Fallacy Files

While following up the subject of “Galton’s Fallacy” (on which I may post some other time) I stumbled across (oh, all right, it came up on Google) a rather nice website, Fallacy Files, which deals concisely with all sorts of fallacies, including many I had never heard of. I haven’t read all the entries by any means, but those I sampled are pretty good. And I didn’t find myself violently disagreeing with anything, which as you know is rare for me!

Posted by David B at 01:35 PM

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Penguin Puzzle

I was watching a BBC natural history programme the other night, and it included an extract from David Attenborough’s Life in the Freezer series about life in Antarctica. This classic scene shows a flock of (male) Emperor penguins huddled together on the ice in a winter blizzard. The commentary explains that the male penguins incubate the egg, holding it on their feet to keep it off the ice. As the icy wind blows, penguins gradually peel away from the windward side of the group and waddle round to the sheltered leeward side. As the voiceover explains, they are ‘taking it in their turn on the outside to suffer the harshest conditions’ (or some such phrase).

But hang on… as good Darwinian I must protest! Why should individual penguins ‘take their turn’ on the outside if they could help it?

I haven’t got the scene on tape to re-view it, but from memory I think the following account would be consistent with the facts: each penguin has the following order of preference: (1) on the inside of the huddle (2) on the leeward side (3) on the windward side. At the beginning of the brooding period the biggest, butchest penguins muscle their way to the inside. But as soon as the wind blows, penguins on the windward side slowly waddle round the outside until they reach the leeward, which means that some of the leeward penguins are no longer on the outside. (They can’t move very fast, because they have to keep the egg balanced on their feet.) This exposes more penguins to the wind, so they in turn move round. Eventually, some of the butch penguins in the middle are exposed, but it is not worth the effort and danger of trying to muscle their way in again, so they also move round. Over a period of time, all the penguins spend some time in each position in the group, but this is not a result of any noble propensity to share the worst positions, just a side effect of the selfish movements of all, combined with the constraints of moving while carrying an egg!

I wonder if any penguin expert has considered this? The test would be to observe the movement of the group as a whole: if I am right, the centre of the group will gradually move in the same direction as the wind. Of course, if the wind shifts, so will the group.

Posted by David B at 07:53 AM

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Less brown slouching than I thought

A week back I commented on the VDARE article “Bad News About Those South Asian Model Immigrants.” Here is data I calculated from a 2002 table titled “Immigrants admitted who were adjusted to permanent resident status by selected status at entry and region and country of birth”:

 FamilypreferenceEmploymentpreferenceImmediaterelativesAslyumDiversityBangladesh  13.4% 18.5% 44.4% 3.3% 20.1%India 16.0% 60.3% 21.2% 3.6% 0.0%Pakistan 21.3% 24.3% 42.7% 3.4% 7.9%Total 16.7% 52.3% 25.9% 2.4% 2.5%

1) Note the differences between the various immigrant streams by nation and the trend of poverty and employment rate correspondingly.2) Seems like the degradation I was assuming isn’t really happening (at least as of 2002). Since Indians are a disproportionate fraction of South Asian immigrants, they are skewing the percentages to look much better than I’d expected for the total. One thing that it tells us is the the “Diversity Lottery” is jack, my personal experience with people who won the lottery is that they are losers (note how many Bangladeshis come that way).

Update-Related brown news: Manish points me to this story about an Indian American who won a Republican primary run-off in South Carolina. A few points….She was born in the U.S. in 1972.Her parents are professionals.She is married to a white American (maiden name “Nimrata N. Randhawa,” she goes by Nikki Haley).You have to look close at her photo to tell she’s brown (she’s from a Punjabi background).Raised Sikh, she “believes in Jesus,” and her children are baptized Methodist, though they still attend a Sikh Temple as well.Finally, her kids are named “Rena” and “Nalin,” not typical southern names. Nalin has a Hindi origin and Rena is Hebrew, though both ambiguous enough to sound South Asia or conventional American. But if she was a “sell-out,” you figure they would be named Rachel and Nathan.Anyway, this woman is probably typical of many of the “first wave” who are assimilating in the United States….

