Orton's Arm

I got a chuckle out of your post. I can't distinguish between the error in the test and the normal distribution of the population? That's rich. In my earlier post, I wrote that the +1 to +2 SD subset contains a higher percentage of lucky people than of unlucky people. Let's expand on that by considering a very thin slice of the measured distribution, as well as an imperfect test. Consider the group of people who scored between 1.99 and 2.01 SDs above the mean. The test is error-prone; so each person 1 SD above the mean has a chance X to get lucky and be scored inside this small slice. Each person 3 SD above the mean also has a chance X to get unlucky and be scored inside this small slice. The number of lucky people from +1 SD = X * the number of people one SD above the mean. The number of unlucky people from +3 SD = X * the number of people three SD above the mean. Because there are more people at one standard deviation above the mean than at 3 SDs; there will be more lucky 1 SDs present in your thin slice, than unlucky 3 SDs. The same logic also holds if you're comparing the number of lucky 1.5 SDs to the number of unlucky 2.5 SDs. Indeed, it applies to any two points on the normal distribution that are equally far from the center of your thin slice. The point that's closer to the population mean will contribute more people to your slice than the point that's farther away. This means that the average member of your thin slice is closer to the population mean than his or her test score would indicate. Retest the members of your thin slice, and they will, on average, score somewhat closer to the population mean the second time around.

I agree with the above paragraph, and I was amused by your long list of excuses to keep calling me an idiot. You clinging to that idea even after it's been disproven is like . . . well, a lot of stuff that happens on these boards. I'd like to add to your above paragraph by pointing out that the effect I'm describing happens even if you have two cutoffs. Suppose you defined your subset as being only those people who scored between 1 and 2 standard deviations above the population mean. In addition to those who ought to be in the cutoff, and are in, you have four groups of people: 1. Lucky people bleeding in from the left cutoff. 2. Unlucky people bleeding in from the right cutoff. 3. Unlucky people who bleeded out through the left cutoff. 4. Lucky people who bleeded out through the right cutoff. The number of lucky people who bleeded in through the left cutoff exceeds the number of unlucky people who bled out through that cutoff. This is because the number of people at, say, 0.9 - 1 standard deviations (available for getting lucky) exceeds the number of people at 1 - 1.1 standard deviations available for getting unlucky. The existence of this cutoff increases the percentage of lucky people in your subset. What happens at the other cutoff? Clearly there are more people at 1.9 - 2 standard deviations available for getting lucky, than there are people at 2 - 2.1 standard deviations, available for getting unlucky. The existence of this cutoff decreases the percentage of lucky people in your subset. However, its effect is much smaller than the other cutoff. This is because everything is happening on a smaller scale as you drift on out toward the right hand tail of the distribution. When the effects of the two cutoffs are averaged out, you're still left with a subset who got disproportionately lucky on that first test. Test them again, and their average will be somewhat closer to the population mean.

It's an effect that happens at the margin of the cutoff. Suppose you were to retest everyone who scored above the population mean. Some of the people you're retesting will be people with true I.Q.s below the population mean who got lucky on that first test. What about the people who should be balancing them out--those with true I.Q.s above the mean who got unlucky and scored below the mean? Those people didn't make the cutoff, and aren't being retested. Hence, more than 50% of the people who made the cutoff got lucky on that first I.Q. test. When you retest the subgroup, its average score will decline somewhat.

You're wrong. Suppose your subset consisted of those who obtained exceptionally high scores. Given that actual scores are due in part to luck, the subset you selected got disproportionately lucky on the first test. Retest them, and on average their score will be somewhat closer to the population mean. But I'm the one who doesn't even have a rudimentary understanding of statistics.

Yet you made them anyway. How nice of you.

Mostly this is a debate about whether a specific statistical phenomenon does or does not exist. I'll avoid answering some of your questions--such as how the mean was determined--to focus on the underlying statistical question. Suppose an underlying, normally distributed population with a mean of 100, and a standard deviation of 10. (This could be a normal distribution for height, I.Q., running speed, whatever. Doesn't matter.) Measure each member of the population with an imperfect test. If someone were to take the test 1000 times, the actual results would be normally distributed around this person's true score, with a standard deviation of 2.5. Give the whole population the test one time. Arbitrarily define a subset of the population, based strictly on their test scores the first time they took the test. For example, you might choose as your subset those who scored between 1 and 1.5 standard deviations above the mean. Or you might choose only those who scored below two standard deviations below the mean. The only restriction on the subset is that it cannot contain both members who scored above and below the mean. Administer a second test to the members of your subset; with the same mean error term of zero and the same standard deviation of error of 2.5. I contend that, as a group, the members of the subset are expected to obtain scores that are closer to the population mean upon retaking the test, than the group averaged when it first took the test. This is because any subset whose members scored below the mean will contain more people who got "unlucky" than "lucky" on the first test; whereas any subset whose members scored above the population mean will conversely contain more people who got "lucky" than "unlucky." When you retest the subset, good luck will presumably balance out bad; and the group as a whole will obtain scores somewhat closer to the population's mean. Whatever view you have of me or of my ideas, you cannot deny that the above is valid statistical reasoning.

