DBrower, idiot at large

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Cranky is right. Here is a brief explanation of positive predictive value. http://en.wikipedia.org/wiki/Positive_predictive_value I don't disagree with the concept. But his numbers cannot be right. To give an analogy - let us have a testosterone test which defines "+ve" when the T/E ratio is say 1.5 or something low like that. It will probably catch 95% of the cheaters in addition to giving a very large false positive rate. By Cranky's definition, the test is 95% accurate, while I would have to disagree.

Au contraire.

P(doping/+ve test) is dependent on P(doper).

That was my point in the example I gave. No, it isn't. It is only dependent on the test. Just in case the use of the term "doper" is unclear in my text, what I was referring to as P(doper/+ve test) = probability that the rider who has triggered the positive test is really cheating. It is 1 minus the false positive rate, which is the probability that a positive test is triggered by a rider who is not doping. P(doper) as in fraction of peloton doping only comes in when you calculate the actual incidence of doping positives.

If the criteria for positives in the test is made more stringent, then it means that any positives determined by the test are more likely to be from real cheaters, which is equivalent to P(cheating/+ve test) going closer to unity. This is the best I can explain this thing; I hope I have cleared some ambiguity.

Go to the extremes - it's easy to see that the probabilities must be dependent upon p(doper). Let's take the probability of a false positive or a false negative with the Bruyneel-test (look into the eyes):

a: All are doped:

Then a false positive is simply not possible (p=0)

b: None are doped:

Then a false negative is simply not possible (p=0)

Assuming the test is the least bit valid, can we all agree that when you have a low percentage of true positives it is unlikely that you will have very many false positives? Add to this the fact that there has been almost certainly a large amount of false negatives, it further decrease the probability that a positive woud be false rather than true.

It also means the "I never tested positive" offers little assurance of a rider being clean.

So the only way the doping appologists can hide behind a generalized false positive argument would be to assert that in reality there was very little doping going on and hence few cases of false negatives, which at this point, seems simply silly, no?

Go to the extremes - it's easy to see that the probabilities must be dependent upon p(doper). Let's take the probability of a false positive or a false negative with the Bruyneel-test (look into the eyes):

a: All are doped:

Then a false positive is simply not possible (p=0)

b: None are doped:

Then a false negative is simply not possible (p=0)I don't think we are contradicting each other. I was referring to the properties of the test itself. Any test has (a) a chance to produce false positives (due to natural variation for example) and (b) the chance to miss true positives. The two are linked together to a large extent - make it more stringent, and you reduce the chance of false positives, but also reduce ability to catch all cheaters. These two quantities (or related quantities) are given in my posts as P(cheating given +ve test) or what I wrote as P(doping/+ve test), and P(+ve test given cheating) or what I wrote as P(+ve test/doping). These two quantities are inherent to the test.

Once you have these conditional probabilities, then, the probability of a false positive when applied to the rider sample is dependent on the fraction of doping in the sample.

Let me give one more shot at this: What we want is P(rider is clean/test is +ve). This is 1 - P(rider is dirty/test is +ve).

By Bayes theorem, P(rider is dirty/test is +ve) = P(test is +ve/rider is dirty) * P(rider is dirty) / P(test is +ve).

P(test is +ve) = P(test is +ve/rider is dirty) * P(rider is dirty) + P(test is +ve/rider is clean) * P(rider is clean).

I think so far Cranky and I are in agreement. The point where we disagree is that P(test is +ve/rider is clean) is not [1 - P(test is +ve/rider is dirty)]. IMO, these two variables are only weakly correlated, and are quantities dependent on the test. For instance, let us consider a testosterone test that triggers a positive when T/E ratio is 100. Then, whether a rider is clean or dirty, the test will not give a positive, and both P(test is +ve/rider is clean) and P(test is +ve/rider is dirty) are essentially zero.

