If you hold a cane in the air, you can move it in any direction, twirl it, and so on. Its motion isn't constrained at all. That is, the top of the cane can move freely in three dimensions.
If you put (and keep) one end on the ground, now its motion is constrained: you can't lift it, or rotate it... although you can swing the top around in a variety of different arcs. That is, the top of the cane can move freely in two dimensions.
If you connect the tops of two canes together and place the other ends on the ground, you can still move the tops, but only along a single (straight) arc, back and forth. That is, the tops of the canes can move freely in one dimension.
If you try the same trick with three canes, now you can't move the tops at all. This is basically what's happening with a three- legged stool. The tops of the cans can move in zero dimensions... which is to say, they can't.
Each time you add a cane, you remove one dimension in which the top can move freely - that is, each new cane removes one 'degree of freedom'.
The specifics of the case are not that nasty, but the direction it leads is nothing but trouble.
Here’s the details:
Matrixx made Zicam and a lot of money. Investors like this.
Matrixx received anecdotal reports that Zicam use led to loss of the sense of smell. It didn’t tell investors because the evidence wasn’t strong.
Good Morning America did tell the public. Matrixx share prices dropped and investors lost money.
Investors sued for security fraud. Matrixx lost, appealed, and has now lost at the highest level. The decision was unanimous.
The case hinges on the requirement that a company release information to investors that is “material” to the value of their investment.
The borderline for what is “material” is squishy. That squishiness has benefitted lawyers in the creation of securities fraud cases. This decision will make this problem worse:
Justice Sonia Sotomayor, writing for the court on Tuesday, roundly rejected Matrixx’s proposal that information can be material only if it meets standards of statistical significance.
“Given that medical professionals and regulators act on the basis of evidence of causation that is not statistically significant,” she wrote, “it stands to reason that in certain cases reasonable investors would as well.”
On the other hand, she said, “the mere existence of reports of adverse events — which says nothing in and of itself about whether the drug is causing the adverse events — will not satisfy” the requirement of materiality. Instead, she said, companies and courts must consider “the source, content and context of the reports.”
Here, Justice Sotomayor wrote, the plaintiffs had accused Matrixx of having received information from “three medical professionals and researchers about more than 10 patients who had lost their sense of smell after using Zicam.” That was enough to allow the case to go forward in its earliest stages, she wrote.
If the accusations are proved true, she said, “Matrixx received information that plausibly indicated a reliable causal link between Zicam and anosmia.”
Reasonable investors would want to know about the reports, she said, particularly given the importance of the product to the company and the risk-benefit calculation consumers might make after hearing of the possibility that using a cold remedy could result in lasting injuries.
In rejecting the proposed categorical rule in favor of a contextual inquiry, the court provided only limited guidance to companies and lower courts.
So, let’s think about this.
Matrixx did invest time to investigate the complaints — duh — that’s how they know they were statistically insignificant.
The Supreme Court has ruled that their objective method of evaluating complaints must be replaced by a subjective one — like what’s the likelihood that an availability entrepreneur out in the legacy media is going to feed their content monster with this.
Now, I have no doubt that investors would really like to know that there are reports of problems with the product they’ve bought a piece of, and I think Zicam is a scam anyway, so I’m inclined to believe there’s some culpability. I’m biased though.
Even so … this is like a theatre owner being told by a patron that there might be a fire in the theatre. That patron may not have a big mouth, but someone else might, and the owner is responsible if they use it. So, the Supreme Court has concluded that the owners must shout “Fire!” in the crowded theatre first to follow the law.
But … we spend a lot of time talking in statistics classes about how everything is a statistical test (whether you view it that way and do the work, or not). And, one of the worst things you can do with a statistical test is to change the level of significance after you run the test. This is precisely what the Supreme Court is suggesting: Matrixx has a significance level for their tests that they used, and the Supreme Court is telling them at a later date that they should have used a higher number. They didn’t specify the number.
Columbia Univ. professor of statistics and political science, Andrew Gelman. Sample:
The USA Today Fallacy: Counting all states (or countries) equally, forgetting that many more people live in larger jurisdictions, and so you're ignoring millions and millions of Californians if you give their state the same space you give Montana and Delaware.
A statistical fallacy perhaps, but not a political one. This is the sort of thing that prevents the tyranny of the majority … you know … like the f***ing Nazis.
A statistician’s wife gives birth to twins. Excitedly, he calls everyone to share the good news. When he tells the church they say, ‘That’s terrific! Bring them down to church this Sunday, and we’ll baptize them!’
’Uh, let’s just baptize one of them,’ says the statistician. ‘We can keep the other one as a control".
A great statistics quote that everyone paraphrases:
One day when I was a junior medical student, a very important Boston surgeon visited the school and delivered a great treatise on a large number of patients who had undergone successful operations for vascular reconstruction. At the end of the lecture, a young student at the back of the room timidly asked, “Do you have any controls?” Well, the great surgeon drew himself up to his full height, hit the desk, and said, “Do you mean did I not operate on half the patients?” The hall grew very quiet then. The voice at the back of the room very hesitantly replied, “Yes, that’s what I had in mind.” Then the visitor’s fist really came down as he thundered, “Of course not. That would have doomed half of them to their death.” God, it was quiet then, and one could scarcely hear the small voice ask, “Which half?
—Dr. E. E. Peacock, Jr., University of Arizona College of Medicine; quoted in Medical World News (September 1, 1972), p. 45 …
Paul Kedrosky put out a bleg a few weeks ago for the original cite for the story that shows up in many statistics books. Here’s Digital Roam:
A nifty little chart was created to show where the maximum number of bullet holes were located on returning aircraft. This chart showed the greatest damage not on the main wing and tail spars, engines, and core fuselage areas, but rather on the aircraft extremities. Based on this, the Air Ministry suggested adding armor to those extremities. Wald suggested they were dead wrong.
Wald said more armor should go on the places that had the least holes. Huh? What was he thinking?
Wald was keeping the Air Ministry from falling into the "survivorship bias": they were forgetting the their data did not include the planes that had been lost. If the returning planes had no holes in their wing spars and engines, the better assumption to make is that even a few holes in those places were deadly: no damage was recorded in those areas because those planes were the ones that had crashed.
The comments to Kedrosky’s piece are interesting: one commenter relates how this technique was used to improve F-4s in Vietnam.
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