DBrower, idiot at large

Crankyfeet

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.

TheDarkLord

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.

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...

Crankyfeet

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.

TheDarkLord

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.

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.

Crankyfeet

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.

TheDarkLord

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.

Crankyfeet

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].

RAGT

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

Cobblestones

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.

italiano

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 today)

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

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Crankyfeet

The original scenario I presented used assumed values to show the effect of changing one variable. There was never any claim that the asumed values were those corresponding to UCI dope tests.

I was trying to work out what DBrower, TBV, whatever was saying. I wasn't agreeing with him, just trying to work out his point. The math example I used after someone questioned what I meant by false positives going down as the percentage of the peloton doping goes up. In essence it is irrelevant because of masking techniques, the effective number of dopers can appear to drop to a low percentage (only a few ever test positive) even though perhaps 80% are doping.

There are some idiots here it seems who perhaps think I am a doping apologiost(??) for stating that I think Floyd feels wronged because he perhaps thought he should have passed the testosterone screen test, even though I believe he he is a 100% doper. No sympathy for Floyd from me. The IRMS test proved he was a doper. Just trying to work out his psychology. To me, he's acting like a criminal who has been done in based on a surprise illegal police raid on his house. He was caught red-handed but believes he shouldn't have been caught if the police followed procedures and the law. And it was just a speculation.

Crankyfeet

Hey VF... what's with pretending to be Italian? Like a child playing little games of espionage on a "cycling forum"...

buckybux

You can not use the same test to measure sensitivity and specificity. They have to be two different and independent tests. The example can be used as done in US Criminal Courts. The test is is the defendant Guity or Not Guilty (Not guilty does NOT mean innocent). The measure is beyond a shadow of doubt, so lots of criminals are declared not guilty. However, if we were to measure innocence, then we should use the standard beyond a shadow of doubt. Lots of defendants would fall in the no mans area where they are neither innocent nor guilty, which we call not guilty.

In statistics, on a test you develop the Alternative Hypothesis (which is what we are trying to prove), then the Null Hypothesis (which is what the test is measuring). Thus we reject the Null Hypothesis only if we feel certain
(.95, .99 or other set limit) and is is not true. Rejecting the Null Hypothesis when in is true (False Positives) is what we are controlling (alpha error). However, failing to reject the null hypothesis (beta error) we can't control. So in favor or calling someone Guilty, we make sure that we don't have false positives by only convicting when we are beyond a shadow of doubt.

So....what this means in cycling: There are a lot of dopers who are not being caught.

It is possible that if they administer a whole lot of tests, and that some may get a false positive. But note, that if the measure of false positive is .01 (99% accurate), that by doubling the tests and making them independent, then that math is .01 x .01 = .0001 probability of a false positive on both tests.
That would be only 1 out of 10,000. How many tests a year are they giving? If they are giving 10,000 tests a year, then you have a 50% probability of 1 false positive.

The last check in the system is the courts. Bottom line, I think that there are a lot of dopers, and that they manage thier biology to keep from testing positive. Problem for cyclists is that biology is a moving target, and especially during races, the biology will change, thus they get caught. Frankly, I think who gets caught is often those who have less money to spend to measure the biology (or those that get desparate for win and take a chance, such as FL).

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