Are referees biased in favor of red uniforms?

A month or so ago, I posted about a study that -- very surprisingly -- found that teams wearing black uniforms tended to be more aggressive, picking up extra penalties. In the comments, an anonymous poster referred me to two additional studies, with even more shocking results.

First, there's a May, 2005 study in "Nature," by Russell A. Hill and Robert A. Barton. (The study isn't online, there's a story about it.)

The authors examined matches in combat sports from the 2004 Olympics. In those contests, one competitor always wears blue protective gear, and the other wears red. Which wears which is chosen randomly.

Hill and Barton found that the guy in red beat the guy in blue a statistically significant portion of the time -- about 55%, roughly the size of the home field advantage in baseball. When the opponents were of roughly similar ability (the study doesn't say how they measured this), the fighter in red won about 62% of the matches. That was significant at 1.4%.

Now, there's a follow-up study by Norbert Hagemann and Jan Leissing, called "Seeing Red." There, the authors ran an experiment to see if referee bias could be the cause of the "red beats blue" effect. They took a bunch of video clips of Tae Kwan Do matches, where one fighter was blue and the other was red. Then, they created duplicates, but electronically switched the colors. They showed the clips to experienced referees, and had the refs score the fights.

Since every clip appears twice, once in each color combination, you'd expect red and blue to score equally. But it turned out that there was indeed a bias in favor of the red combatant. The table/figure is missing from the online version of study, but the authors say that

"The competitor wearing red protective gear was awarded an average of 13% (0.94 points) more than the competitor wearing blue protective gear."

So the average score was 8-7 for the red guy. That seems like a pretty huge advantage (it's significant at .01).

I'm still shocked that this is happening; I didn't expect referees to be so biased. I'm hoping that it's not really true. I'm hoping that what's actually going on is that 100 different researchers tried this experiment, and this was the only one that came up significant at 1%.

But if not, and this is a real effect, well, at least it explains why the Habs have more Stanley Cups than God's Team.

Hat tip: anonymous

Consumer Reports: it's bad to lend people money for chemotherapy

Note: non-sports, non-numbers post. Proceed at your own risk.

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I subscribe to Consumer Reports magazine, and I trust them when they test material things. I think their engineers know what they're doing when they figure out ways to see how well microwaves work. And I believe that they are unbiased in the sense that they don't favor one brand of hair dryer over another.

But when it comes to issues that don't involve testing products in a laboratory, their opinion isn't necessarily any better than anyone else's. A case in point is an article in the July, 2008 issue called "Overdose of Debt."

Apparently financial companies have come out with new lines of credit to enable people to pay for medical treatment. Consumer Reports is aghast. Why? I'm not sure -- there's no actual argument as to why using credit to pay for medical care is a bad idea. There's anecdotes and innuendo, but no actual logic on the point.

The impression I get is that CR wishes everyone had insurance, or that government paid for all health care (and if I'm not mistaken, they have called for a Canadian-style health-care plan). But they don't address that point explicitly.

The closest I can get to an actual argument is that, if consumers didn't have the ability to pay for health care on credit, hospitals and doctors would have to provide their services for free. And patients would benefit from that kind of charity. But, again, that's not stated explicitly. And CR does write that

"Medical providers should certainly be paid for reasonable costs …"

So they don't seem to be arguing for more charity.

What they *are* arguing are several points unrelated to the issue of consumer credit as applied to health care. I'll deal with them one at a time.

1. Credit card interest rates are high.

CR tells us that one medical credit card charges "as much as" 27.99 percent interest. But so what? Isn't that what department store cards charge, and what some Visa or Mastercards charge? That's not completely out of line, especially with the "as much as" clause that CR weasels in.

And why is this a special drawback of medical credit? The implication is that it's OK to pay high interest rates for a flat-screen TV, but not for chemotherapy. Shouldn't it be the other way around? If you haved to borrow money at high rates, shouldn't it be for something that can save your life?


