Statistical question - can you answer this?

[LinusKS]

Suppose you're at the doctor's office for your annual exam, and the doc mentions she's testing you for rare disease.

"Don't worry," she says, as she's drawing your blood, "I'm sure you don't have it. But the test is very accurate, and your insurance will cover it."

A few days later, you get the bad news.

The test is positive. "I'm sorry," the doc says, "There's no treatment. Only one in a million have this disease, but the test is very accurate - 99.5%. It will only give a false positive 0.5% of the time."

The disease is fatal.

What are the chances that you are sick?

[elitegimp]

Is this a quiz where you don't want people who know how to do it to answer? Or are unsure of the answer yourself?

Because I _can_ answer it, but it's a fun one for people to try on their own, so I don't want to ruin the game.

[olavfo]

I think you've forgotten one small detail. You must assume that the test is always positive when the person actually is sick, right?

olavfo

[LinusKS]

[ QUOTE ]
Is this a quiz where you don't want people who know how to do it to answer? Or are unsure of the answer yourself?

Because I _can_ answer it, but it's a fun one for people to try on their own, so I don't want to ruin the game.

[/ QUOTE ]

It's a quiz. I think I do know the answer.

[LinusKS]

[ QUOTE ]
I think you've forgotten one small detail. You must assume that the test is always positive when the person actually is sick, right?

olavfo

[/ QUOTE ]

No, not necessarily. You can assume the test has a 0% false-negative rate if you want. Or you can assume the false-negative is the same as the false-postive -- 0.5%.

I don't think it changes the answer much.

[BruceZ]

[ QUOTE ]
I think you've forgotten one small detail. You must assume that the test is always positive when the person actually is sick, right?

olavfo

[/ QUOTE ]

That's right, that should be stated. Then the chance that you are sick is essentially <font color="white"> 1 in 5000 </font> <font color="white"> </font> (in white).

[olavfo]

Bayes theorem, right?

Define:

P(sick|pos) = probability of sickness when test is positive (the number we want to calculate)

P(pos|sick) = probability of positive test when sick (we assume the test has +/- 0.5% accuracy, so this number is 99.5%)

P(sick) = probability of being sick (1 in 1 million)

P(pos) = probability of a positive result when you carry out the test on a randomly chosen person.

P(pos) is given by the sum of the probabilities of a positive result when sick (99.5%) and a positive result when well (0.5%), weighted with the probabilities of being sick and well respectively (1/1000000 and 999999/1000000):

P(pos) = 0.995*0.000001 + 0.05*0.999999 = 0.050001

Bayes theorem tells us that

P(sick|pos)

= P(pos|sick)*P(sick)/P(pos)
= 0.995*0.000001/0.05001
= 0.00002
= 0.002%

If you assume that the test is always positive when the person is sick you get no significant difference and the probability of being sick is still 0.002%.

Now, where can I claim my prize? /images/graemlins/smirk.gif

olavfo

[LinusKS]

Good answer olavfo (and Bruce).

Sorry, no prizes. /images/graemlins/smile.gif

It is an interesting problem, though, because it comes up in rw situations where docs test people for rare - and even not-so-rare - diseases. Any time the incidence of the disease approaches the error rate of the test it turns out the test doesn't really tell you all that much - despite a seemingly high rate of accuracy.

It has an application to poker as well, if you assume only a small minority of players are long-term winners, and the "error rate" over medium size samples approaches the random chance a player is really that good.

For example - what is chance a random player is a winner given that -

- he has a valid sample of 10,000 hands showing +2BB/100

- only 5% of players actually able to sustain a rate that high

- the chance that a losing player would win 2BB/100 over 10,000 hands is 2.5%?

[elitegimp]

[ QUOTE ]
Good answer olavfo (and Bruce).

Sorry, no prizes. /images/graemlins/smile.gif

It is an interesting problem, though, because it comes up in rw situations where docs test people for rare - and even not-so-rare - diseases. Any time the incidence of the disease approaches the error rate of the test it turns out the test doesn't really tell you all that much - despite a seemingly high rate of accuracy.

It has an application to poker as well, if you assume only a small minority of players are long-term winners, and the "error rate" over medium size samples approaches the random chance a player is really that good.

For example - what is chance a random player is a winner given that -

- he has a valid sample of 10,000 hands showing +2BB/100

- only 5% of players actually able to sustain a rate that high

- the chance that a losing player would win 2BB/100 over 10,000 hands is 2.5%?

[/ QUOTE ]

Wow, I never thought of applying that to poker! Neat!

For those wondering, this is called "Simpson's Paradox", and Google (http://www.google.com/search?hl=en&amp;q=simpson%27s+paradox&amp;btnG=Google+Sea rch) turns up a ton of information about it.

[Pokerscott]

1,000,000 people take the test.
1 is sick
5,000 get a false positive.

I guess the chance is 1 in 5000 you are sick.

Pokerscott

[BruceZ]

[ QUOTE ]
For those wondering, this is called "Simpson's Paradox", and Google (http://www.google.com/search?hl=en&amp;q=simpson%27s+paradox&amp;btnG=Google+Sea rch) turns up a ton of information about it.

[/ QUOTE ]

There is quite a bit more to Simpson's Paradox. It usually involves counterintutive experimental data due to a flaw in the way test subjects are partitioned. Here is an old thread (http://archiveserver.twoplustwo.com/showthreaded.php?Cat=&amp;Number=28197&amp;page=&amp;view=&amp;sb= 5&amp;o=) I started about it with some good examples.

[elitegimp]

[ QUOTE ]
[ QUOTE ]
For those wondering, this is called "Simpson's Paradox", and Google (http://www.google.com/search?hl=en&amp;q=simpson%27s+paradox&amp;btnG=Google+Sea rch) turns up a ton of information about it.

[/ QUOTE ]

There is quite a bit more to Simpson's Paradox. It usually involves counterintutive experimental data due to a flaw in the way test subjects are partitioned. Here is an old thread (http://archiveserver.twoplustwo.com/showthreaded.php?Cat=&amp;Number=28197&amp;page=&amp;view=&amp;sb= 5&amp;o=) I started about it with some good examples.

[/ QUOTE ]

good call -- I should have said "this is an example of Simpson's paradox"

[motorholdem]

at 1 in 5000, you are drawing "dead" ...literally, I guess

[SumZero]

It is impossible to answer this exactly without knowing the false negative rate. If you assume a 0 false negative rate than for every 1,000,000 people 1 of them will test positive and have the disease, and 999,999*0.005 will test positive and be false positives. Therefore the answer is 1 / (999,999*0.005+1). That is 1 in 5000.995. Any increase in the false negative rate will make it even less likely.

[EliteNinja]

Very thought provoking thread =)

Good thing I live in Canada.
Free health care!

[tylerdurden]

[ QUOTE ]
Good thing I live in Canada.
Free health care!

[/ QUOTE ]

A) You *pay* for it, just not directly.

B) You get what you pay for.

[Dead]

[ QUOTE ]
Very thought provoking thread =)

Good thing I live in Canada.
Free health care!

[/ QUOTE ]

TANSTAAFL(There ain't no such thing as a free lunch)

The first concept that an econ student learns is that for every benefit there's also a cost. While the person who receives the benefit might not necessarily pay or even be aware of the cost, you can be sure that there is a cost paid by someone.

In this case, Canadian taxpayers foot the bill for the universal health care system.