Lance Armstrong's Doping Claim: A Probabilistic Calculation

This Sunday CBS' 60 Minutes did an expose on 7 time Tour de France winner Lance Armstrong where his former teammate and close friend Tyler Hamilton accuse him on camera (and the piece also says that teammate George Hincapie testified this under oath secretly to a grand jury) of using the performance enhancing drug erythropoietin or EPO.  Part one of the 60 minutes interview can be seen above.  This is just the latest of many accusations made against him over the years.  His response has always been that he has been tested many times but no drugs have been found.  His most recent quote on his Facebook page is "20+ year career. 500 drug controls worldwide, in and out of competition. Never a failed test. I rest my case." 

Using probability theory, it is possible to compute the chance of him never testing positive assuming that he was using EPO.  First we look at the probability of testing positive for the drug when you are really taking EPO.  This is determined by the test maker.  Finding this information is difficult.  The World Anti Doping Agency or WADA which oversees the testing of athletes does not readily provide this data on their webpage.  They do provide testing positive rates for each of their labs worldwide.  The probability equation for one test is given by:

 P(-EPO test when using EPO) = 1 - P(+EPO test when using EPO)

A group of researchers in Denmark in 2008 gave EPO to a group of eight male non athlete volunteers, put them on exercise tests, took urine and blood samples from them, sent them to two WADA labs.  Post exercise one lab had no positive tests and another had 8 positives out of 40 samples or 20%.  Using the better case scenario lab (WADA does not agree with these results) that would mean that if this probability were accurate across his 20 year career and the outcome of each test was independent of the others, the probability of testing negative on each test over this period when he was using EPO equals 

P(-EPO test when using EPO on one test)^(Number of tests)

Where ^ means raised to the power of the number of tests.  The logic is like when you toss a coin once the chance of it coming up heads is 0.5.  If you toss it twice the chance of it coming up heads twice is 

0.5 x 0.5= (0.5)^2 = 0.25 = 1/4.

Plugging the Danish probability into the equations above there is an 80% chance of him testing negative when he was using EPO on one test and the chance of him testing negative on 500 tests is (.80)^500.  You can plug that into a calculator to see that that is a really small probability of him always lying (3.50*10^-49 to be exact).

This example is of course an oversimplification.  The accuracy of screening tests does change over time as do doping drugs and masking agents.  This does not prove conclusively that Lance Armstrong never used EPO but it does illustrate how hard it would be to hide if WADA were doing an adequate job even with a test with 20% accuracy.  Part II of Hamilton's interview with 60 Minutes (abbreviated version below) can be seen here where Hamilton claims that Armstrong did fail a test in 2001 in the Tour of Switzerland. The report does raise questions about the integrity of WADA.

In the worst case scenario where there is a 0.1% chance of getting caught when doping on one test when doping and thus a 99.9% chance of not getting caught when doping on one test then there would be a 61% chance of never getting caught when doping across 500 tests.  If Tyler Hamilton is right then Lance Armstrong falls into that 39% who did get caught and had it covered up.

**Update**

The news that Armstrong has been stripped of his seven tour titles and all of his other titles dating back to 1998 shouldn't be that surprising.  The interviews that I posted last year are available in abbreviated version above (they were originally available in full format) from CBS with no mention of his alleged failed 2001 drug test.  The full transcript of the 60 Minutes interview with Hamilton can be read here.  On CBS This Morning there was a discussion of the charges also with no mention of the 2001 failed drug test.  It seems strange that there is no mention of this as this allegation should not be hard to check out.  Armstrong was cleared of criminal wrongdoing.

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