2. Companies offer low rates that disappear if you miss a payment.

CR writes,

"Interest rates can jump to as much as 27.99 percent retroactively. That's the rate ChaseHealthAdvance's zero-interest plan charges, for example, if you … don't pay off the debt in the promotional period. By contrast, the average fixed-rate credit card charges 11.9 percent …"

Aha! So the "up to" 27.99 percent occurs if you don't pay off the balance in time. The true situation is that the credit card will waive your interest -- all of it! -- if you pay off the balance within a certain time. That's pretty normal … here in Canada, it's what furniture and electronics stores do when they offer you zero interest. And if you have good enough credit to qualify for the 11.9% card, here's what you do: you take the zero interest, and at the end of the promotional period, you take an advance at 11.9%, pay off the debt on the other card, and start paying it off at 11.9% starting today. That way you avoid the 28% rate, you pay no retroactive interest, and you got an interest-free loan for a year for medical care that you needed! (In fact, CR actually advocates doing this, in a sidebar.)

So why aren't they *advocating* the use of credit, given that it's being offered at 0%?


3. Consumers feel pressured to borrow.

"Consumers report that they sometimes feel pressured by medical providers to finance needed medical care, in some cases when sedated or recovering from treatment."

What does this have to do with credit? What about consumers who pay cash? Don't they sometimes feel pressured by medical providers to *approve* needed medical care, for which they will have to pay? What's the difference if it's cash, cheque, or credit?

And don't doctors discuss the possibility in advance that further treatment might be required? If you put your colonoscopy on your credit card, doesn’t it cross your mind that if the doctor finds something malignant, it's going to wind up costing more? Shouldn't you discuss this possibility with your doctor before undergoing the first procedure? If you don't, why is it the credit card's fault?

And, unfortunately, God wasn't kind enough to suspend the requirement to make medical decisions "while recovering from treatment." It would be nice if, after your exploratory surgery that found cancer, you had a couple of months for your scars to heal before you had to take the next step. But cancer doesn't take time off while you heal. It's perfectly usual, if unfortunate, that you may have to make critical medical decisions while unwell.

As for the "while sedated," some doctors might be jerks that way. But what does that have to do with the financial arrangements?


4. The more patients can afford, the more expensive treatments doctors will recommend.

"Doctors and dentists have financial incentives under these arrangements to encourage patients to sign up for more expensive treatments …"

Again, what does this have to do with credit cards? Does the conflict of interest somehow disappear when the patient pays cash? (Boy, I'm glad my used car salesman doesn’t take Visa, or he'd have tried to sell me a more expensive car!)

And wouldn't doctors have more incentive to extract money from rich people, rather than poor people relying on financing?

Most importantly: if credit was not available, CR thinks that doctors would, out of necessity, prescribe cheaper treatments. But what if the more expensive treatments are better? How many patients' lives are being saved by giving them the ability to finance their treatment? Why does CR see only unnecessary expensive procedures, and not the necessary ones? How many unnecessary debts does it take, in CR's view, to balance each unnecessary death?

I see it the opposite way CR sees it. The availability of credit allows patients to afford more of the treatment they need. This is a BENEFIT of medical credit, not a cost. In fact, this is the main reason that patients would borrow money -- so they are more likely to get well! I am totally perplexed at why CR doesn't see this.


5. Doctors will not offer the best financing alternatives to their patients.

"Doctors and dentists have financial incentives … to steer [patients] into extended financing plans that take a smaller cut of the practitioner's fee."

So, doctors try to get as much money out of their patients as they can. So do lawyers, and electronics salespeople, and car dealers, and furniture stores. Consumers understand that they have to make the right decision on financing their TV, and not to blindly accept the first one the salesman offers; it's not a big step to apply the same diligence to financing medical care.

I can see how it might offend CR that doctors are trying to squeeze a few extra dollars out of their patients. And, to be honest, it does bother me that some patients may be too intimidated to look for financial options other than what the doctor offers. But the answer isn't to eliminate one of the patient's choices -- because, after all, it might wind up being the best choice. A better solution would be to require the doctor to lay out the financial options he offers, and give the patient enough time to evaluate them all.


6. CR has some kind of vague discomfort with using credit to pay for medical expenses.

How else can we interpret this:

"But paying for chemotherapy on credit isn't quite the same as charging a plasma TV …"

Really? It seems the same to me? Why wouldn't it be? What's the difference, apart from the fact that it appears somehow to offend CR's sensibilities?

