“Random”

You probably remember the NFL quarterback Colin Kaepernick who started the protest against racism in the US by kneeling during the national anthem. In an earlier post  I discussed how his statistics suggested he was being shunned by NFL teams due to his political stance. And in a joint triumph for decency and marketing, he subsequently became the current face of Nike.

Since I now follow Kaepernick on Twitter, I recently received a tweet sent by Eric Reid of the Carolina Panthers. Reid was the first player to kneel alongside Kaepernick when playing for the San Francisco 49ers. But when his contract expired in March 2018, Reid also struggled to find a new club, despite his form suggesting he’d be an easy selection. Eventually, he joined Carolina Panthers after the start of the 2018-19 season, and opened a dispute with the NFL, claiming that, like Kaepernick, he had been shunned by most teams as a consequence of his political actions. 

This was his tweet:

The ‘7’ refers to the fact that Reid had been tested seven times since joining the Panthers in the standard NFL drug testing programme, and the “random” is intended ironically. That’s to say, Reid is implying that he’s being tested more often than is plausible if tests are being carried out randomly: in other words, he’s being victimised for the stand he’s taking against the NFL

Reid is quoted as saying:

I’ve been here 11 weeks, I’ve been drug-tested seven times. That has to be statistically impossible. I’m not a mathematician, but there’s no way that’s random.

Well, let’s get one thing out of the way first of all: the only things that are statistically impossible are the things that are actually impossible. And since it’s possible that a randomised allocation of tests could lead to seven or more tests in 11 weeks, it’s certainly not impossible, statistically or otherwise. 

However… Statistics is almost never about the possible versus the impossible; yes versus no; black versus white (if you’ll excuse the double entendre). Statistics is really about degrees of belief. Does the evidence suggest one version is more likely than another? And to what extent is that conclusion reliable?

Another small technicality… it seems that the first of Reid’s drug tests was actually a mandatory test that all players have to take when signing on for a new team. So actually, the question is whether the subsequent 6 tests in 11 weeks are unusually many if the tests are genuinely allocated randomly within the team roster.

On the face of it, this is a simple and standard statistical calculation. There are 72 players on a team roster and 10 players each week are selected for testing. So, under the assumption of random selection, the probability that any one player is tested any week is 10/72. Standard results then imply that the probability of a player being selected on exactly 6 out of 11 occasions – using the binomial distribution for those of you familiar with this stuff – is around 0.16%, while the probability of being tested 6 times or more is 0.17%. On this basis, there’s only a 17 in 10,000 chance that Reid would have been tested at least as often as he has been under a genuinely random procedure, and this would normally be considered small enough to provide evidence that the procedure is not random, and that Reid has been tested unduly often.  

 

However, we need to be a bit careful. Some time ago, in an offsite talk (mentioned here) I discussed the fact that 4 members of the quant team shared the same birthday, and showed that this was apparently an infinitesimally unlikely occurrence. But by considering the fact that it would have seemed surprising for any 4 individuals in the company to share the same birthday, and that there are many such potential combinations of 4 people, the event turned out not to be so very surprising after all.

And there’s a similar issue here… Reid is just one of 72 players on the roster. It happened to be Reid that was tested unusually often, but we’d have been equally surprised if any individual player had been tested at least 6 times in eleven weeks.  Is it surprising, though, that at least one of the 72 players gets tested this often? This is tricky to answer exactly, but can easily be done by simulation. Working this way I found the probability to be around 6.25%. Still unlikely, but not beyond the bounds of plausibility. A rule-of-thumb that’s often applied – and often inappropriately applied – is that if something has less than a 5% probability of occurring by chance, it’s safe to assume that there is something systematic and not random which led to the results; bigger than 5% and we conclude that the evidence isn’t strong enough to exclude the effect just being a random occurrence. So in this case, we couldn’t rule out the possibility that the test allocations are random.

So we have two different answers depending on how the data is interpreted. If we treat the data as specific to Eric Reid, then yes, there is strong evidence to suggest he’s been tested more often than is reasonable if testing is random. But if we consider him as just an arbitrary player in the roster, the evidence isn’t overwhelming that anyone in the roster as a whole has been overly tested,

Which should we go with? Well, each provides a different and valid interpretation of the available data. I would argue – though others might see it differently – that it’s entirely reasonable in this particular case to consider the data just with regard to Eric Reid, since there is a prima facia hypothesis specifically about him in respect of his grievance case against the NFL. In other words, we have a specific reason to be focusing on Reid, that isn’t driven by a dredge through the data. 

On this basis, I’d argue that it is perfectly reasonable to question the extent to which the allocation of drugs tests in the NFL is genuinely “random”, and to conclude that there is reasonable evidence that Eric Reid is being unfairly targeted for testing, presumably for political reasons. The number of tests he has faced isn’t ‘statistically impossible’, but sufficiently improbable to give strong weight to this hypothesis. 

 

 

 

2 thoughts on ““Random”

  1. there’s a typo here I guess:

    “On this basis, there’s only a 17 in 1000 chance that Reid would have been tested…”

    should be 10 000?

    <3

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