Statistics Challenge

[Zosimus]

It is commonly argued here in this forum that there is overwhelming evidence in favor of Darwinism that makes the chance that Darwinism is true quite high.

However, when I start talking to people here about statistics and calculations and working out the probabilities, most people don't seem to know that much about statistics, probabilities, or how to calculate them.

To that end, I am posing this challenge:

A certain woman goes to her doctor and gets a routine blood test. She tests positive for a disease, which we'll call Disease X. The test is 90 percent accurate, meaning that if 100 people who do not have the disease are tested, 10 of them will incorrectly test positive, and 90 of them will correctly show negative.

The doctor insists on redrawing her blood and retesting it. The woman again comes back positive for Disease X. What are the chances that the woman has disease X?

 

[Loudmouth]

According to your denial of evidence, you could repeat the test 1,000 times and you would still deny the diagnosis because it affirms the consequent.

 

[Zosimus]

According to your denial of evidence, you could repeat the test 1,000 times and you would still deny the diagnosis because it affirms the consequent.

I'm sorry, but that's not the correct answer.

 

[Papias]

You haven't given enough information. This is the classic false positive exercise. We would need to know the rate of disease X in the population. If it is very low, say, 1 in 1E6, then it is still much more likely to get 2 false positives on this rather crude test (90% is pretty bad), than to actually have the disease.

Papias

PS - you also haven't given the rate of false negatives - of 100 people who *DO* have the disease, how many will the test show as negative, and how many as positive?

 

[Loudmouth]

I'm sorry, but that's not the correct answer.

Am I wrong? Would you ever accept the diagnosis due to the fact that the tests commit the logical fallacy of affirming the consequent?

 

[selfinflikted]

Am I wrong? Would you ever accept the diagnosis due to the fact that the tests commit the logical fallacy of affirming the consequent?

 

[keith99]

You haven't given enough information. This is the classic false positive exercise. We would need to know the rate of disease X in the population. If it is very low, say, 1 in 1E6, then it is still much more likely to get 2 false positives on this rather crude test (90% is pretty bad), than to actually have the disease.

Papias

PS - you also haven't given the rate of false negatives - of 100 people who *DO* have the disease, how many will the test show as negative, and how many as positive?

Thanks for doing the heavy lifting.

I'd just add that one also needs to know just what yields a false positive. Specifically if there are any persistent conditions that give false positives. To have any chance to answer one needs to know just how independent the two tests are.

 

[bhsmte]

Am I wrong? Would you ever accept the diagnosis due to the fact that the tests commit the logical fallacy of affirming the consequent?

And besides, Zos doesn't believe in doctors.

 

[whois]

i would say 50%, she has a 50-50 chance of having the disease.

the doctor insisting on a retake is significant.

 

[Zosimus]

You haven't given enough information. This is the classic false positive exercise. We would need to know the rate of disease X in the population. If it is very low, say, 1 in 1E6, then it is still much more likely to get 2 false positives on this rather crude test (90% is pretty bad), than to actually have the disease.

Papias

PS - you also haven't given the rate of false negatives - of 100 people who *DO* have the disease, how many will the test show as negative, and how many as positive?

That is the correct answer. Before we could calculate the a posteriori probability, we would need to have an a priori probability to work with. If, for example, the woman tested positive twice for prostate cancer, we could safely work out that both tests were false positives because her a priori probability of having prostate cancer is extremely low.

P.S. The rate of false negatives is not relevant as the woman has not tested negative.

 

[Loudmouth]

That is the correct answer.

How did you rule out God creating a false positive?

Also, how can the medical test support the diagnosis? All it can do is rule out other diagnoses, correct?

 

[Zosimus]

How did you rule out God creating a false positive?

Also, how can the medical test support the diagnosis? All it can do is rule out other diagnoses, correct?

The correct answer is that there is not enough information.

Now if we wanted to delve more deeply into the analogy, we might ask about the premises. How do we know, you might ask, that the test in question is 90 percent accurate? That is a very good question. In fact, the answer to that question leads to an infinite regress.

Test A is 90% accurate, because it agrees with Test B.
Test B is accurate because it agrees with Test C.
Test C is verified by Test D... etc., etc.

Using a real life example, we might talk about AIDS, HIV, and the Western Blot.