Well, I can think of one big difference. Chemotherapy is a necessity; a plasma TV is not. If you have no other options, having a source of financing for your chemotherapy is a matter of life or death.

And that's why it's a GOOD thing. Indeed, CR immediately admits that credit is a matter of life or death!

"[credit is] … last-resort financing for uninsured or under-insured patients with urgent medical needs [says health-care activist Claudia Lennhoff] … "The consumers who come to us for help when they are struggling with [medical credit] debt are desperate people who were required to provide payment for an appendectomy or cancer treatment at the time of service.""

Exactly! The medical credit card *saved these people's lives*! And now the patients are "struggling" to pay it off. So let me get this straight:

-- these patients had medical emergencies that were life-threatening.
-- they had no money to pay for treatment.
-- the credit card companies came to the rescue and offered them financing.
-- the credit card company assumed all the risk, which means the doctor got paid in full and was willing to treat the patients.
-- the patients got the treatment they needed.
-- but the patients are so poor that they can't even pay off 2% of their debt per month, so the credit card company is probably going to take a loss.

Sure, the patient is in financial straits trying to pay off as much as he can. But the alternative was worse -- perhaps death -- as evidenced by the "desperate" (CR's word!) patient taking on the debt rather than refusing treatment. What made the patient poor was not the credit card company, but the illness. If anything, the credit card company is the only one losing on the deal, probably having to settle with the patient for less than the amount owed. The patient gains, by getting his care, and getting it cheaper (although admittedly the cost is still a financial burden). The doctor breaks even, getting paid for his work. And the credit card company takes the loss.

So why is Ms. Lenhoff slagging the credit card company? I suspect it's because she doesn't like credit card companies, and she hasn't thought it through. She adds:

"Health-care providers steered them into these finance plans with rates they didn't understand."

How does she know the patients didn't understand the rates? Are sick people dumb? Are poor people dumb? Is there any evidence that these lines of credit weren't the best option available?


7. Many patients can't take advantage of the low rates, and credit card companies lie about this.

"GE Money spokeswoman Cristy Williams says almost 80 percent of its medical-financing customers pay off the full amount in the promotional period. But … figures show almost 60% of households with regular cards carry a revolving balance."

Those numbers are fully consistent if you assume, unlike CR, that consumers have a brain and know how to use it. Patients are rationally choosing to pay off the medical card that would otherwise charge them large amounts of retroactive interest, but keeping balances on the card that charges non-retroactive low interest. This is not rocket science.


8. Patients will lose their promotional zero interest rate if they're not careful.

CR writes,

"… there is plenty to be cautious about. Chase's online marketing aimed at patients proclaims that "you can enjoy the option of paying no interest at all." It could be a good deal if you pay it off within the promotion time. But if you don't or miss a payment, you'll be hit with interest rates of up to 27.99 percent."

Wow. Free interest if you pay on time every month? I suppose you have to be "cautious" that you don't forget to mail the cheque, but that's a small bit of vigilance to receive 0% interest. Wouldn't you take a mortgage like that? I'd just hire three separate people to call me every month and remind me to pay.

Geez, the things that CR finds to complain about.


9. Doctors promoting these credit cards are abusing patient trust.

Specifically:

"Some experts worry about blurring traditional lines of responsibility. "When you have doctors promoting cards and loans with unconscionable finance terms … it raises serious ethical issues, given the trust patients place in physicians …" says Gina Calabrese, a clinical law professor at the nonoprofit Elder Law Clinic at St. John's University School of Law in New York City."

-- What makes Ms. Calabrese an "expert"? This is a matter of philosophical opinion, not fact. (Who is the abortion "expert," Roe or Wade?)

-- 28 percent is not an "unconscionable" rate, especially when (a) it's zero percent if you pay it early, and (b) it's financing an urgent medical procedure.

-- As I mentioned, there is a larger conflict of interest in that doctors can prescribe unnecessary treatment to rich patients, and make a lot more dirty money than pushing credit on poorer patients.

