Chance News 52

Quotations

Correlation coefficients are now about as ubiquitous
and unsurprising as cockroaches in New York City.

In Stephen Jay Gould's The Mismeasure of Man,
Second edition, 1996, Page 286.

Forsooths

Growing up in China [Yale Assistant Professor of Genetics] Jun Lu might have pursued a career in math if his father, a mathematician, hadn't advised against it.

His reasoning: Math can be done with little more than a pen, paper and your mind. For 2,000 years, thinkers have had those tools to contemplate the questions of mathematics. "Any questions left behind by them are probably very hard to address," Lu said.

But biology, his father said, capitalized on advances in technology. His son could have the chance to explore a new frontier.

Arielle Levin Becker, "Weeding Out Cancer"
The Hartford Courant, July 12, 2009

A Princeton professor of economics/public affairs, who is also vice chairman of the Promontory Interfinancial Network and former vice chairman of the Federal Reserve Board, argues that the economy has bottomed out.

[T]here is a reasonable chance—not a certainty, mind you, but a reasonable chance—that the second half of 2009 will surprise us on the upside. …. [T]his seemingly high growth scenario … [would follow] directly from the arithmetic of hitting bottom. …. Eventually those huge negative numbers [associated with the decline of some GDP components] must turn into (at least) zeroes … [and] … almost certainly do better. …. None of these events are probabilities; they are all certainties. The only issue is timing, about which we can only guess.

“The Economy Has Hit Bottom” [1]
The Wall Street Journal, July 23, 2009

Wednesdays, summer/spring seasons associated with increased suicide rates

“National Study Finds Highest Rate Of Suicide On Wednesdays”
by Arielle Levin Becker, The Hartford Courant, July 11, 2009

A University of California at Riverside study, published in Social Psychiatry & Psychiatric Epidemiology, found, surprisingly, that suicides are more likely to occur on Wednesdays than on any other day of the week. The study’s results were based on the 131,636 suicides in U.S. death records for the period 2000-2004.

One chart [2] gives the percentages of all U.S. adult suicides by day of the week: Sunday 11.8%; Monday 14.3%; Tuesday 12.7%; Wednesday 24.6%; Thursday 11.1%; Friday 11.2%; Saturday 14.4%.

A social worker suggested that workplace stress might mount up through the week, with weekend relief seeming too far away by Wednesday. A reader [3] blogged that a person in crisis might be upset to find that his/her therapist has Wednesday off, while he/she has to work. One of the researchers advised mental health workers to schedule more patient appointments on Wednesdays.

The researchers also found that summer and spring were more common seasons for suicide than fall or winter, another counterintuitive finding. A second chart [4] gives the percentages of all U.S. adult suicides by season: Autumn 23.8%; Winter 24.4%; Spring 25.8%; Summer 26%.

A third chart [5] shows the trend in the number of U.S. adult suicides (eyeballed approximations): 24,200 in Year 2000; 25,100 in Year 2001; 29,000 in Year 2002; 26,100 in Year 2003; 26,900 in Year 2004.

A Hartford Courant analysis of the 966 reported adult suicides in Connecticut for the period 2001-2004 showed less variation in rates among the days of the week. “Most suicides – 16.7 percent — occurred on Tuesday, while 16.4 percent occurred on Monday and 14.5 percent on Wednesday. Thursday had the lowest occurrence, 12.1 percent.”

However, Connecticut records agreed with the national study with respect to summer and spring being the more common seasons for suicides. A Connecticut mental health worker hypothesized that:

“People think it’s normal to be depressed in the winter. “Spring is the time of year when people are supposed to be rejuvenated and outside and enjoying themselves, and if you’re not, it makes you feel comparatively worse than everybody else, which may make you feel more hopeless,” he said.

Discussion

1. With respect to the national study, do you think that the differences among days of the week and/or among seasons of the year were statistically significant?

2. Were you surprised that the Connecticut analysis showed less variation than the national study, in rates among the days of the week, from a statistical point of view?

Two new anti-aging studies

“Two Mammals' Longevity Boosted”
by Keith J. Winstein, The Wall Street Journal, July 9, 2009

In the journal Nature, anti-aging researchers from Maine, Michigan, and Texas reported on a study that found that a chemical (Wyeth’s rapamycin) used to treat organ transplant patients increased the life span of mice. Because the chemical suppresses the immune system, humans are advised against taking the drug to prolong their lives.

Mice given rapamycin -- starting when they were 600 days old, or roughly the equivalent of 60 human years -- lived longer on average than mice who didn't get the drug. Their "maximal life span" -- meaning the age at which 10% of the mice were still alive -- increased to 1,245 days for females, compared with 1,094 days for those not fed the drug, or a 14% increase. For males, the maximal life span was 1,179 days, a 9% increase over the 1,078 days for those not fed the drug.

In an upcoming issue of Science magazine, University of Wisconsin scientists will publish results of a study that shows that reducing the calorie intake of monkeys extends their lives. A person who has seen the study's results said that "after 20 years, only 20% of the calorie-restricted monkeys had died, compared with half of the monkeys on a normal diet."

The Wisconsin study, which began in 1989 with 30 monkeys and added 46 more in 1994, is an effort to test calorie restriction in an animal genetically closer to humans. Researchers have known since the 1930s that eating 30% fewer calories than normal lengthens the life span of mice. Half the monkeys were given a normal diet, and half had their food intake cut back by 30% at roughly age 10.

A British gerontologist commented, "Aging is, unequivocally, the major cause of death in the industrialized world and a perfectly legitimate target of medical intervention."

