Lies, Damned Lies, and Statistics

mzimmers

So...you're claiming that, after you know the test results, your chances are the same as before?

kshegunov

Yeah, pretty much, I guess.

mzimmers

So, here's the deal. As Mike Caro (a brilliant professional gambler) has observed, "in the beginning, everything was even money." In other words, lacking any other information, one's best guess as to the probability of ANYTHING is 50-50.

Now, consider the problem I posed. If all I told you was a certain population was (partially) afflicted with a disorder, and I asked you what the chances were that a given individual in that population is afflicted, your best guess would be 50-50, because you have absolutely NO other information upon which to base an estimate.

So, now I feed you another datum: the population is afflicted with an incidence of .1%. You immediately change your answer from 50-50 to 1 in 1000.

Nothing has changed except the amount of information you possess, yet you've just profoundly altered your estimate (and correctly so).

So, I ask you, why would my giving you a second datum (your test result) not cause you to further revise your answer?

kshegunov

@mzimmers said in Lies, Damned Lies, and Statistics:

So, I ask you, why would my giving you a second datum (your test result) not cause you to further revise your answer?

The second piece of information relates to the accuracy of the test, not the incidence level. The incidence level is unchanged by the reliability of the test.

I am one person, not a population to base measure on. So with some probability (99%) the test is correct and if you average the test measure you'd get that from the 0.1% of people that have the condition 99% were correctly diagnosed and 1% were incorrectly diagnosed (have had false positives). Still, this does not affect the incidence level, just the reliability of the testing.

mzimmers

But I'm not asking what the incidence level is -- I'm asking, what are the chances that you have the disorder? Your goal is to use the available information to make the best guess/estimate possible.

With no other information, your best estimate is 50-50.

With knowledge that your population has an incidence rate of .0%, your best estimate is 1 in 1000 (or 999-1 against to express it as odds).

With knowledge that your test came back positive, your best estimate is...?

kshegunov

Yeah, I got it now, but I have to point out I really hated statistics in the university and Bayes' theorem wasn't one of my favorite topics. I would have the particular disease with probability of 1% and change ...

mzimmers

I'll wait to see if anyone else wants to hazard a guess before I give the answer.

kshegunov

Okay but you do realize this is different from gambling (i.e. the lottery), where every run is independent.

JonB

@mzimmers said in Lies, Damned Lies, and Statistics:

I'll wait to see if anyone else wants to hazard a guess before I give the answer.

Can you wait 24 hours on that? I want to read & get my head around what you're saying so I can try to answer, but it's way too late tonight now .... :)

mzimmers

@JonB heh...sure, I'm not going anywhere. Anyone who can't wait for the answer can message me...

JonB

@mzimmers ... tell me tomorrow how many ppl messaged you ... :)

JonB

@mzimmers
Right, let's start my logical analysis :)

First, let me see if I've got the figures from what you have said:

Out of every 1,000 people, 1 has the affliction.
The test will always identify that one person as being afflicted.
Additionally, the test will report 10* other people as being afflicted who in fact are healthy.

[* Actually, the remaining population is 999, so really 9.99 rather than 10.0. This would affect my final figure, but I imagine you're not looking for that degree of accuracy, so my answer will be right to nearest couple of decimal places!]

Obviously I have misunderstood them I reserve the right to be corrected by you and then re-analyse! Otherwise, please continue....

So, I take the test, and it reports me positive. (I knew it! Just my luck :( This is about my smoking, isn't it?)

Well, in this case, the test has reported 11 people as positive. 1 is genuinely positive, while 10 are false positive.

My conclusion:

Before the test result I had 1 in 1,000 chance of the terminal illness you are imposing.
After the test I have a 1 in 11 chance of being the positive one, and a 10 in 11 chance of being one of the falsies.

If it helps any, you can also think of this as balls in a bag:

There is 1 black ball, which has "You're toast" on a piece of paper inside it.
There are 10 black balls, which have "Only kidding" on a piece of paper inside them.
There are 989 white balls.

You put your hand in the bag and pull out a ball. It's black :( Given that, until you open the ball and look at the piece of paper, there's a 1 in 11 chance it contains the fateful news.