Posted by razib at 04:07 PM

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What's in a face

In recent years cognitive scientists have identified “domains” within the brain that are geared toward processing certain kinds of information. One of the most obvious instances of a cognitive specialization where humans can access gestalt knowledge is the uncanny ability we have to recognize faces. Here is a bizarre case from Survival of the Prettiest (page 165):

For the past ten years I have been conducting an intensive case study of a man in his mid-forties who suffers from the rare syndrom called prosopagnosia. Prosopagnosics are unable to recongize people by face, including their children or their own faces in a mirror. This man’s wife has to wear a special ornament-a ribbon of a certain color or a distinctive hairclip-when they attend public events so that he will be able to find her. As I drove him home once, I saw two childre in his driveway. I asked him if they were his children and he replied, “Must be, they are in my driveway.”

Posted by razib at 05:56 PM

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Does this matrix make sense?

Check it:   Particularistic Universalistic Exclusivistic Judaism   Islam/conservative Christianity  Pluralistic Hinduism/Daoism/Shintoism   Liberalism/liberal Christianity/Buddhism   

Does this make any sense? Just a general impression that I wanted to recast in matrix form. There seems a contradiction in pluralistic universalism….

Addendum: Please note that I’m making idealized generalizations about the systems of belief. In practice, the typical Jew, Muslim, Hindu and Christian are quite similar in their day-to-day behavior, to a far greater extent than a axiomatic reading of their religious system would indicate. The differences begin to manifest themselves when you sum up the collective behavior of groups of humans and you intregate over time

. Posted by razib at 12:52 PM

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Regression to the Mean: a puzzle

We all like to think we understand regression to the mean, as it arises frequently in statistics, genetics, psychology, economics, etc.

Roughly speaking, wherever there is a correlation between two variables, we expect the more extreme values of one variable to be associated with less extreme values of the other, which is said to ‘regress’ towards its mean value.

More precisely, for any sets of variable x and y, measured in units of their own standard deviation, if there is a correlation other than 0, 1, or -1 between x and y, then, for any given value of x, the mean value of the y’s corresponding to that value of x will be closer to the mean of all y’s than that value of x is to the mean of all x’s. Since correlation is a symmetrical relationship, the same proposition will be true if we substitute y for x and x for y throughout.

This, or something like it, is the definition usually given of ‘regression to the mean’, and it is commonly said that it is a necessary consequence – or even a ‘mathematical necessity’ – wherever there is an imperfect but non-zero correlation.

So I was disconcerted to find an example that apparently violates the general rule: a case where there is a non-zero correlation (and in fact quite a strong one) but no regression to the mean…

Godless comments: There is a fallacious assumption here, which is that the linear conditional MMSE formula applies to the case of non-normal random variables. See inside .

David B comments: My original post anticipated this objection, saying: “I suppose one response to this puzzle, or paradox, is that the relationship between the variables in this case is not linear, so the standard Pearson formulae for linear regression and correlation are not appropriate. I would agree that the Pearson formulae are not ideal for this case, but I don’t see that in any strict sense they are invalid. The correlation does account for about half of the total variance, which is better than many correlations that are accepted as meaningful.”

So I don’t accept that the use of a linear regression formula is strictly a ‘fallacy’. I note that the statistics text that Godless links to seems to take a similar view, saying: “Such linear estimators may not be optimum; the conditional expected value may be nonlinear and it always has the smallest mean-squared error. Despite this occasional performance deficit, linear estimators have well-understood properties, they interact will [sic: presumably the author means ‘well’] with other signal processing algorithms because of linearity, and they can always be derived, no matter what the problem.”

In the present case, a linear regression is not ‘optimum’, but it accounts for about half the variance, which is not bad, and I’m not sure that any other formula would do much better.

Draw, or imagine, a scattergram as follows.

First draw the x and y axes, with x = y = 0 at the origin, and mark off 2 units (in inches, or whatever) along each axis in both directions from the origin.

Then draw a square with sides of length 2 units in the upper right quadrant, with its lower left corner on the origin. Draw a similar square in the lower left quadrant, with its upper right corner on the origin.

Now fill each square evenly with dots, except that no dots are to fall on the axes themselves.

Let each dot represent a pair of associated x and y observations.

