Chance News 19
Like dreams, statistics are a form of wish fulfillment. - Jean Baudrillard
Everyone believes in the normal law of errors: the mathematicians, because they think it is an experimental fact; and the experimenters, because they suppose it is a theorem of mathematics. However, it is neither an experimental fact nor a theorem of mathematics - Garbriel Lippmann.
This Forsooth was suggested by Paul Alper.
The Auburn football team appeared to be the biggest benefactor of Professor Petee's directed-reading offerings. The 18 football players received an average G.P.A. of 3.31 in the classes ...In all of their other credit hours at Auburn, their average was 2.14.Top grades and no class time.
New York Times, July 14, 2006
P.S. The star running back Carnell (Cadillac) Williams, now playing in the National Football League, said the only two classes he took during the spring semester of his senior year were one-on-one courses with Professor Petee. One of these two courses was a statistics class. Williams described the class this way: 'You're just studying different kinds of math. It's one of those things where you write a report about the different theories and things like that.'"
Another Look at "The Kindness of Strangers?"
In a recent wiki, The Kindness of Strangers, based on a paper Study of the Therapeutic Effects of Intercessory Prayer (STEP)in cardiac bypass patients by Herbert Benson et al., you will find a commentary regarding this latest statistical attempt to foist intercessory prayer--IP as it is now known--into the realm of science. Nevertheless, despite the excellence of the wiki, some additional comment is in order. As stated, the $2.4 million dollar waste of time was sponsored by the foundation of the billionaire John Templeton; for more on the individual, his son and the foundation, and why so many "American medical schools now offer courses on links between health and spirituality," the reader is directed to The Templeton Foundation: A Skeptic's Take. In short, the answer is money for the asking.
The same issue of The American Heart Journal [vol. 151, Issue 4, April 2006, Pages 934-942] which contained the paper by Benson -- there are, believe it or not, 15 other authors!--also has an editorial by Krucoff, Crater and Lee [Pages 762-764] which states "the STEP investigators' interpretation of the study results appears to reflect more the cultural bias that healing prayer...is only capable of doing good if it does anything at all." Unfortunately, the editorial while being skeptical, fails to note some other failures inherent in the article.
For one thing, unlike real medicine, there is no notion of dosage as in amount of time spent per individual praying. For another, in defiance of physical laws, distance between patient and prayers [St. Paul, MN, Worcester, MA and Lee's Summit, MO] appears to be irrelevant. And then, there is the statistical difficulty of going from a sample to a population. As is virtually always true, the people doing the praying are Christians. Consequently, while the patients who were prayed for in this study did worse than those who weren't prayed for, it is conceivable that other religions would score higher. However, Templeton is not a Moslem, Shintoist or a Hindu so we will never know because I suspect his foundation is not eager to pursue this line of reasoning.
The Annals of Behavioral Medicine, June 13, 2006 has an excellent article, "Are There Demonstrable Effects of Distant Intercessory Prayer? A Meta-Analytic Review" by Masters, Spielmans and Goodson. STEP is not included but 14 other studies are, including the discredited one by Lobo, Cha and Worth--Lobo withdrew his name and Worth is in prison. Based on their meta-analysis, Masters, Spielmans and Goodson write,
There is no scientifically discernable effect for IP as assessed in controlled studies. Given that the IP literature lacks a theoretical or theological base and has failed to produce significant findings in controlled trials, we recommend that further resources not be allocated to this line of research.
1. Why is the following phrase cherished by statisticians and other scientists? "Extraordinary claims demand extraordinary evidence."
2. If one assumes that IP is absurd, what is it about conventional prayer--prayer by the patient, prayer by his loved ones, etc.--that distinguishes it from IP?
3. If IP has some effect, is it ethical to prayer for someone without his knowledge?
4. Benson claims "We were unable to locate other Christian, Jewish or non-Christian groups that could receive the daily prayer list of this multiyear study." Suppose they did locate these other groups. Speculate on the outcome if these other groups were included.
5. According to The Columbia University 'Miracle' Story when referring to Daniel Worth, "A good rule of thumb for a medical journal is that anyone who uses the names of dead children in order to fraudulently obtain bank loans, jobs and passports is not a reliable source of data." However, the other two authors are medical doctors so should suspicion diminish?
6. Steven Weinberg is a Nobel Laureate in Physics and an atheist. He once said, I am all in favor of a dialogue between science and religion, but not a constructive dialogue." What do you think he means?
7. Discuss how the interest in IP reflects the shift to the right in American politics and religion.
8. What sort of pardox is implied when people of faith need statistics to buttress their beliefs?
Submited by Paul Alper
Frederick Mosteller, 1916-2006
Dr. Frederick Mosteller, 89, put statistics to a world of uses.