-- Patients place trust in physicians because the doctor is an expert, and they're not. On finance, patients are actually more 'expert' in their own financial situation than the doctor is. I am certain that I know as much about finance as my own doctor. The idea that patients need to place blind trust in doctors on non-medical matters is one I find troublesome.

-- Even if some doctors are unethical, I'd bet that the overwhelming majority of doctors who do promote those credit cards do so with their patients' welfare in mind. If you need treatment, and you don't have money, those cards are a godsend.


10. The cards are promoted to doctors as a way for them to make more money.

I think it's OK for doctors to make money, and I think it's OK for doctors to want to make sure they get paid. But CR doesn't:

"The potential conflict is underscored by how the financing plans are pitched to health care providers.

For example … "Patients are more likely to book full comprehensive treatment plans" because of the financing" … [another card ] emphasizes [to doctors] how offering the card "can significantly impact
your bottom line."

[another brochure tells doctors] patients are likely to delay treatment because "the average American has only $300 available credit on their consumer credit cards."

Right! With the cards, patients can finance all the treatment they need, doctors can be sure they'll get paid, and patients could only afford part of the treatment without them.

Isn't this a good thing, providing patients with means to receive treatment? Is it somehow a bad thing that doctors make more money because they don't have to spend valuable time as part-time bankers and collection agencies to their patients?

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Anyway, I'm getting tired and frustrated, and I'm only halfway through the article … I'm just going to quote and comment until this is done.

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"If a patient were to finance … [on a] no-interest, 18-month payment plan, the dentist would pay 13.5 percent of the total as a 'processing fee'. The processing fee for merchants is usually 2 percent or less."

It's "usually" 2 percent or less because when you use your MasterCard at Best Buy, they don't offer 0% financing for 18 months. Isn't that kind of obvious?

"If the patient opted for an extended-payment plan over two to five years, CareCredit would take only 5 percent of the dentist's fee, and the patient would pay interest at an initial annual rate of 11.9 percent that could rise to 23.9 percent if he or she failed to pay off the balance in the specified time."

See? When it's not zero percent, the doctor pays less. And 11.9% is below market for a typical credit card, which, again, is why the fee is more than two percent.

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"Doctors thus are given a financial incentive to have patients stretch out payments at double-digit interest rates from the start."

1. Virtually all credit card rates are "double digit". CR should get used to it. Also, cars normally cost "five digits" and Happy Meals cost "three digits."

2. Doctors also have an incentive to charge $500 for a checkup instead of $300. But they're limited to what the market will bear. That's because most patients have some idea of what things cost -- even medical services. And most patients will understand that zero percent is better than 11.9%. And that using their home equity line of credit, if they have one, costs them only 5 or 6 percent.

3. CR has a financial incentive to have subscribers renew every year instead of every three years, since the three year subscription costs less. Do we need an independent third party to advise us on our subscription, since CR has a conflict of interest?

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"The American Dental Association is among many professional medical groups that endorse CareCredit, adding to patients' perception that financing is a wise choice."

For God's sake, it IS a wise choice! "Geez, you know, I like that zero percent interest on the chemotherapy, but if I miss a payment it'll cost me a couple of hundred dollars. So I think I'll just opt for the palliative care and hope for a painless death. Nice try, Doc, but you can't fool me. I know about your conflict of interest!"

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"When Siya Kuweza … took her 17-year-old daughter to a dentist …" [long story ensues about how she applied for credit and then the dentist did bad work.]

Which has nothing to do with the credit card, she just got a crappy dentist. Why is this story even in the article?

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"Hospitals increasingly are checking patients' credit reports … [which] allows hospitals to determine whether patients have any available credit on existing credit cards. Thus they can try to persuade them to put the charges on their cards [instead of giving them charity]."

Why is that bad? Hospitals necessarily have a limit to how much charity care they can give, no? Shouldn't they concentrate on patients who have no way to pay at all? And not everyone who needs to use credit is poor. I've known people with six-digit incomes living paycheck-to-paycheck. If they get sick, isn't it right that they put it on a credit card instead of expecting free treatment?