We don't know that HIV causes AIDS.
We don't know that the Western Blot test detects HIV. In fact, we know that Western Blot tests usually show positive if the person has tuberculosis. Western Blot is never used for HIV screening in Peru, because we have one of the highest numbers of tuberculosis infections in the world. This doesn't stop people from claiming (see HIV test accuracy, results and further testing | Guides | HIV i-Base ) that Western Blot is 100% effective.

All positive laboratory tests in the UK are routinely confirmed using a second type of test called western blot that is 100% accurate.

Since all our decisions are made in an environment of uncertainty, Decision Theory is very important.

 

[sfs]

P.S. The rate of false negatives is not relevant as the woman has not tested negative.

Incorrect. You can't know the probability she has the disease unless you know how many sick people test positive. That number depends both on disease incidence and on the test sensitivity (rate of false negatives).

 

[Loudmouth]

The correct answer is that there is not enough information.

Do you think there is a 50% chance that there is a false positive due to God in the same way that you view evolution and creation?

How do we know, you might ask, that the test in question is 90 percent accurate?

Why do you even allow the use of the test in the first place when it commits the logical fallacy of affirming the consequent? For example, an ELISA test for HIV will have a colored dot for positive or a dot that is not colored for a negative. Colored dot, therefore HIV. That is affirming the consequent, is it not?

We don't know that HIV causes AIDS.
We don't know that the Western Blot test detects HIV. In fact, we know that Western Blot tests usually show positive if the person has tuberculosis. Western Blot is never used for HIV screening in Peru, because we have one of the highest numbers of tuberculosis infections in the world. This doesn't stop people from claiming (see HIV test accuracy, results and further testing | Guides | HIV i-Base ) that Western Blot is 100% effective.

So you are saying that we shouldn't accept the results of ELISA HIV tests as meaning anything, just as you do for the evidence that supports evolution?

 

[SteveB28]

From my dim recollection of using two-way tables, it is necessary to know the data for both false positives and false negatives, before reliable probabilities can be calculated. Am I correct, statisticians?

 

[Zosimus]

Incorrect. You can't know the probability she has the disease unless you know how many sick people test positive. That number depends both on disease incidence and on the test sensitivity (rate of false negatives).

The number of sick people who test positive does not depend on disease incidence. It depends entirely on how accurate the test is and the total number of people tested. At any rate, the number doesn't concern us. We are far more interested in the percentage.

 

[Loudmouth]

The number of sick people who test positive does not depend on disease incidence. It depends entirely on how accurate the test is and the total number of people tested. At any rate, the number doesn't concern us. We are far more interested in the percentage.

The accuracy of the test depends on how many test positive v. how many are positive. In order to determine how many people in the trial reall are positive, you have to know the percentage of people in the trial who really are positive.

But in the end, you don't accept the idea that any scientific test is valid.

 

[Zosimus]

Do you think there is a 50% chance that there is a false positive due to God in the same way that you view evolution and creation?

You have zero comprehension.

Why do you even allow the use of the test in the first place when it commits the logical fallacy of affirming the consequent? For example, an ELISA test for HIV will have a colored dot for positive or a dot that is not colored for a negative. Colored dot, therefore HIV. That is affirming the consequent, is it not?

Correct, therefore a positive ELISA test doesn't prove that you have AIDS or even that you are HIV positive. Multiple positive ELISA tests from different laboratories might (arguably) raise the subjective probability that a person is HIV positive, but even then it is not proof.

So you are saying that we shouldn't accept the results of ELISA HIV tests as meaning anything, just as you do for the evidence that supports evolution?

You should apply Decision theory, which depends far more on the potential outcomes than on the probability of those outcomes. Not that you understand decision theory, or even that you are interested in finding out. You are more interested in ignoring or misrepresenting standard principles of information theory so as to make it seem that the data support your preconceived notions than in anything else.

 

[Zosimus]

From my dim recollection of using two-way tables, it is necessary to know the data for both false positives and false negatives, before reliable probabilities can be calculated. Am I correct, statisticians?

Not really. You need to know false positives vs. true positives, so false negatives help us only in the sense that we can use that information to determine the true positives because the sum of false negatives and true positives is (supposedly) known, assuming, of course, that the question has enough information – something that this question did not have.

 

[Zosimus]

But in the end, you don't accept the idea that any scientific test is valid.

Liar, liar, pants on fire.