Boston Globe, 26 July, 2006
Fred Mosteller is well known for his many contributions to teaching and research in probability and statistics, and the application of probability and statistics to real world problems. But this obituary also reminds us what a great guy Fred was. Here are a few quotations from the Globe article:
A roll of the dice set Frederick Mosteller on the path to becoming the preeminent statistician of his time.
When three dice are rolled, a professor asked when Dr. Mosteller was a college sophomore, what is the chance the numbers on the faces will add up to 10? Intrigued, the young man sought advice from another professor, who showed him an advanced way to solve the probability equation if even more dice were involved.
"It was the most marvelous thing I had ever seen in mathematics," he said in the book "More Mathematical People." "It used mathematics that, up to that time, in my heart of hearts, I had thought was something that mathematicians just did to create homework problems for innocent students in high school and college."
And one of Dr. Mosteller's papers in The American Statistician, not a journal many turn to for a chuckle, included this observation about teaching: "In spite of the 20th-century visual aids, blackboard and chalk are still a mainstay for most of us. . . . Some people, like me, have rather poor handwriting, and to them I say just one thing, write large. This helps a great deal. Then there are those who write beautifully and small, and very lightly and swiftly, so that students can't keep up. I assume that there is a special part of the Inferno waiting for them, but I cannot imagine a sufficiently severe punishment. For those who stand in front of what they have written and block it, no doubt the afterlife holds still worse."
Born in Clarksburg, W.Va., he moved as a child with his family to a suburb of Pittsburgh. To earn money for college, he worked for his father's highway building company on a road crew that played poker when rain delayed a project. Showing an early knack for the laws of probability, he became a good enough player that decades later several professor friends presented him with an honorary siphon because he had siphoned away so much money from them at the poker table.
His work in a variety of disciplines led Dr.Mosteller to achieve an unprecedented feat. He helped found Harvard's statistics department and was its first chairman. He went on to serve as chairman or acting chairman for three other departments, along with teaching courses at Harvard Law School and the Kennedy School of Government.
Even after nominally retiring to professor emeritus status, he kept doing significant work. As recently as the mid-1990s, when he was approaching 80, he influenced national policy by using statistics to show that smaller class sizes allowed students to learn more and perform better.<\blockquote>
The IQ of Ashkenazi Jews
Steven Pinker is a well known Harvard Psychologist. In this New Republic article he discusses the claim that Ashkenazi Jews have an advantage in average intelligence. Ashkenazi Jews, also called Ashkenazic Jews, are descendents from the medieval Jewish communities of the Rhineland. There are about 6 million Jews in the US of which approximately 5 million are Ashkenazi Jews. You can obtain more information about the history of Ashkenazi Jews here.
The appearance of an advantage in average intelligence among Ashkenazi Jews is easier to establish than its causes. Jews are remarkably over-represented in benchmarks of brainpower. Though never exceeding 3 percent of the American population, Jews account for 37 percent of the winners of the U.S. National Medal of Science, 25 percent of the American Nobel Prize winners in literature, 40 percent of the American Nobel Prize winners in science and economics, and so on. On the world stage, we find that 54 percent of the world chess champions have had one or two Jewish parents.
Does this mean that Jews are a nation of meinsteins? It does not. Their average IQ has been measured at 108 to 115, one-half to one standard deviation above the mean. But statisticians have long known that a moderate difference in the means of two distributions translates into a large difference at the tails. In the simplest case, if we have two groups of the same size, and the average of Group A exceeds the average of Group B by fifteen IQ points (one standard deviation), then among people with an IQ of 115 or higher the As will outnumber the Bs by a ratio of three to one, but among people with an IQ of 160 or higher the As will outnumber the Bs by a ratio of forty-two to one. Even if Group A was a fraction of the size of Group B to begin with, it would contribute a substantial proportion of the people who had the highest scores.
We illustrate this in terms of one of the many studies carried out to compare the IQ's of Jews with Gentiles (non-Jew whites). For this we use a study The Intelligence of American Jews by Richard Lynn reported on the web in 2004. Lynn is a well known researcher in the field of Intelligence. You can read about his work here. In this article, Lynn reviews the results of studies carried out to compare the IQ's of Jews and Gentiles. Lynn writes:
The existing state of the research literature on the IQ of American Jews is ...that some studies have shown that their verbal IQ is about the same
as that of gentile whites while other studies have shown that it is considerably higher at 107.8 (Backman, 1972), 112.6 (Herrnstein and Murray, 1994) and 112.8 (Bachman, 1970). However, the last of two of these studies have sample sizes of fewer than 100. There is room for more data on the IQ of American Jews,
and it is to the presentation of this that we now turn.