I understand little but Cranky is right.....false positive CAN be conditined like that......funny....http://cyclingforums.com/images/smilies/biggrin.gif.....his argument sound as Dbrauer, idiot at large according to bro http://cyclingforums.com/images/smilies/biggrin.gifhttp://cyclingforums.com/images/smilies/biggrin.gif

I understand little but Cranky is right.....false positive CAN be conditined like that......funny....http://cyclingforums.com/images/smilies/biggrin.gif.....his argument sound as Dbrauer, idiot at large according to bro http://cyclingforums.com/images/smilies/biggrin.gifhttp://cyclingforums.com/images/smilies/biggrin.gifTroll.

Troll.Sciense star….gazer..check TBV yourself....easy
http://cyclingforums.com/images/smilies/tongue.gifhttp://cyclingforums.com/images/smilies/biggrin.gifhttp://cyclingforums.com/images/smilies/biggrin.gifhttp://cyclingforums.com/images/smilies/biggrin.gif

Let me give one more shot at this: What we want is P(rider is clean/test is +ve). This is 1 - P(rider is dirty/test is +ve).

By Bayes theorem, P(rider is dirty/test is +ve) = P(test is +ve/rider is dirty) * P(rider is dirty) / P(test is +ve).

P(test is +ve) = P(test is +ve/rider is dirty) * P(rider is dirty) + P(test is +ve/rider is clean) * P(rider is clean).

I think so far Cranky and I are in agreement. The point where we disagree is that P(test is +ve/rider is clean) is not [1 - P(test is +ve/rider is dirty)]. IMO, these two variables are only weakly correlated, and are quantities dependent on the test. For instance, let us consider a testosterone test that triggers a positive when T/E ratio is 100. Then, whether a rider is clean or dirty, the test will not give a positive, and both P(test is +ve/rider is clean) and P(test is +ve/rider is dirty) are essentially zero.Agreed. One clearly cannot calculate or estimate P(test is +ve/rider is clean) as [1 - P(test is +ve/rider is dirty)]. It's the other way around: A higher cut-off would yield both a less probability of a false positive, and a less probability of a correct positive.

The probability that a positive test is from a clean rider is in simplified(?) language

% false positive * % clean riders / [% false positive * % clean riders + % correct positive * % dirty riders]

The ting is: If a person believe the tests are shitty, then she might have a very good "statistical" reason to believe the peloton is mostly clean and that a rider with a positive test might be a clean rider.

Say she thinks the tests are producing say 0.5 % false positives from a clean rider because of natural variation (so the B-sample-test would also be positive if the lab is OK), and that the test is only catching say 10 % of the cheaters, then what should she think about the peloton and the positive tests?

Well, that depends upon how many positives there are.

If 0.5 % of the tests are positive, then all positives are false - the whole peloton is clean.
If 1 % of the tests are positive, then 5,3 % of the peloton is dopers, and 47 % of the positives are a false positive.
If 10 % of the tests are positive, all are dopers.

Let me give one more shot at this: What we want is P(rider is clean/test is +ve). This is 1 - P(rider is dirty/test is +ve).

By Bayes theorem, P(rider is dirty/test is +ve) = P(test is +ve/rider is dirty) * P(rider is dirty) / P(test is +ve).

P(test is +ve) = P(test is +ve/rider is dirty) * P(rider is dirty) + P(test is +ve/rider is clean) * P(rider is clean).

I think so far Cranky and I are in agreement. The point where we disagree is that P(test is +ve/rider is clean) is not [1 - P(test is +ve/rider is dirty)]. IMO, these two variables are only weakly correlated, and are quantities dependent on the test. For instance, let us consider a testosterone test that triggers a positive when T/E ratio is 100. Then, whether a rider is clean or dirty, the test will not give a positive, and both P(test is +ve/rider is clean) and P(test is +ve/rider is dirty) are essentially zero.The probabilities I am using are P(rider is dirty/test is +ve) = 1 - P(rider is clean/test is +ve). Given a positive test, there are only two mutually exclusive sets of possibilities. Either the rider who tests positive is in reality clean or he is dirty. The probablility that a guy who tests positive is a doper = 1 - probability that the guy who tests positive is in fact not a doper.

The (rider is clean/+ve test result) is my definition of a false positive.

But my point that I used the math to illustrate, was that the probability of a false positive goes down as the percentage who actually dope goes up. There is a dependant mathematical relationship between that variable and the incidence of false positives. That was my original point with which you seemed to have disagreement.