It is evident that:

a. The mean of all the x observations is 0. Similarly, the mean of all the y observations is 0.

b. The mean of all the positive x observations is 1. Similarly, the mean of all the positive y observations is 1, while the mean of the negative x and y observations is -1.

c. The standard deviation (sd) of the x’s is equal to the standard deviation of the y’s. They are both greater than 1. (I estimate that they are around 1.2, but the precise value does not matter.)

d. The pairs of x’s and y’s have a positive covariance, since they all fall in the ‘positive’ quadrants of the scattergram. The covariance is approximately 1.

e. There is a positive correlation between the x’s and y’s. Assuming 1.2 for the standard deviation of the x’s and y’s, the Pearson product-moment correlation coefficient is about 1/1.44 = approx .7.

f. However, there is no regression to the mean. For each positive x value, the mean of the corresponding y values is 1, while for each negative x value it is -1. For half of the x values (those between 1 and 2 or between -1 and -2), the mean value of the corresponding y’s is closer to the mean of all y’s (0) than the x value is to the mean of all x’s (also 0), so it could be said that for these values there is a regression towards the mean, but these are exactly balanced by the other half of the x values, where there is a ‘regression’ of equal size away from the mean. So overall there is no regression. This conclusion is not affected if we measure each variable in sd units.

So it appears that we have correlation, but no regression to the mean. Of course, we can still formulate a ‘regression equation’ to predict the value of x given y or y given x. Since the sd’s of the two variables are the same, the regression coefficients are equal to the correlation coefficient. The predicted values of the dependent variables are always closer to their means than are the given values of the independent variables. So there is a predicted regression. But the actual observed values show no regression to the mean, as usually defined.

I suppose one response to this puzzle, or paradox, is that the relationship between the variables in this case is not linear, so the standard Pearson formulae for linear regression and correlation are not appropriate. I would agree that the Pearson formulae are not ideal for this case, but I don’t see that in any strict sense they are invalid. The correlation does account for about half of the total variance, which is better than many correlations that are accepted as meaningful.

Perhaps a better response is that the ‘population’ of observations is really a combination of two different populations, within each of which the correlation is zero, but which have different means. It is known that a combination of populations with different means gives rise to a correlation sometimes described as ‘spurious’, or an ‘artifact’. However, real-life populations are also often a mixture of heterogeneous sub-populations, and it seems to be a matter of taste how far it is legitimate to combine them together.

Anyway, I thought the puzzle might be of interest, so I would welcome any comments. Of course, the example is a very simple one, but there may be more complicated real-life examples where there is less ‘regression to the mean’ than might be expected simply on the basis of correlation coefficients.

Godless comments:

The mistake here is in assuming that E[Y | X] = r X for arbitrary *non*-normal random variables X and Y. The conditional MMSE minimizer is not a linear function of the measurement (= rX) in the case where X and Y are not jointly-normal random variables.

Reference on the MMSE (= minimum mean square estimator) for Y given X. Note that the estimator for Y given X only has the simple form “rX” in the case where Y and X are correlated standard normal random variables with correlation coefficient r. In the general bivariate normal case, Y = aX+b where a and b are complicated terms. [1]

In the general case, where X and Y are (say) correlated gamma random variables, E[Y|X] need not take on the form of a simple linear function. In general, E[Y|X] = f(X) is a nonlinear function of the measured variable X with which we are predicting the expected mean of the variable Y, and may exhibit behavior such that E[Y|X] = f(X) is GREATER
than the measurement X. This is the opposite of the expected behavior in the linear formula when r is less than 1 (as |r| |X| is less than |X| always). That is, it violates the regression to the mean rule, which is only strictly valid when E[Y|X] = f(X) and |f(X)| < |X| for all X, of which E[Y|X] = rX with |r| less than 1 is a special case.

Reference on linear vs. nonlinear MMSE.

A further subtlety: E[Y|X] need not be the same as argmax P(Y|X). That is, the mean need not equal the mode in the conditional probability distribution.

This situation is purely mathematical and has nothing to do with Galton’s “fallacy”. Perhaps I will make a fuller discussion of this in a later post.

[1] They aren’t *that* complicated. You can easily remember them through the projection formula, as the set of zero-mean random variables is an inner product space and all the standard formulas apply (with the inner product (X,Y) = E[XY]).

Posted by David B at 03:10 AM

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Much ado about what?