Paying for medical care is not cruel and unusual punishment. Neither is paying interest.

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"If you finance medical bills on a credit card, you lose leverage to negotiate payments with health-care providers, who may charge self-paying patients up to five times more."

Huh? Why would they refuse to negotiate? Do they only negotiate with poor people? That's not negotiation, then, is it? It's charity.

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This is a long story, but I have no idea what CR's point is in recounting it:

"The pressure to 'charge it' may come when you're most vulnerable … [continues with story of man who was taken by ambulance to hospital A, which charged him, instead of hospital B, which had previously offered him free care.] After he returned home, hospital representatives began calling him several times a week bout the $28,000 bill … they did not discuss whether he qualified for the hospital's charity care program or offer to negotiate a reasonable monthly payment plan. The hospital obtained a copy of Wilkinson's credit report, which showed he had … available credit of $13,000 [on a credit card he had forgotten he had]. …

'The people from the hospital were threatening to put a lien against our home or freeze our bank account if we didn't agree to use the card for the hospital bill," he says. The couple agreed to charge $13,000 on the card, which the hospital accepted as payment in full. They made the first two months' payments of about $260 but could not keep up and sought legal help."

"I've been doing legal aid work for 20 years and I've never seen anything like this," says Dale Pittman, their attorney. "This is a couple with a good credit history, raising two kids and dealing with a devastating illness, yet still managing to hold it together until the hospital puts the wolf at their door by pushing them into a card with predatory terms. The credit card's annual percentage rate is up to 29.99 percent …"

Okay, this story confuses me in many ways.

1. If the couple owns a house -- the house the hospital threatened to put a lien against -- why can't they get a home equity loan for $13,000? Maybe it's a very small home?

2. The "predatory" interest rate is on the couple's own credit card, not one of those special medical credit cards that CR is decrying.

3. The hospital accepted $13,000 as payment in full on a bill of $28,000. That's better than half price. Isn't that quite a bit of goodwill on the part of the hospital?

4. What's a "reasonable monthly payment plan" on $13,000? I'd think $260 a month qualified, but the couple can't afford even that.

5. The article said that the couple had two children, and a combined income of $18,000 a year. On that basis, 30% is a reasonable interest rate, given the high probability of default (and, in fact, the couple did default after only two months). It doesn't matter if you have a "good credit history" if you now can't afford to pay.

6. Is CR suggesting that the hospital was obligated to have given Mr. Wilkerson free care?

7. Is CR suggesting that things would have been better if Mr. Wilkerson had refused treatment or the hospital had refused to treat him?

8. If the couple has a "good credit history," why can't they get a better interest rate than 29.99 percent?

And the bottom line: who should pay for the $28,000 in care Mr. Wilkerson received? What, exactly, is CR's objection? I'd be happy to agree that it's tragic that he's fighting cancer, and it's unfortunate that his financial situation is so grim. But he did receive $28,000 in services from the hospital, and the hospital did agree to cut that down to $13,000. Isn't Mr. Wilkerson responsible for the other $13,000? Isn't it reasonable to ask him to do what he can to pay it? And isn't it nice that the credit card company was willing to lend it to him?

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Finally, the last bit of the article is the story on a couple who fell into dire financial straits after their daughter got very sick. It's a sad story, and illustrates how expensive medicine can be. But it seems to me that the family is very lucky to have access to credit to pay for their daughter's medicines. I'm glad they found someone to lend them money for that. Because it wouldn't be me. I sure wouldn't risk lending them money, even at 29% interest -- they're now broke and are in the process of losing their home.

Why CR chose to run this anecdote, I'm not sure. If it were an argument for government health care, I could understand it. But as an argument against medical credit, the story serves to undermine its own argument.

In fact, the moral of CR's last story is:

11. Health care is expensive and the cost can really screw up your life.

Well, with respect to CR, we already knew that. And that's what credit is for -- to help spread out the cost, to make treatment affordable, and to help patients save their lives!

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So I'm not sure where CR is coming from in this article. Yes, health care is expensive. Yes, if you don't have money and want to borrow some, you have to pay interest. Yes, interest rates can be high if you have bad credit. Yes, doctors and hospitals want to be paid. And, yes, unexpected medical expenses can be a great hardship.