Right?

======================================================

Meanwhile....
You also wrote:

As Mike Caro (a brilliant professional gambler) has observed, "in the beginning, everything was even money." In other words, lacking any other information, one's best guess as to the probability of ANYTHING is 50-50.

I don't know if there was a context in which he wrote this which you have omitted, but that's a very strange statement. Lacking any information at all, one's "best guess" of a probability should not be anything like "50-50". I can only think a gambler might think that way!

BTW, a quick analysis:

I tell you I have a bag of balls, which you cannot see.
I ask you to guess how many balls are in the bag.
This is an example of "you have absolutely NO [other] information upon which to base an estimate".
You say: There are 23 balls in the bag.
According to you/him, the odds of this being correct are 0.5.
You decide to guess again. This time you predict 587.
Again, you/he claim the odds of this being right are 0.5.
Finally, you decide to change your mind to 77.
One more time, it's 0.5 likely you're right.

3 guesses, each of which has a 0.5 chance of being right? I don't think so!

Now, we could re-analyse precisely what you mean by "one's best guess as to the probability of ANYTHING is 50-50", because perhaps you didn't have just the case above in mind.

But the point is: "lacking any other information, one's best guess as to the probability of ANYTHING is 50-50." is not a "good guess". The correct answer is: "Lacking any information, a 'probability' is simply meaningless." Probability requires some information in order to have anything to say.

J.Hilk

Ok, I give it a try myself

We have the starting position, you either have the illness or your don't, with a 0.1% chance that you have it.
The test always has a result, but there's a 1 % chance the result is the exact opposite.
it is asked only for the cases that the test says "You have it"

you have it 0.001 and the test shows it 0.99 => 0.00099
you don't have it 0.999 but the test says you have it 0.01 => 0.00999

=> 0.01098 ~ 1.1 % chance you're diagnosed with the illness when only 0.1% off all people have it ?

JonB

@J.Hilk
The question posed is: "Given that your result is reported as positive, what is the probability that you actually do have the disease?"

Are you claiming that the answer to that is your "1.1%"? I say it's ~ 1 in 11, more like "9.09%".

J.Hilk

@JonB said in Lies, Damned Lies, and Statistics:

@J.Hilk
The question posed is: "Given that your result is reported as positive, what is the probability that you actually do have the disease?"

Are you claiming that the answer to that is your "1.1%"? I say it's ~ 1 in 11, more like "9.09%".

well it is 0.099 % you have it and it is diagnosed
to 0.999 % you don't have it and it is diagnosed

=> ~10% chance you actually have it, when it is diagnosed ?

JonB

@J.Hilk
Well, your "~ 10%" is not far off my "~ 9.1%", so we're close, though I'll stick (as per my black-balls-in-bag) to my 9.1% being closer than 10%.

BTW, your:

well it is 0.099 % you have it and it is diagnosed

is slightly off. We know (before the test) there is a 0.1% chance you have the ailment, and that the test "correctly diagnoses all [actual] cases". So, depending on your phrasing, this should remain at 0.1%. The bit where you originally wrote:

you have it 0.001 and the test shows it 0.99 => 0.00099

should have read:

you have it 0.001 and the test shows it 1.0 => 0.001

Then you have:

The test always has a result, but there's a 1 % chance the result is the exact opposite.

Not quite. It does not do "the exact opposite". There is a 1% chance it reports positive when it should be negative. But the opposite is not the case: it does not report negative when it should be positive ever.

There is a 0.01% I have the disease, in which case I will deffo be told I do.
There is a 0.1% I don't have the disease, but will be told I do.
[Note that the above 2 cases are mutually exclusive, with no dependencies.]
The test will report 0.11% total positives. 11 people out of 1,000. 10 will be incorrect, 1 will be correct. You have a 1 in 11 chance of being the positive one, and a 10 in 11 chance of being one of the false ones. Period.

mzimmers

JonB nailed it. His analysis was spot-on, with one minor nit: The problem stated:

Someone devises a test for this disorder which, in correctly diagnoses all cases, but also reports a false positive exactly 1% of the time. As stated, the false positive rate is given without regard to any true positives -- it occurs at a rate of 1%. In a population, it will "lie" about 10 individuals. So, the correct answer is exactly (not nearly) 1 in 11.