Thanks for making assumptions about what I do or don't understand.

You use "valid logic"?

There's a difference between a test that's "imperfect" and a test that "has no scientific method behind it whatsoever." In the imperfect test I've described, someone's "true I.Q." is defined as what that person's average score would be if they took the I.Q. test 1000 times. A person's actual test scores will be normally distributed around his or her "true I.Q." with some non-zero standard deviation. Again, if a group of people who scored a 140 on an imperfect I.Q. test find themselves being retested, is the group expected to average a 140 on the retest, or is it expected to average a lower score?

Don't quit your day job.

I expect stupidity from Ramius, and he helpfully delivers. Thanks, man.

I'd expect this level of pure stupidity from Ramis, but not from you. I'm not saying I expected you to reach lofty intellectual heights--the last 50 pages have proven otherwise. But this is a new low for you.

Misinformed and partisan insults coming from pinkos such as yourself don't bother me a whole lot. But what your post did not contain is a refutation of a very specific statistical claim I've been making for the last 30 - 50 pages. Nobody's interested in whether you like me, or what opinions you've formed of eugenics. They want to know where you stand on one specific issue. If you're going to post anything at all in this thread, you need to address that issue. That issue is this: suppose you have a population that's normally distributed with respect to intelligence. Suppose you were to give every member of the population a mildly imperfect I.Q. test. It's mildly imperfect in the sense that a given person's score is expected to vary a little from one test taking to the next. Suppose you were to take a group of people who scored a 140 on an I.Q. test; and were to ask them to retake the test. Would the group as a whole be expected to average 140 on the retake, or would it be expected to get a lower average? That's just about the only thing we've been arguing about these last 50 pages. If you can come down off your high horse to address this issue, I'm sure a lot of people would appreciate it.

I'm afraid you need to go back and reread the example. I wrote that of the people who actually scored a 140 on the first I.Q. test, 1000 of them had true I.Q.s of 130 and got lucky, 800 had true I.Q.s of 140 and were scored incorrectly, and 10 had true I.Q.s of 150 and got unlucky. Imagine those 1810 people sitting down to take the retest. If you were omniscient in kind of a weird way, you'd see 1000 people with little bubbles over their heads which read, "I'm a true 130, who got lucky on the first test and scored a 140." You'd see ten people with little bubbles which read, "I'm a true 150, and I got unlucky on the first test and scored a 140." You'd expect the 1000 true 130s to score an average of 130 upon being retested; just as you'd expect the 10 true 150s to score a 150 upon being retested. Whoever is collecting those retests will get 1000 from true 130s, 800 from true 140s, and 10 from true 150s. The average value for these test scores will be less than 140; so the group's I.Q. will appear to have regressed toward the population mean between the first and second tests. The above logic holds true for any population distribution where there is measurement error, and where people tend to be clustered more toward the middle than the extremes. If people with I.Q.s of 150 were as common as people with I.Q.s of 130, the phenomenon I'm describing would not take place. But there are more people with true I.Q.s of 130 available for getting lucky, than there are people with I.Q.s of 150 available for getting unlucky. Any group of people who scored a 140 on an I.Q. test is expected to contain more lucky 130s than unlucky 150s. More generally, any given group of people who scored above the population mean on an imperfect test is expected to contain more people who got lucky on the test, than people who got unlucky.

That's the best you could come up with? Look. What I'm saying here should be obvious to anyone. Suppose you were to take the population of, say, New Zealand, and gather up everyone who scored a 140 on an I.Q. test. In the group of people you gathered, do you think lucky 130s and unlucky 150s would be equal in number, or do you think lucky 130s would outnumber unlucky 150s? It's a simple, easy question, and the answer leads directly to the phenomeon I've been describing. There's no way Coli's going to be dumb enough to destroy his own credibility by pretending that there will be as many unlucky 150s as lucky 130s.

I'm sure Coli would love to help you guys. He's known you and Ramius a lot longer than he's known me. In addition, your political views are probably a lot more similar to his than mine are. He'd love to help you. He just can't.

