The conceptual basis of classical hypothesis testing is useful in a wide variety of areas.* Pity most students get too caught up in following the recipes for constructing test statistics to realize this.
A Type I error is foreclosing against a good borrower. A Type II error is letting a bad borrower avoid foreclosure. I will readily grant that a Type I error is worse than a Type II error, so we should tolerate some of the latter in order to avoid the former. However, I contend that we have let this bias get completely out of hand, resulting in a huge pileup of Type II errors with catastrophic effects on the housing market.
This is in regard to a Maryland family that has gamed the system and not made a payment on the house they continue to live in for over 5 years.
* A Type I error is when you make the mistake of rejecting something that is true. A Type II error is when you make the mistake of not rejecting something that is false. For example, a Type I error is convicting someone who is innocent, while a Type II error is letting someone go who is guilty.
The probability of being hit by lightning is low, but the conditional probability* may be much higher.
As a Utah resident, I can tell you why the death rate is higher in these dry western states.
First, lightning often strikes exposed hilltops, mountaintops, cliffs, and so on. Utah has a lot of those. Also, they’re scenic and accessible, so we have a lot of people out on them.
Second is virga. Easterners don’t usually even know what this is. Virga is when a storm drops rainfall and it evaporates before hitting the ground. We have a lot of virga in Utah. The thing is with virga, even though you’re dry, that’s a real live storm up there.
Third is visibility. Where I live, I can see for 60 miles on a bad day. It’s routine for me to be able to see multiple storms, separated by clear sky. And, it can be really easy to stop worrying about lightning when you see lightning, and count to over 100 before you hear thunder. Lightning routinely strikes as much as 10 miles from a storm, and up to 50 miles (who knew!). So, in Utah, not only can you see multiple storms, you can be hit by them too.
Fourth, there is a lot more lightning in sub-tropical areas, but there’s also a lot more rain. I lived in Louisiana for almost a decade: if there is a thunderstorm outside, you simply don’t go outside because it is like walking into a running bathroom shower with your clothes on. You wouldn’t jump into a pond fully clothed, and in Louisiana you don’t go out in the rain if you can avoid it because sometimes you’ll get that wet in a minute or two. This doesn’t happen out west, so people are outside more when there’s lightning.
* Probability is the chance of something happening. Conditional probability is the chance of something happening when you know some other information. Ben Franklin wasn’t dumb because the probability of being hit by lightning was high, but because the conditional probability of being hit while flying a kite is much higher.
† Do you ever wonder if some people’s teenage years would have been easier if they changed their names?
This is like not using batting average (at all) to compare baseball players because there is much more to the game.
Repeating the paraphrase of Orwell “Some ideas are so stupid that only intellectuals believe them.”*
Here’s Bryan’s result:
… If you mention ability bias, however, labor economists will quickly point you to a massive literature that supposedly debunks it.
But if you pay close attention, there's a bizarre omission. Despite their mighty debunking efforts, labor economists almost never test for ability bias in the most obvious way: Measure ability, then re-estimate the return to education after controlling for measured ability. For example, you could measure IQ, then estimate the return to education after controlling for IQ.
When I ask labor economists about their omission, they have a puzzling response: "IQ is a very incomplete measure of ability." True enough. But the right lesson to draw is that controlling for IQ provides a lower bound for the severity of ability bias. After all, if the estimated return to education falls sharply after controlling for just one measure of ability, imagine how much it might fall after controlling for measures of all ability.
What happens to the return to education after controlling for IQ? I've done the statistics myself on the NLSY, and found that the estimated return to education falls by about 40% …
So, the reported returns to education — how much extra one gets paid for more schooling — are almost double the corrected estimates because they’re ignoring the basic observation that smart people are likely to get paid more anyway.
The flip side of this is that the value of teachers and schools is overstated by ignoring the quality of the students put into the system.
Gee … do’ya think there’s any grant funding out there interested in establishing and promoting that result?
Statistically, Bryan is pointing out that conventional parameter estimates are biased upward, but he’s missing a second problem. The big three problems in statistics are bias, consistency, and efficiency — and the conventional estimates are not just biased, they’re also inefficient.
Inefficiency is a lot more subtle than bias. Frankly, I don’t think I understood it until about 8 years into my academic career when I had to explain in seminars to finance Ph.D.’s why they should listen to an economics Ph.D. about the problem with one of their techniques.
In short, an inefficient estimator is one that’s just dumb: like forecasting the weather without looking at the sky. There’s a lot more to it than that, but it means that you’re not doing something which might be helpful.
One way to think about this is that bias is about parameter estimates that are off in one direction or the other, while efficiency is about standard error estimates that are off in one direction or the other.
In practice, this suggests that not only are the typical estimates of the returns to education biased upwards, but they probably reported to be a lot more precise than they actually are. That is, Bryan is pointing out that the parameter estimates are biased upward, and I’m pointing out that the standard error estimates are probably biased downwards.
Bias and inefficiency also have distributional consequences for the conventional results. Because those results are based on leaving something out, they suggest that the things that are left in are more important than they would be in a better model. The thing is, you don’t know which of your variables is more or less seriously affected until you run the better model. Maybe you’re lucky, and the variable of interest is the one that’s least affected … and maybe you’re not.
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.
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