Regarding the "everything is 50-50" assertion: this has its roots in philosophy as much as it does in probability, but it's still valid IMO. I'd love to hear how Mike Caro
would respond to your interesting point.

JKSH

I like algebra, so here's some notation from probability theory:

P(A): Probability that A occurs
P(A ∩ B): Probability that A and B both occur
P(A | B): Probability that A occurs, given that B occurs

In this example,

A: Have the disease
B: Get a positive test result

@JonB said in Lies, Damned Lies, and Statistics:

Out of every 1,000 people, 1 has the affliction.

P(A) = 0.001

The test will always identify that one person as being afflicted.

P(B | A) = 1

By the axiom of probability, P(B∩A) = P(B|A) * P(A) so P(B ∩ A) = 0.001

Additionally, the test will report 10* other people as being afflicted who in fact are healthy.

[* Actually, the remaining population is 999, so really 9.99 rather than 10.0. This would affect my final figure, but I imagine you're not looking for that degree of accuracy, so my answer will be right to nearest couple of decimal places!]

P(B | ¬A) = 0.01

Similarly to before, P(B ∩ ¬A) = 0.00999

Well, in this case, the test has reported 11 people as positive. 1 is genuinely positive, while 10 are false positive.

P(B) = P(B ∩ A) + P(B ∩ ¬A) so P(B) = 0.01099

My conclusion:

Before the test result I had 1 in 1,000 chance of the terminal illness you are imposing.

Yep, before the test, nothing was given, so you can only use P(A). We already know P(A) = 0.001.

After the test I have a 1 in 11 chance of being the positive one, and a 10 in 11 chance of being one of the falsies.

Given that you got a positive test result, what is the probability that you have the disease?

P(A | B) = P(A ∩ B) / P(B) = 0.001 / 0.01099 so P(A | B) = 0.0909918... which is a teensy bit more than 1 in 11.

If it helps any, you can also think of this as balls in a bag:

There is 1 black ball, which has "You're toast" on a piece of paper inside it.

There are 10 black balls, which have "Only kidding" on a piece of paper inside them.

There are 989 white balls.

You put your hand in the bag and pull out a ball. It's black :( Given that, until you open the ball and look at the piece of paper, there's a 1 in 11 chance it contains the fateful news.

Right?

Haha, awesome analogy!

As Mike Caro (a brilliant professional gambler) has observed, "in the beginning, everything was even money." In other words, lacking any other information, one's best guess as to the probability of ANYTHING is 50-50.

I don't know if there was a context in which he wrote this which you have omitted, but that's a very strange statement. Lacking any information at all, one's "best guess" of a probability should not be anything like "50-50". I can only think a gambler might think that way!

I don't think that "50-50" means "Each answer has a 50% chance to be correct". Rather, it means "Each answer has the same chance of being correct as every other possible answer".

So, if you're guessing heads or tails, there are only 2 possible answers so you have a 50% chance of getting it right. However, with the bag of balls, if the bag is big enough to hold 99 balls then there are 100 possible answers, so you have a 1% chance of getting it right.

mzimmers

JKSH got it right as well (with the same very minor glitch as JonB).

@J.Hilk said in Lies, Damned Lies, and Statistics:

well it is 0.099 % you have it and it is diagnosed

Actually 0.1% (as discussed above).

to 0.999 % you don't have it and it is diagnosed

1%.

=> ~10% chance you actually have it, when it is diagnosed ?

10 false positives, one real positive: your chances are 1 in 11, or about 9%. You were pretty close.

mzimmers

Since you guys did so well on that one, here's another: I hand you a bag, inside which are three coins. The coins appear identical, but while two are "fair," one will always land heads-up.

You pull a coin from the bag, and toss it three times. You get a head every time. What are the chances you pulled the unfair coin?

(Those who get this right might be ready for the extremely unintuitive Monte Hall problem...)