We know all this. What I want CR to tell me is why suddenly blame all this on the credit card companies -- the only ones that will finance urgent care for people who would otherwise be at the mercy of their illness.

You know what I think is happening? I think CR just doesn’t like the idea that medicine costs money, and that people make money off sickness. I think CR recoils at the idea of having to paying high interest for necessities. I think CR can't get over its aversion to the fact that poor people have to pay higher interest rates. And so I think they strung a bunch of anecdotes and pseudo-arguments together without thinking too hard about the issue.

And, in my opinion, they wound up wrong on almost every point.

Hall of Fame voters: too easily influenced by their colleagues?

Chris Jaffe takes a novel look at baseball Hall of Fame voting in a new Hardball Times article today.

He finds that the more votes a player gets one year, the more likely the writers who didn't vote for him will come around and vote for him next year. And the reverse: the fewer votes a player gets, the more likely the writers who voted for him will change their minds next year and drop him from the ballot.
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Here's Chris's chart. The left column is the percentage of votes this year; the right column is the percentage are of dissenting voters who change their minds next year. (Actually, that's not quite true: the right column is the *net* percentage – even though the top group of candidates gained the equivalent of 31.9% of the previously-non-voting-for-them writers, that could be a combination of 45% gained and the equivalent of 13% lost, or some such.)

70%-75%: 31.9%
65%-70%: 21.0%
60%-65%: 17.7%
55%-60%: 11.3%
50%-55%: 12.4%
45%-50%: 3.8%
40%-45%: -2.0%
35%-40%: 4.6%
30%-35%: 0.03%
25%-30%: -2.8%
20%-25%: -2.8%
15%-20%: -6.0%
10%-15%: -7.7%
05%-10%: -7.3%

So it looks like voters are easily influenced by their colleagues.

What to make of this? One possibility is that the writers don't actually have strong opinions, and so they're easily swayed when their colleagues vote differently – that is, they're kind of reluctant to be dissenters. Another possibility, the one that Chris champions, is that the writers ARE fairly confident in their opinions, but that they are convinced and persuaded by the give and take of the "grand national conversation" about who deserves to get in:

"Some would consider this a knock on the BBWAA and call it herd mentality. I beg to differ. It shows a willingness to listen to others, reexamine their position, and accept arguments others espouse. Simply put, it shows a willingness to listen, something I find commendable."

I don’t know much about psychology, so I'm not sure what's going on, but I suspect it's the same thing that drives changes in opinions on social issues. Thirty years ago, the idea that gays should have the right to get married was laughable ... but, over time, and very slowly, people started coming around to the idea, to the point where it's now mainstream (and legal, here in Ontario).

On gay marriage, I don't think people sat down and thought about it in any formal way ... it just kind of happened, as they slowly absorbed the chaging norms over time. Perhaps that's what's happening here.

Or perhaps not. Other suggestions are welcome.

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UPDATE: here's another suggestion. Voters might think player X should be in the HOF, but, in a year with lots of candidates, they voluntarily limit the number of players they vote for. So 60% might vote for X now, and some of the other 40% join in next year, when there are fewer candidates and they have a vote to spare.

For instance, suppose every voter decides to choose only three candidates. But there are four worthies. The first year, B gets 90%, C gets 80%, D gets 70%, and E gets 60%. The next year, D and E are joined by A and F. Now A gets 90%, D gets 80%, E gets 70%, and F gets 60%.

So D and E have gained 10% each, just as observed. The 10% was the result of many writers wanting them in, but just not feeling they could spare a vote that year.

For this to work, I think you need the maximum number of votes per writer to exceed the (long-term average) number of worthy inductees.

"The Blind Side:" I still can't figure out why left tackles are so valuable

Yesterday, Steven Levitt recapped "The Blind Side" over at his Freakonomics Blog. Levitt points out that while quarterbacks are the highest paid NFL players, at about $5 million per year, left tackles are second, at $4 million. He writes,

"The book highlights how N.F.L. teams only slowly became aware of the immensely important role that offensive left tackles play in protecting the quarterback from blind-side hits."