For his study, Lynn uses data from the annual surveys on approximately 1,600 individuals in continental United States carried out by the American National Opinion research Center (NORC). Lynn says "previous studies suggest that Jews tend to excel in verbal and numerical ability but not unexceptionally in spatial or perceptual problems with most convincing advantage in verbal ability".
In the years 1900,1991, 1992, 1994 and 1996 NORC surveys included a 10 word vocabulary test. Lynn used the results of these vocabulary tests for his data, remarking that previous studies suggest that vocabulary is a good measure of both general intelligence and verbal intelligence. Here are his results for Jews and Gentiles:
Ethnic Group N Mean Sd IQ Jews 150 7.32 2.16 107.5 Gentiles 5300 6.28 2.03 100.0
In this study Lynn is comparing the average IQ's of different groups with Gentiles as a reference group. The raw scores are normalized by a linear transformation to make the raw scores of the reference group have mean 100 and standard deviation 15 as follows.
From our table we see that the raw scores of the Gentiles had standard deviation 2.03 and mean 6.28. To make the standard deviation 15 we multiply these scores by a = 15/2.03. This makes the mean a x 6.28. To make the mean 100, we add b = 100 - 6.28 x a = 53.60 to the raw scores. This does not affect the standard deviation so the Gentile scores have mean 100 and standard deviation 15.
We use the same normalization for the Jews. From our table we see that they had standard deviation 2.16. With normalization it is 2.16 x a = 15.96. The mean was 7.32 so now it is 7.32 x a + b = 107.8. So the Jews have an average IQ of 107.8 Note this agrees with the Backman, 1972 study mentioned by Lynn in his survey of previous studies. Many consider this study to be the most reliable study of the IQ of American Jews.
Pinker makes the point that the modest difference in means can lead to a significant difference in the probability of tail events. For example consider the probability that a randomly chosen member of one of the two groups has an IQ greater that 145. Using a normal distribution applet we find that this probability is .0013 for the Gentiles and .0095 for the Jews. Thus we can expect about 7 times more Jews than Gentiles to have an IQ of at least 145. Similarly we can expect about 16 times more Jews than Gentiles to have an IQ of at least 160. However, consistent with our second quotation, it is well known that the normal approximation is not a good approximation for such extreme tail probabilities.
This method of comparing the IQ and performance of different groups is used in a number of essays on a web site named La Griffe du Lion (The claw of the lion).These essays are written by one or more anonymous persons. We will call the authors Griffe. In an essay called Assessing the Ashkenazic IQ , Griffe compares Ashkenazi Jews with Gentiles using data from the annual USSR Chess Championships and the winners of the mathematics Fields prize. Griffe summarizes the results by writing:
In sum, from the frequency of appearance of Ashkenazic Jews in the annual USSR Chess Championships, we assess the mean Ashkenazic IQ to be 116. And from the frequency of Ashkenazic Fields medalists, we infer an Ashkenazic "math IQ" of 111. On these numbers, gentlemen, you can hang your yarmulkes.
As Pinker remarks, it is easier to show that Ashkenazi Jews have higher IQ's than Gentiles but much harder to explain why this is the case. The big issue is obviously heredity vs environment. Pinker writes:
A team of scientists from the University of Utah recently strode into this minefield with their article Natural History of Ashkanzi Intelligence, which was published online in the Journal of Biosocial Science a year ago, and was soon publicized in The New York Times, The Economist, and on the cover of New York magazine.
The Utah researchers Gregory Cochran, Jason Hardy, and Henry Harpending proposed that Ashkenazi Jews have a genetic advantage in intelligence, and that the advantage arose from natural selection for success in middleman occupations (moneylending, selling, and estate management) during the first millennium of their existence in northern Europe, from about 800 C.E. to 1600 C.E. Since rapid selection of a single trait often brings along deleterious by-products, this evolutionary history also bequeathed the genetic diseases known to be common among Ashkenazim, such as Tay-Sachs and Gaucher's.
The Lion's Claw
Griffe's provocative essays combine data, social science, math, and statistics. Results are described in colorful ways. Here is a graphic from an essay called The color of death row showing the surprising result that, when it comes to death penalties, Southern states are the ones that would appear to have a bias against whites.
Of course it is natural to wonder who Griffe is. Those who have made suggestions on the web believe that Griffe is one person, a man. Suggestions that we have seen on the web are Richard Lynn whom we have just met, Charles Murray co-author of The Bell Curve (see how deftly he explains one of Griffen's essays in an OpinionJournal editorial), and Dan Seligman, currently a columnist for Fortune Magazine but better known to Chance News readers for his Mr. Statistics column in Fortune Magazine column in the 1990's.