Doing a search on PUBMED, I get:

6 responses for spandrel
28 responses for punctuated equilibria
107 responses for evolutionarily stable strategies
110 responses for inclusive fitness
263 responses for kin selection

“Not scientific” as they say. But when I hear people saying that “Stephen Jay Gould was a world famous evolutionary thinker,” I wonder if they have ever heard about punctuated equilibria. Stephen Jay Gould benefited from the Julia Roberts Effect, despite being horsey, she is one of America’s beauties, because she is America’s A-list actress, while she is an A-list actress because she gets big roles…because she is one of America’s beauties…. Stephen Jay Gould was one of the century’s greatest evolutionary thinkers because he had so many columns and was such a public presence, and he was a public presence who got to comment on baseball because he was one of the century’s greatest evolutionary thinkers (something milder seems to have happened to Stephen Hawking, though his ideas are not nearly as kooky from what I gather [and can admit he was wrong], and at least he suffers from a disease which makes his survival somewhat miraculous).

Posted by razib at 06:52 PM

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The eternal recurrence of elite betrayal

Steve Sailer’s latest column considers why the black American elite is now turning against immigrant blacks and their children as far as affirmative action at elite universities goes, while neglecting to say a word about the allotment toward Hispanics. The short answer is that it doesn’t help them out to reduce the Hispanic allotment, since their own is fixed at around 8%.

But what does this have to do with most of black America? It seems arguing over who gets into Harvard is beyond the pale of most black Americans as far as everyday experience, let alone most Americans. But it matters to the elites, and in this case, the black elite. I think what we are seeing is a heightening of the differences in priorities between the black elite, and the masses of African Americans they claim to represent.

It is a paradox of “mass movements” that they need leaders, and so birth their own elites. Senator Robert C. Byrd might be of working-class Appalachian antecedents, and those might be the folk he represents, but over the decades he has been given an education by those same people, and now waxes eloquently with the diction of America’s elite. One might wonder if the descendents of Robert Byrd will have the same emotional attachment to their ancestor’s working class origins, or whether they will join the “elite” that he so despised in his youth? In victory does the power of the masses drain away, suborned by the ambition toward advancement that lurks in the hearts of all men, most of all the “leaders” of the meek not born to power.

About 6 months ago I was reading the history of the Han dynasty, China’s defining epoch, the dynasty which gave the majority Chinese their term of self-appellation. Near the end of the “Former Han,” and during the interregnum of the “usurper” Wang Mang, a series of natural disasters let to a host of uprisings. Peasants and brigands roamed the country-side, and the great lords of the previous age fell from grace. But a time came when they needed to select a leader, and who did they choose? A variety of great men rose, and one of them was a distant scion of the old Han house. He went on to found the Latter Han, and that dynasty in the end recapitulated the fall of its predecessor.

I can give a host of similar examples cross-culturally. But in the end, the moral is the same. From the masses rise men and women who are fueled by their fury at injustice, wishing to make a better world, who wish to do good for the toiling common man, their own kith and kin, in words and deed if not their self-interested hearts. A new order rises, tight in execution and focused on higher moral purpose. As the generations progress, the initial ardor diminishes, and the Mandate of Heaven, the acclamation of the populous, fades. The made leaders of the first generation give way to the born leaders of the next, and so forth. The masses and those who they chose to lead them drift apart. And so another generation of made leaders is born, and the terminal generation of the last dispensation is tossed aside.

Between the cross-purposes of human nature we fabricate the abstract mental superstructure of ideologies and religions that give purpose and compel us toward the “greater good.” But in the end, there is the foundation of our nature that drags us down as time works its so-called wonders. The traditional Chinese acknowledged both elements and fused it into a compound ideology, lionizing the atomic importance of family, but scaffolding it with the importance of public virtue, and reformulating non-kin relations as ones of familial import. The Mandate of Heaven was nothing more than a word, a force that encapsulated the entropic unwinding of the Chinese social order over a period of centuries and its resurrection with the rebirth of the Great Idea of the Son of Heaven and father of the nation.

As far as this nation is concerned, the black American elite seems to be complacent in its assured role, ignorant or contemptuous of the fates of elites who naturally over time betrayed their mandate. Clever with words, gifted with connections with a patronizing white companion class and rich accumulated social merit, they swim on the glory of past deeds. But memories fade, interests change, and there is always a new righteous elite ready to take on the mandate of the people and do “justice,” unhesitant and brutal in their sense of purpose.

As they say, “Only the paranoid survive.”

Update: Thanks to David Orland on some editing advice. Though I would ask for the indulgence of readers, this post was hacked out in 20 minutes.

Posted by razib at 04:30 PM

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