But I still have a problem with this idea of Lewis's, that left tackles are so valuable because they save the quarterback from injury. How many quarterbacks get injured? Say, 25% a season? (That seems high to me, but let's give the left tackles the benefit of the doubt.)

And of those 25%, how many were injured because of a blind-side hit that the left tackle could have prevented? Again, let's say 25%.

25% of 25% is about 6%, so about one quarterback in 16 can be saved by the left tackle.

But, now, what's the difference between a typical left tackle and the replacement-level left-tackle? Let's suppose the average left tackle is twice as good as preventing QB injury as the backup. That means that the good LT could save only 3% of the quarterbacks under his care.

Now, 3% of a $5 million quarterback is only $150,000. (Of course, some injured QBs are out more than one season, so you'd have to multiply the $150K by the number of seasons. But many injuries are for only a few games, so it seems reasonable to figure one year as the average.)

So how can you argue that the LT is worth $4 million because of only $150,000 in quarterbacks saved? Obviously, if left tackles deserve to be the second-best paid position in the NFL, they must be doing something else right.

Maybe it's sacks. But as I wrote two years ago,

"The Hidden Game of Football" (p. 104) estimates that the difference between a good quarterback and an average one is four completions per game. An eyeballing of 2005 NFL sack statistics shows that the difference between a good individual sack total and an average one is only about four sacks per *season*. That’s a difference of 1500 percent in favor of the quarterback."

So if a good QB is worth $5 million a year for those 60 extra completions, then a good LT should be worth only about $350,000 more for those 4 saved sacks.

So $150,000 in injuries saved plus $350,000 in sacks prevented adds up to only $500,000. Where's the other $3.5 million?

Well, there's also the unmeasured effect of giving the QB more time in the pocket. But that would have to be huge, wouldn't it, to add up to $3.5 million? You'd need the LT to be responsible for 2 or 3 extra completions per game (or the equivalent in yards). That seems unlikely, considering that the rest of the offensive line has a big stake in the outcome as well.

The bottom line is: I can't really figure out why left tackles are considered as valuable as they are.

One possibility: the famous career-ending injury to Joe Theismann came from the blind side. Could that gruesome example of what can happen to a blind-sided quarterback have caused GMs to overestimate the value of the left tackle's job?

(UPDATE: "Doc" points out in the comments that it's not the full $4MM we have to explain, just the premium above other offensive linemen. He's right. That premium appears to be only $1MM or so. So it's $1 million we have to explain, not $4 million. But, still, it's hard to see how that million dollars is just for the value of protecting the QB from injury. It seems more likely that only a small portion of that has to do with injury, and the rest has to do with LT being the most important blocking position, and teams putting their best lineman there.)

A nine-yard gain is better than a first down

What's better on first-and-10: gaining 10 yards for a first down, or gaining 9 yards for second-and-1?

Brian Burke shows us that the nine-yard gain is better. That's because, effectively, it gives the offensive team a free second-down pass. If they fail to connect, they just run on third-and-1, which has a very good chance of succeeding (and so does fourth-and-1, if they fail to make it on third down).

Brian studied all first-down plays between 2000 and 2007 outside of field-goal range in the first 28 minutes of the game. From that database, he was able to figure that the difference is quite large: almost a whole point on the scoreboard.

So should teams deliberately choose to gain nine yards instead of 10? Yes, in theory. In practice, of course, it doesn’t make sense to deliberately go down at 9, because you don't know for sure that you'll be stopped at 10 – you could wind up at 16 or 17. Also, as Brian points out, coaches are risk-adverse:

" ... the first time anyone actually did it intentionally, and his team failed to convert the 1st down, the criticism would be merciless and it would never be done again."

However: what's to stop teams from doing it "kind of" deliberately and hoping nobody notices? Nothing. And the evidence does suggest that it does happen. Although gains of more yards are generally less frequent than gains of fewer yards, the drop from 9 yards to 10 is particularly steep. Furthermore, 10-yard gains are actually less frequent than 11- and 12-yard gains.

