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The problem with professional cycling is that so many riders have used drugs that for someone to win consistently they either have to take drugs or be so much better than anyone else that they can win without drugs. A low dose of EPO will put someone just that bit ahead of the pack.

I wonder what is the chance of a false positive?

If you take 500 tests, shouldn't the chance for at least one false positive be quite large?

So when there are no false positives, perhaps that indicates tampering with the results, or something similar?

I am just curious. Statistically, you should get false positives, one or the other, if you test enough times. Regardless of how strict you are exchanging the risk for false positives with the risk for false negatives.

I am simply considering statistics here as I have little interest in cycling in general.

Lets set up a toy situation to see how strong Armstrong's argument is, based on the fact that he "never failed a test" of 500.

Suppose he /was a doper/ and that each test is independent of the next, conditioned on the fact that we know he is a doper (not an unreasonable assumption). Then, suppose that the test only correctly identifies a doper 1% of the time. That is, 99% of dopers go uncaught. Then, since each test is independent, the probability of Armstrong passing every test is (0.99) * (0.99) * ... 500 times. (0.99)^500 = 0.00657. So, even if there is only a 1% chance of identifying him as a doper in a single test, after performing 500 of them there is only a 0.6% chance that he passed all the tests.

Your comments about the tests not being independent don't make very much sense. Clearly, even if the tests are independent but identically distributed, information about some will tell you something about new unseen tests. Imagine tossing an unfair coin 500 times. The fact that you can learn the weighting of the coin from the first 499 tosses does not make the 500th toss dependent.

Anyways, it seems like a pretty strong argument to me. If you were a betting man and you bet that he was a doper then you'd have a 99.4% chance of losing.

@Travis: But never failing any number of tests is no kind of exoneration, as Kaiser points out, if any the bullet points he lists are true of a situation. E.g. in the case where you're using a drug with no test, your equation is not (0.99)^500 = .6%, but (1.00)^500 = 100%. If you're using a measure that has no validity, it tells you nothing.

@Jason: Of course, if your tests are unable to detect whatever drugs he is using then it is clear that the chance of him failing even one of any number test trials is zero (barring false-positives).

The point I was trying to make is that, even if your tests are highly ineffective-- with a success rate of only 1% (but not zero)-- not failing any of five hundred tests is a strong sign that you do not use what the test is testing for.

The bulk of Kaiser's article is about how tests are designed to have fewer false positives and that this fact makes Armstrong's 500 negative tests meaningless. And that is garbage.

I can't stop you from believing that he has magical, undetectable drugs.

Travis: That's why the independence assumption is the key here. If I were a doper, and I pass the test, this tells me that my doping regimen is pretty good; if I pass two tests, it increases my confidence that my doping regimen is good; the more tests I pass, the more I feel good about the expertise of my doping advisors.

Another way to think about this is the fact that every athlete who have confessed and/or failed a positive test will have had a long string of negative tests prior to failing. Unless one believes these athletes (like Andy Pettite) who claim that the only time they took steroids was the time they got caught, it is very difficult to make the case that a string of negatives means much.

Well, if someone is accusing you of doping, and you haven't and you have passed 500 some doping tests, exactly what else is left to you to prove your innocence? It's not possible.

I sort of understand your point about passing a drug test isn't proof you don't take drugs, but really, what else do you want from the guy?

I stumbled upon this blog today while looking for other things. I'm loving the lucid discussion of statistics, but want to comment on something else about this post.

Athletes (cyclists, tennis players, golfers, footballers, etc) are entertainers. They are entertainers in exactly the same sense as rock musicians, classical musicians, actors, models, dancers, etc.

We expect our entertainers to do lots of "unnatural" things to their bodies in order to enhance the entertainment value of their performances - botox, breast enhancement, tooth whitening, constant dieting, etc. And our entertainers do these things to enhance their performances.

Cyclists doping are enhancing their performances. It's all entertainment, folks. They're entertainers. If doped cyclists make for a more entertaining race, then it's their prerogative to dope all day long for all I care.

A recent article in Science Daily on testing baseball balls and bats also makes this point:

"The issue of juiced balls emerged in 2000, when the first two months of the major league baseball season saw substantially more home runs than the same time the previous year.
"...
"They found the balls' coefficients of restitution -- their ability to bounce -- were nearly identical. In retrospect, Smith speculates that it may have been the players, not the balls, that were juiced."

http://is.gd/M5q3lp

This post stuck in my head and bubbled to the top again yesterday with the Casey Anthony verdict. We'll never know the truth about that situation either. But the outrage over the Anthony verdict is an important reminder that we don't live in a binary world where "found not guilty" automatically translates into "must be innocent."

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