Baseball Musings: Lowell Tackles Probability

January 17, 2008

Lowell Tackles Probability

World Series MVP Mike Lowell is willing to give blood if that's what it takes to be tested for human growth hormone. But only if the test is 100 percent accurate. Not 99 percent.
"If it's 99 percent accurate, that's going to be seven false positives," the Red Sox third baseman said Thursday before the annual dinner of the Boston chapter of the Baseball Writers' Association of America. "Ninety-three percent is 70 guys. That's almost three whole rosters.

"You're destroying someone's reputation. What if one of the false positives is Cal Ripken? Doesn't it put a black mark on his career?"

Mike doesn't quite get the probabilities right. This is a perfect place to use Bayes Rule. What we want to know the probability of not using HGH given a positive test, p(~HGH|+). Let's say we know this (I'm making up the numbers):

The probability of a player using HGH is 30%, p(HGH) = .3.
The probability of a player not using HGH is 70%, p(HGH) = .7.
The probability of a player testing positive given that he is using HGH is 99%, p(+|HGH) = .99.
The probability of a player testing positive given that he is not using HGH is 1%, p(+|~HGH) = .01

Note that the test is 99% accurate in both direction. It detects 99% of players who use HGH, and it detects 99% of players who don't use HGH. By Bayes Rule, p(~HGH|+) = .01*.7/((.01*.7) + (.99*.3)) = .023, or 2.3% of the positives are going to implicate clean players. In fact, you should get five false positives if you test 700 players, not seven.

You can play with the number if you like. If only 10% of players are using HGH, then the chance of a false positive is 8.3%. In that case, you end up with about six false positives when you test 700 players. The smaller the number of players who actually use HGH, the more clean players end up testing positive!

Lowell's overall point is correct. We need to know the rates of false positives and true positives to really understand the results. There will be players caught who are clean.

Posted by David Pinto at 11:15 PM | Cheating | TrackBack (0)

Comments

I think he means even if there were NO players taking HGH. The test would still return 7 (1% of 700) false positives.

He's saying, in essense. "I'm cloean. If everyone was clean, then 7 people would be falsely accused. And that's not fair to them."

Larry

Posted by: Larry at January 18, 2008 01:10 AM

I'd love to see the look on Mike Lowell's face when someone told him that he's off by a couple thanks to his misunderstanding of Bayesian probability.

And yeah, I don't understand it either, but I think he meant if there were 0% taking HGH.

Posted by: p at January 18, 2008 03:04 AM

This problem is exactly why HIV test results are treated with confidentiality; the consequences of a false positive can be devastating.

As you note, the lower the rate of occurrence of the tested-for condition, the greater the relative percentage of false positives. If the test is 99% accurate, but there's one guy in a 1000 who you're looking for, you'll catch that guy and falsely implicate nine others. (You also have a miniscule chance of missing that guy.)

Posted by: Subrata Sircar at January 18, 2008 04:53 AM

If 0% were taking HGH, there would be no need to test.

Posted by: David Pinto at January 18, 2008 06:57 AM

But Cal Ripken took 'roids...I don't understand how that helps Lowell's point.

Posted by: Gordon at January 18, 2008 07:28 AM

True, David, but how do you know unless you test? And if you test, how do you know all of the positives aren't all false? Now we've got to start talking 95% confidence intervals (although I think Lowell has made it clear that he thinks even a 99% CI wouldn't be good enough).

Posted by: mraver at January 18, 2008 08:43 AM

If such a test were developed, it would be important to study whether false positives were simply random occurrences or something specific to a person's blood chemistry. If they are random, you could virtually eliminate the chance of error by performing multiple tests before deeming a person guilty.

Also, in the medical field, doctors often use more than the results of single test to diagnose existence of a condition. Symptoms also play an important role. A positive test along with a recent injury, spike in performance, change in body structure, muscle mass, etc. could also be taken into consideration. A single positive test does not need to be the only deciding factor in determining guilt.

Lowell sounds like someone who has something to hide.

Posted by: Scott Segrin at January 18, 2008 12:31 PM

No, Lowell sounds like someone who's familiar with the limitations of blood testing. He's a testicular cancer survivor, so he most likely has regular blood work done, and knows all too well how it can't be relied upon alone.

Regardless of the correctness of his math, I totally agree with Lowell on this point.

Posted by: gordon at January 18, 2008 02:47 PM

I you are only willing to use tests that are 100% accurate, then you will never have a test for anything. If 200 players are taking HGH and a 99% accurate test comes along that cuts that number down to 20, then the test will do far more good than harm.

Posted by: Scott Segrin at January 18, 2008 06:18 PM

If you were the lucky recipient of a false positive in a system that makes first offenses public, I think you'd disagree.

Posted by: gordon at January 18, 2008 06:27 PM

How can anyone argue that finding HGH users is worth ruining the careers of 5-10 non HGH users?
I completely disagree. This is the very reason the MLBPA wanted first test failures to not be revealed to the public and just warnings and additional testing requirements for those that failed. There is almost no chance that someone that failed a 99% confidence test twice is not a user and therefore he should be both identified and punished. But the impact of false positives on the innocent should not be ignored. This is not the America I want to live in. The goal should not be to catch all the bad guys even if it means some good guys are caught too. The goal should be to catch no good guys and live with however many bid guys you can catch without falsely punishing the innocent.

Posted by: giantsrainman at January 18, 2008 08:12 PM

I don't know if I agree with the above comment. Perhaps there is something in a person's body chemistry that throws off tests. Many steroids and other PEDs are organic and just change the levels of different chemicals in the blood. You test by checking the ratio of these chemicals. If a normal ratio is 2, and a person tests at 12, he's probably on something. But what if he tests at 4? He might be someone who had an abnormal body chemistry, since we are not all alike, or he might be someone with normal chemistry and some extra-curricular enhancement. If it's the first guy, he could certainly fail a pair of 99% tests.

Larry

Posted by: Larry at January 18, 2008 11:46 PM

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