When the eugenics debate started, Coli joined in. He made it clear he was going to present just one post. In that post, he presented the extreme pro-nurture side of the debate, in such a way as to make it seem it was the only view any reasonable scientist had even considered. But he made it clear that would be his last post on the topic of eugnenics. This particular thread isn't intended to be a debate about eugenics, but about a specific statistical principle. Coli doesn't have to spend endless time debating anyone. He could just create a single post stating whether he agrees or disagrees with the specific phenomenon I've described. His refusal to do so speaks volumes.

This sounds more like something you came across, than something you wrote yourself. Unless you have multiple personality disorder or have started referring to yourself in the royal plural, that is. In any case, I don't disagree with anything in the above post. The phenomenon I've been describing is different from the above, in that measurement error or natural fluctuations in a person's underlying I.Q. are necessary for it to take place. The fact that both phenomena were labeled "regression toward the mean" certainly makes the discussion more confusing. Wraith, at least, felt that the phenomenon I've been describing exists, but should not be labled "regression toward the mean." He could well be right, though there are other sources such as Hyperstats and the Wikipedia article which would indicate a broader definition of regression toward the mean. Whether the phenomenon I've been describing meets the technical definition of regression toward the mean is far less interesting to me than the fact that it exists in the first place. Suppose you have a group of people who scored a 140 on an I.Q. test. This group will have a greater number of lucky 130s than unlucky 150s. When everyone in the group retakes the test, the group's average score is expected to be lower than the original 140. Call this phenomenon whatever you want (except you, Bungee Jumper), but know that it exists.

When on earth did I say I'd "shamed" Coli? I wrote that Coli knows that if you have a normally distributed population, and if you administer an imperfect test to said population, those who obtain extreme scores will tend to score somewhat closer to the mean upon being retested. I wrote that Coli would probably love to feed some humble pie to a guy who's advocating eugenics, but that he can't do so in this case because he knows the specific claim I've outlined above is correct. So he's keeping his mouth shut.

To think that we got rid of Ted Cottrell to have this guy's system installed. The Wade Phillips/Ted Cottrell system seems to be doing just fine over in San Diego.

I'd point out that the Bills' offense isn't the only one that sometimes gets good field position from its defense or special teams. Well, you might reply, perhaps the Bills' offense is getting more than the usual share of help from defense or special teams. Hey, maybe you're right. But I'd point out that the Bills' offense has returned the favor by keeping turnovers low. As for those deep drops for Losman, I'll agree that with the offensive line situation we had earlier in the season, a deep drop is a low percentage proposition. Let's say that for every four deep drops, you'd get two sacks, an incompletion, and a 40 yard gain. Is it worth it? Arguably so, for two reasons. One is that the 40 yard gain will probably turn into points. Without an offensive line, it's really tough to drive the ball the length of the field in short chunks. If you get the occassional 40 yard gain, you'll at least be able to put something up on the board. In addition to that, your offense becomes more one-dimensional if you never go for the deep throw. Teams playing Flutie realized they didn't have to respect the deep play, and were able to crowd defenders up front to take away the running game and the short passing attack.

I can't believe there are still people criticizing Steve Fairchild. Dude, wake up! The Bills' offense has been a lot more effective lately than it was earlier in the year. The players are learning the system. There's a strong commitment to the running game. The playcalling has created a dangerous passing game, prone to taking shots deep down the field. Losman's been playing the best football of his career. (Not that he's had a long or illustrious career, but still.) No name players are being plugged into positions, and are having success. The Bills scored an average of 26 points in their last four games, and won three of those games in the process. Why on earth you have a problem with the architect of the Bills' young and improving offense is completely beyond me.

Ralph will never see that $750 million unless he sells the team. Assuming he holds onto it, that $750 million number doesn't mean anything.

Thanks. That was hilarious.

To which "stats experts" are you referring? Because I haven't seen a whole lot of stats knowlege in this discussion from anyone except myself and Wraith (and on occassion Dave B). That doesn't make me a stats expert, but it does mean that I understand the fundamentals. Which is a lot more than can be said about a windbag like Ramius. Edit: maybe you're thinking about Coli. He's a guy who knows about stats, but he's also a partisan liberal. When I started arguing the case for eugenics, he made the claim that I was interpreting statistical concepts incorrectly, due to a general lack of knowledge about stats. He had no earthly idea how I was interpreting or not interpreting those statistical concepts, so it was an easy cheap shot. Circumstances were vague enough he could make it without significantly decreasing his own credibility. But then I started claiming something specific. I wrote that if you have an I.Q. test with measurement error, and if you have an underlying normally distributed population, those who obtain extreme scores on their first tests will, on average, score somewhat closer to the mean the second time around. Given our political differences, I think Coli would have jumped at the chance to exploit any given chink in my armor. But he couldn't contradict what I wrote about the appearance of regression toward the mean without destroying his own credibility. So he kept his mouth shut.