over Frequentists: Bayesians see probabilities as degrees of belief, whereas Frequentists see them as an expression of actual occurrences of real events.
Both have their application. -
@uri, I'm sure you're very good at biology, since you are demonstrably very with math and statistics. I'm terrible with "hard science" as it were, but I certainly respect your command of it. My point has been. and maybe I have been clear enough, not that the math is wrong, it's surely right. And as FP pointed out, Bayes has its uses. But when you rely on base numbers like that, they are inherent;y drived from research. Your formula works out great if there is actually 1 in ten sick people and actually a 90% accuracy, and no other factors at play. But when you are sick and go to the doctor, this is far from the case, even those base numbers are questionable since they are derived from research (even the best of which isn't totally accurate). There are a lot more factors to consider, and it really can come down to experience and intuition of a good doctor, which can make the difference. For a good example of what I mean, check out the TV show called "House." Doctors are more like a plumber or some other technician than they are like mathematicians. So when you you say things like "Maybe you should do the math yourself and not trust your doctor when assessing your survival chances..." it implies that the math is a good way of assessing survival chances, and that isn't the case.
Also, I love this riddle, and the Monty Hall/Tiger and Bush riddles, all lots of fun and way over my head intellectually! Great threads!
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I decided not to let a good decision tree go to waste, so I blogged about this riddle:
necrofiles.blogspot.com/2009/06/bayesian-analysis-to-probably-be-or.html-
cool!
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Um so what was the answer? I didn't read everything...sorry,2 long-
I believe it was as simple as flipping a coin - 50%, either you're sick or you're not
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No!
Not a simple "either you're sick or you're not" so it's 50%. It's not simple at all, just that using these specific probabilities (10% and 90%) the answer is exactly 50%. -
Hi Uri. Minor point but a medical screening test is never described by one percentage for accuracy: there are separate percentages for false negatives and false positives (or sensitivity and specificity). Your riddle - which is a classic riddle - does rely on the assumption that for this particular test sensitivity = specificity; nothing wrong with that in a riddle but in real life that isn't usually true. -
Yes, I was trying to make things a little bit simpler, so I used the same percentages. Even in the simplest form most people get it wrong.
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The link on my blog has a riddle which does have separate accuracies for the tests.
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Ok, didn't notice this right away but let me just parse this to give you an idea of our misunderstanding uri. Uri said:
[[The accuracy of the test is still 90%. A smoker takes the test and the result comes back positive. By Poodle’s article, most doctors will tell this smoker that he’s got 90% chance of dying (instead of the real 69% in this case).]]
This is incorrect. According to poodle article, and based on your facts, most doctors would say there is a 90% chance the guy's test is right. Not a 90% chance of actually succumbing.
The thing is, a medical diagnoses is based on a whole number of things, not just the Bayesian probability of a test, which as Deray pointed out above there, is never based on a single positve test in real life. Rather diagnoses draw from a vast array including the doctor's own experience, which is ineffable and cannot be reduced to a statistical representation.
That's the thing about bayes, it is just a representation, a mental construct, and when the reference is flawed, there can be a drastic difference between reality and the bayesian prediction drawn from the reference.-
"According to poodle article, and based on your facts, most doctors would say there is a 90% chance the guy's test is right. Not a 90% chance of actually succumbing."
- Same thing. I was just making it more dramatic. Anyway, there's a 85% chance that your doctor will say you've got a pessimistic 10% of surviving, when actually you've got a semi-optimistic 50%.
p.s. And if you still think Bayes is not important to real life:
books.google.com/books?id=Ua-_5Ga4QF8C&pg=RA2-PA254&lpg=RA2-PA254&dq=o+j+si...
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82% of all statistics are made up on the spot.
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So what are the chances that the answer of 50% is made up on the spot then?
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14% chance.
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Dead or Alive - one or the other, and you better pray like halvah that it ain't the first one!
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