As for the significance of the pseudonym "the claw of the lion", here is an extract from a chapter called "The Lion's Claw" of John Derbyshire's recent book Unknown Quantity:
I cannot resist at this point telling my favorite Newton Story, though I think it is quite well known. In 1696 the Swiss mathematician Johann Bernoulli posed two difficult problems to the mathematicians of Europe. Newton solved the problems the day he was shown them and passed on his solutions to the president of the Royal Society in London, who sent them Bernoulli without telling him who had supplied them. As soon as he read the anonymous solutions Bernoulli knew them to be Newton's---"tanquam ex ungue leonem," he said ("as by [his] claw [we know] the lion")
(1) Do you think a Stylometric expert could determine who Griife is? How would you go about trying to solve this mystery? Could you at least eleminate some of the guesses?
(2) What are the 95% confidence intervals for Lynn's IQ's estimate?
(3) Lynn referred to the article: Patterns of Mental Abillities: Ethnic, Socioeconomic, and Sex Differences by Margaret E. Backman, American Educational Research Journal, Vol, No 1, pp. 1-12, 1972. He summarizes the results of this study:
There is only one study of the
intelligence of American Jews in the last half century which appears to be representative and had a reasonable sample size. This is Backman’s (1972) analysis of the data in Project Talent, a nationwide American survey of the abilities of 18- year-olds carried out in 1960. The study had sample sizes of 1,236 Jews and 1,051 white gentiles (in addition to 488 blacks and 150 Orientals). IQs for six factors were calculated. The mean IQs of the Jews in relation to gentile white means of 100 and standard deviations of 15 were as follows: verbal knowledge (described as “a general factor, but primarily a measure of general information” and identifiable with Carroll’s (1993) gc or verbal comprehension factor - 107.8; English language – 99.5; mathematics – 109.7; visual reasoning (“a measure of reasoning with visual forms”) – 91.3; perceptual speed and accuracy –
102.2; memory (short term recall of verbal symbols) – 95.1.
Is it clear from this that American Jews are smarter than Gentiles?
Submitted by Laurie Snell
Probability of Pregnancy
On June 24th, the UK newspaper The Guardian contained the following quote:
A friend of mine got pregnant the first time she slept with her (now) husband, at the age of 43; this was after she had made a documentary on infertility and had been repeatedly told by fertility experts that her chances of conceiving naturally were "less than 5%". This may be right as a general statistic, but it wasn't right in her case: her chances of conceiving, provided she had sex at the right time of the month, were 100%.
(Full article available here).
- What do you think the author meant by "a general statistic". What other sorts are there?
- Any person's chance of getting pregnant is 0% or 100%, so what does a 5% chance of getting pregnant mean?
- Are there different (but equally correct) interpretations of that 5% probability?
Financial ties to industry cloud depression study.
Wall Street Journal, Tues. July 11, 2006
Another example of what the most important question a consumer of statistical information should ask can be found in this Wall Street Journal article. The front-page piece is a lengthy commentary on a JAMA article dealing with the desirability of prescribing antidepressant medication to pregnant women. ("Relapse of major depression during pregnancy in women who maintain or discontinue antidepressant treatment", Cohen LS, Altshuler LL, Harlow BL, et al JAMA. 2006; 295: 499-507).
Turns out that "most of the 13 authors are paid as consultants or lecturers by the makers of antidepressants." And apparently paid well, although none chooses to reveal the amount to the WSJ. Further, as the lead author, Dr. Lee S. Cohen, of the JAMA article put it, "'it didn't seem relevant' for him and several of his co-authors to disclose their industry relationships in the JAMA paper in part because the study was funded by the government, not drug makers." JAMA's editor-in-chief "says the journal wasn't aware of the relationships" to the pharmaceutical industry but Dr. Cohn's explanation "will be published very soon in an upcoming issue of JAMA." In any event, he says, "we are not talking about megabucks" although "He declined to specify what he does in his consulting role for the companies or how much he is paid."
1. Defend Dr. Cohn's distinction regarding source of funding. Criticize Dr. Cohen's distinction regarding source of funding.
2. Justify JAMA's editor-in-chief's delay in publishing Dr.Cohen's explanation.
3. Speculate on why there are so many authors.
4. Suppose a medical doctor with ties to antidepressant pharmaceutical companies would publish a study which revealed that antidepressants were harmful in certain situations. What, if any financial consequences would he suffer?
5. The JAMA article states, "the current investigation used a nonrandomized design." Further, the study was completely without blinds, that is, the physicians and the patients were aware of which arm of the study was applicable. How do these facts affect the faith in the study?
Submited by Paul Alper