OK, quick quiz. You’re an employer who has either fallen prey to the stupid Nancy Reagan delusion or is required to profess fealty because of contracts or regulation. The emphasis is heroin, since that’s been grabbing the headlines. You have a drug test, done by one of the “certified” labs that has greased the correct palms; this test is 98% accurate for heroin detection. You randomly grab one of your workers, force her to piss in a cup, and you get a positive result. What is the probability she’s a heroin user?
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17 responses to “Drug Testing and Math”
98% is the false negative rate. It will catch 98 out of 100 heroin users. Without knowing the false positive rate I can’t do the math.
So, according to http://www.drugabuse.gov (assume it’s at least a reasonable estimate), there are about 335,000 Americans who have admitted to heroin use “in the past month”. For round numbers, figure it’s 1 in 1000 Americans. So now take 1000 Americans, one of whom is a heroin user. Since the test is 98% accurate (which I take to mean that there will be a false positive in 2% of trials), that means there will be about 20 non-heroin-using folks who test positive. So, of the 21 people that test positive (the other 98% accurately test negative), only one is a heroin user. So the chances are about 5% that your employee uses heroin, and 95% that the test was wrong. How’d I do?
*narrows Bayes*
Aw, Max! You beat me to it! My first instinct was to investigate the priors as well.
Irrelevant, what they do off the clock isn’t my concern.
Did I win?
How would one start a new post? Assuming I wanted to, that is.
This is a smarter bunch than the average mouth-breathers. I had to explain Bayes to a guy with a PhD in STEM today and he still didn’t get it. So of course, our politicians don’t have a prayer of getting it right.
That makes a red flag pop up for me. I would say chances are slim.
Immediate thought – 98% accurate for heroin detection is nonsense. What is the test? How many other substances will it detect? Does this chick have a prescription for any opiates? Anything that mimics opiates?
Average doctors don’t get that one right. I have sympathy, though – ever try to explain Monty Hall to an unbeliever? I think it’s just so counter-intuitive that unless you have specifically learned it somewhere, you have little chance of getting it right.
How would one start a new post?
First you register as a commenter, look for “login” on the column to the right. Then they approves you, then you can comment.
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I find our cognitive biases related to the reckoning of the probability of events to be fascinating.
The drug-testing thing isn’t driven by the math of harm reduction so much as the maths (Sorry, Commonwealth English pukes) of do-somethingism, lawyerism and actuarialism. IOW, Risk-Reduction Theater, just like the TSA. Also, social control and “helping” the junkies get treatment.
Max explains it very well. Even if you assume 98% accurate, the false positives absolutely overwhelm real positives if you’re drawing from a more or less random population with respect to drug use.
But most folks see the “98%” and that’s that.
Something very much like this question was a problem in my first semester Statistics class. The problem used a terminal disease as the thing the test was looking for.
I remember explaining it to a friend of mine. Smart guy. I point to him when I interact with folks that say education and smarts go hand in hand. The guy was not educated past high school, but is smart and knows a lot of stuff. He didn’t get it at first until I went through the math. Then he got it.
@Max, re the Monty Hall problem, I tried explaining it to a guy with a Six Sigma Black Belt, so presumably conversant with statistics. I finally resorted to diagramming it for him. “OK, I see what you’re saying, but I just can’t believe it.”
You know how I’ve learned to explain it? Give them 6 to 5 on a bet and play it with them until they stop being stubborn… 🙂
The other way is to say something like, Monty Hall has 1000 doors, pick one. OK, you picked door #7. Monty then removes all doors except #7 and #394, do you want to switch now?
That is an excellent way of explaining it! Many thanks.
“98% accurate” is not enough information. More importantly, what are the odds I just happen to lose that report if she’s a good employee?
God, I fucking hated six sigma. I could see it being useful in certain applications, but forcing people to do it just to check a box results in a lot of bullshit monkeywork Mickey Mouse projects.
I rather like the Monty Hall problem.