The probabilities I am using are P(rider is dirty/test is +ve) = 1 - P(rider is clean/test is +ve). Given a positive test, there are only two mutually exclusive sets of possibilities. Either the rider who tests positive is in reality clean or he is dirty. The probablility that a guy who tests positive is a doper = 1 - probability that the guy who tests positive is in fact not a doper.

The (rider is clean/+ve test result) is my definition of a false positive. Ok, explain the equation you used to calculate the quantity that you want, and define all quantities. There is something wrong in your math example, and it has to do with the definitions used.

But my point that I used the math to illustrate, was that the probability of a false positive goes down as the percentage who actually dope goes up. There is a dependant mathematical relationship between that variable and the incidence of false positives. That was my original point with which you seemed to have disagreement. I will agree with you about that point. The equations that I had in my last post confirm that.

Since the equations in my previous post are clear (to me at least), I shall re-do your math: Assume that the test catches 50% of the dopers; i.e. P(test is +ve/rider is dirty) = 0.5. Also, assume that the probability of false positive in the test is 1%; i.e. P(test is +ve/rider is clean) = 0.01. [Note that there are two assumptions you have to make regarding the test, not one as you did in your example.]

Scenario 1: 80% of peloton is doping. Then, P(rider is dirty/test is +ve) = 0.5*0.8 / (0.5*0.8 + 0.01*0.2) = 0.995. Thus, false positive rate of the test when applied to the riders is 0.5% - acceptable.

Scenario 2: 0.5% of peloton is doping. Then, P(rider is dirty/test is +ve) = 0.5*0.005 / (0.5*0.005 + 0.01*0.95) = 0.21, or 79% false positive rate, which is a nightmare.

So, I agree with you, but not the math you showed in your post...

But this exercise is very illuminating. It shows that if the peloton is clean overall, the probability of the test triggering a positive when the sample is clean (i.e. just due to natural variations) better be really small. 1% doesn't cut it. Now, given the number of tests that are actually done in real life, cover-ups not withstanding, I think this number is really really small in actual tests. I expect it to be less than 0.1%. I know that people aim for this to be of the order of 10^-5 in other tests (not medical/doping where I don't know these numbers). The overall P(rider is dirty/test is +ve) better be greater than 0.95 or so for a test to be acceptable...

TDL... read my post above again. The probability of a false positive is the posterior probability that a rider is clean given the prior event of a positive test. This is written as P(rider is clean/+ve test). That means out of the pool of positive tests, what percentage (by probability) of riders are expected to be clean (false positive) and what percentage are the riders expected to be dirty (true positive).

The way you have defined a false positive [P(+ve test/rider is clean)] is the probability that a positive test occurs given the event of a rider being clean. It is back to front and not the same.

And I'm struggling to see the difference in your math?? You have just assumed different values (50% specificity/accuracy for the test instead of 99% specificity/accuracy) for the variables to show the same conclusion it seems???

post edit - and I think one of your variables in scenario 2 should be 0.995 (= 1- 0.005) rather than 0.95 on a cursory skim.

TDL... read my post above again. The probability of a false positive is the posterior probability that a rider is clean given the prior event of a positive test. This is written as P(rider is clean/+ve test). That means out of the pool of positive tests, what percentage (by probability) of riders are expected to be clean (false positive) and what percentage are the riders expected to be dirty (true positive).

The way you have defined a false positive [P(+ve test/rider is clean)] is the probability that a positive test occurs given the event of a rider being clean. It is back to front and not the same. Cranky, read my posts again. I define the test's tendency to produce a false positive as P(+ve test/rider is clean). That is the property of the test. And I derive the posterior probability in my equation. Maybe the use of two terms with false probability is confusing. But please read my post again - I have defined everything.

And I'm struggling to see the difference in your math?? You have just assumed different values (50% specificity/accuracy for the test instead of 99% specificity/accuracy) for the variables to show the same conclusion it seems??? The difference is that you assume that if a test has 99% chance of triggering a positive if the rider is doping, then the test has a 1% chance of triggering a positive if the rider is clean. That is wrong as the two quantities are not related by 1 minus the other.