Of course, that might not have anything to do with offensive strategy. As Brian says, it could have to do with the way the referee spots the ball. Or it could be that defenses are guarding the first-down marker so well that the offense is brought down at nine yards.

So we don't know whether the offense is being smart, or the defense is unwittingly playing into the offense's (perhaps also unwitting) hands.

Do players become more aggressive when they wear black?

Here's one of the most surprising legitimate findings I've seen in sports analysis. According to a 1988 study by Mark G. Frank and Thomas Gilovich, wearing black uniforms makes players more aggressive. Moreover, referees and fans subjectively rate those players as more aggressive, even when they're not.

The study seems pretty thorough. (And it's an pretty decent read.)

First, Frank and Gilovich (F&G) gave volunteers pictures of all NHL and NFL uniforms, and had them rate the jerseys for "malevolence." It turned out that jerseys with black in them were rated the most aggressive.

In the NHL, the top six were: Vancouver (black at the time in 1988), Philadelphia (orange with black trim), Boston (black), New Jersey (non-black; black trim would appear a few years later), Pittsburgh (black), and Chicago (black trim).

In the NFL, the top five were the Raiders (black), Pittsburgh (black), Cincinnati (black trim), New Orleans (black), and Chicago (dark blue, but the authors classify as black because the volunteers thought they were black).

I don't think there are any other teams with black in their uniforms, but I might have missed some. If I'm right, then black uniforms took all the top slots, with the exception of one spot taken up by the New Jersey Devils.

(By the way, the least aggressive ratings went to the Miami Dolphins (aqua and orange) in the NFL, and the Hartford Whalers (blue and green) in the NHL. My beloved Toronto Maple Leafs didn’t scare anyone either, finishing third last.)

Having now established that black uniforms signify malevolence, the authors now checked whether the black-clad teams actually were more aggressive. It turned out that they were. In terms of penalties, the black teams out-offended the non-black teams, with 98% signficance. In the NHL, the same was true at a 99.5% significance level.

A quick glance shows that the effect couldn't be caused by just one team. See for yourself; in visual terms, here are how the black teams ranked:

NHL: XXX--X---X-------------
NFL: X-X---XX---X----------------

Moreover, there were a couple of natural controls in the NHL in the late 1970s, and those were the ones where the results really shocked me. In the middle of 1979-80 season, the Pittsburgh Penguins abruptly changed from blue uniforms to black. Astonishingly, their penalty minutes immediately rose by 50%, from 8 minutes per game in blue uniforms to 12 minutes in black. Furthermore, after having been average or below in PIMs for eight of the past nine seasons, the Penguins went on a streak of four above-average seasons.

The case of the Vancouver Canucks is similar. After the 1977-78 season, the Canucks went from blue uniforms to (ugly!) black ones. The next season, their PIMs shot up from almost exactly average, to 2 SD above average (and this is before their trade for Tiger Williams). Like the Penguins, they maintained an above-average rate for three more seasons after that.

What could be causing this? One explanation is that the black uniforms made the players more aggressive. Another is that the black uniforms made the team *appear* more aggressive, and so the referees penalized them more. A third explanation is that teams with black uniforms have management that wants to play more aggressively (although this explanation is more reasonable for the Penguins and Canucks, whose management actually chose the team colors rather than inheriting them).

The authors ran little experiments to test the first two possibilities.

First, they recruited some experimental subjects, and randomly assigned them black or white T-shirts. Then, they let the subjects choose one out of several competitions to play. The black-shirted subjects chose more aggressive games.

Second, the authors staged several aggressive football plays, one where the white-shirted team was the aggressor, and (an identical) one where the black-clad team was the aggressor. They then showed tapes of those plays to referees and casual fans, asking them to decide whether the play was a penalty. Both groups called significantly more penalties when the aggressor team wore black. Further, when they showed the plays in washed out black and white, so it was no longer obvious that the dark shirts were black in color, the subjects called penalties at an identical rate for both teams.

So it does seem that wearing black (a) makes competitors more aggressive, and (b) makes them *appear* more aggressive. At highly significant levels. I am blown away.

Hat Tip: Daniel Engber of Slate