Here's my first post with the math and I've added in more definitions for the variables... I apologize to everyone but TDL for boring people with a debate over math.


Assume that a dope test is 99% sensitive and 99% specific. Therefore P(+ve test/rider is dirty) = 0.99



=> Scenario 1: The peleton has a low real incidence of doping and only 0.5% of cyclists are doping. Therefore P(dirty) = 0.005... and P(clean) = 0.995

=> Scenario 2: Vastly different and 80% of the peloton are doping. Let's assume that the doping corresponds to what's being tested. Therefore P(dirty) = 0.80... and P(clean) = 0.20



If we wish to determine the posterior probability that a given positive is a false positive, in each case, we can apply Bayes Theorem.



Scenario 1. (low real incidence of doping - only 0.5% dope)


Chance that a positive is in fact a real positive = P(dirty/+ve test) = [P(+ve test/dirty) x P(dirty)]/[P(+ve test/dirty) x P(dirty) + P(+ve test/clean) x P(clean)]


= (0.99 x 0.005)/[(0.99 x 0.005) + (0.01 x 0.995)]

= 0.332

Therfore... P (dirty/+ve test) = 0.332

and P(clean/+ve test) = 1 - P(dirty/+ve test) = Chance of false positive = 0.668

= 66.8%




therefore only 33.2% of positive tests are actually dopers and there is a 66.8% chance (1 - 0.332) that a positive test is a false positive.




Scenario 2. (relatively high real incidence of doping - 80% dope)

similarly as in the variables defined above in Scenario 1.


Chance that a positive is in fact a real positive = (0.99 x 0.80)/[(0.99 x 0.8) + (0.01 x 0.20)

= 0.997

therfore... P (dirty/+ve test) = 0.997

and P(clean/+ve test) = 1 - P(dirty/+ve test) = Chance of false positive = 0.003

= 0.3%


therefore 99.7% of positive tests are actually dopers and there is a 0.3% chance (1 - 0.997) that a positive test result is a false positive.

You are making the same mistake that I have been trying to point out in so many posts. You say P(+ve test/rider is dirty) = 0.99. You then assume that this implies that P(+ve test/clean) = 0.01. I have already given an example illustrating why this is wrong. But if you insist on sticking with it, well so long. This will be my last post on this math.

You are making the same mistake that I have been trying to point out in so many posts. You say P(+ve test/rider is dirty) = 0.99. You then assume that this implies that P(+ve test/clean) = 0.01. I have already given an example illustrating why this is wrong. But if you insist on sticking with it, well so long. This will be my last post on this math.99% sensitive means that the test will correctly identify a doper 99% of the time... and 99% specific means that the test will correctly identify a non-doper (clean) testing negative 99% of the time.

That means that due to the sensitivity of the test P(+ve test/doper) = 0.99 ... and P(-ve test/doper) = 0.01 [there are only two possible outcomes given the conditional prior event and they are mutually exclusive]

Likewise, due to the test's specificity, P(-ve test/clean) = 0.99 ... and P(+ve test/clean) = 0.01 [there are also only two possible mutually exclusive outcomes given the conditional event].

Yes, but there is absolutely no reason whatsoever to believe that the sensitivity and specificity are equal (and certainly the sensitivity is not .99 %).

Yes, but there is absolutely no reason whatsoever to believe that the sensitivity and specificity are equal (and certainly the sensitivity is not .99 %).

This is true and TDL was right to point this out. If you characterize the problem in terms of percentages, you can give three completely independent numbers. Sensitivity, specificity, and say the percentage of dopers. Now, if you chose the numbers as Cranky did (although in general you don't need to chose two of the numbers to be equal) then his math is correct and proves his original point.

It started in challenge. … it end in old dilema…..in context to his post, Crank correct. …….in context to testing efficency, TDL correct……different definitions due to different endpoint concerns….. one - view of accused …. another - fair testing system to most who clean.

Check this…it clear all controversy……doper and his apologist (floyd and duckstrap) argue well agaist fans who know well. (can not post link..crazy system todayhttp://cyclingforums.com/images/smilies/cool.gif)

Check dpf thread ‘How good does the testosterone test need to be?



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