A blogger [6] provided the following hypothetical dialogue.
Joe: Do you want to live to 100?
Pete: Don't ask me; ask the guy who's 99.

THE FLAW OF AVERAGES

Sam L. Savage
John Wiley & Sons, published June 2009.

This is a fascinating book and I will review it here but with the slowness that comes with old age. This ola age gives me fond memories of Sam's father, I. Richard Savage. In his Memorial we read:

Savage is one of a few mathematical statisticians of his generation who chose to pursue the application of statistical principles and concepts to problems of public policy.

Now his son Sam is one of the few of his generation who choose to pursue the application of probability principles and concepts to problems of public policy.

The book starts with an explanation of the Flaw of Averages. A humorous example of this involves the statistician who drowned crossing a river that was on average 3 feet deep. It is pretty clear from this what Flaw of Averages means, but we are also invited to go to flawofaverages.com. Here we read that "Plans based on average assumptions are wrong on average!"

The book assumes no statistical background, but for those with statistical training the author claims he can repair the damage within the first few chapters. This type of humor appears throughout the book. We find here other examples of the Flaw of Averages including this one:

Suppose you are on time to your appointments on

average, and so is your partner. When you go someplace together, the Flaw of Averages ensures that you will be late on average. In fact the FOA pretty much explains why everything is behind

schedule, beyond budget, and below projection.

Sam give many more serious applications of FOA to illustrate the use of computer tools such as spreadsheets and simulations. Sam also makes some of these available on the internet. Here is one of these. We go to http://flawofaverages.com/book/ and we see a simulated version of the following problem to be continued

Laurie Snell

To be continued

Sabermetrics

“Baseball Veers Into Left Field”, by Austin Kelley, July 17, 2009

This article describes some results of researchers in the field of “sabermetrics,” apparently the “science of baseball analysis,” in which “no topic is too small or hypothesis too unlikely” to be investigated due to the huge database available. Phil Birnbaum edits the statistical-analysis newsletter for the Society for Baseball Research (SABR).

From Wayne State University researchers, we learn that:

[M]ajor-league players who have nicknames live 2 ½ years longer, on average, than those without them.

[B]aseball players live a little longer than average folks.
[P]layers who debut at a young age have a shorter life expectancy than their slower-developing teammates.
[P]layers with "positive" initials have a longer lifespan than those with initials like "P.I.G."
[S]outhpaws are a little shorter than right-handers.

[M]ajor-leaguers are more likely to die on their birthdays than chance would predict.

From a pair of researchers, at University of California-Berkeley and Yale University, we have:

[P]layers whose first or last name begins with "K" strike out more than those without "K" initials.

From another pair of researchers, at Pennsylvania State University and Washington University, we are told:

Democrats support the designated-hitter rule more than Republicans.

The website directs readers to some articles:
"The Etiology of Public Support for the Designated Hitter Rule” [7]
"Moniker Madness: When Names Sabotage Success” [8]
"The ‘K’ study for real” [9]
"Underestimating the Fog” [10]

"Strong Evidence" standards

“The Final Report of the National Mathematics Advisory Panel”
U.S. Department of Education, 2008

This report is the result of two years of work by a panel of mathematicians, educators, and foundation representatives, formed in 2006 “with the responsibilities of relying upon the ‘best available scientific evidence’ and recommending ways ‘… to foster greater knowledge of and improved performance in mathematics among American students.’” Their main focus in seeking national standards was on the “delivery system in mathematics education,” especially the high school algebra sequence and pre-secondary math courses leading up to it.

In compliance with the President’s Executive Order forming the group, the Panel’s “assertions and recommendations were based the “highest-quality evidence available from scientific studies.” The Panel reviewed “more than 16,000 research studies and related documents. Yet, only a small percentage of the 16,000 research studies reviewed met the standards of evidence and could support conclusions.”

The Panel placed strongest confidence in studies that “test hypotheses, that meet the highest methodological standards (internal validity), and that have been replicated with diverse samples of students under conditions that warrant generalization (external validity)….” The Subcommittee on Standards of Evidence developed relatively detailed standards for rating the quality of potential evidence as strong, moderately strong, suggestive, inconsistent, or weak. This section of the report begins on page 81 of the document.

Here is an excerpt relative to “Strong Evidence”:

All of the applicable high quality studies support a conclusion (statistically significant individual effects, significant positive mean effect size, or equivalent consistent positive findings) and they include at least three independent studies with different relevant samples and settings or one large high quality multisite study. Any applicable studies of less than high quality show either a preponderance of evidence consistent with the high quality studies (e.g., mean positive effect size) or such methodological weaknesses that they do not provide credible contrary evidence. Factors such as error variance and measurement sensitivity clearly influence the number of studies needed to support a conclusion ….

See the United States Coalition for World Class Math [11] for more information about efforts to mitigate the decline in math performance among U.S. students, as well as its ranking of individual states with respect to math performance.

A return to coin tossing

In Chance News 50 we discussed a talk given by statistician Peter Donnelly. In this talk Peter asked us to consider two patterns of heads and tails, HTT and HTH, that would occur if we tossed a coin a sequence of times. He asked us if we thought that, on average, HTT would take longer to appear than HTH, or if HTH would take longer, or if they would take about the same number of tosses. He said that most people would think they would on average occur at about the same time. He said that this is wrong and in fact HTT would, on average, take less time to occur than HTH. This seems strange because it is obvious that they both have an equal chance of being the first to occur. However, it is strange but true. It turns out the expected number of tosses until the pattern HTT occurs is 8 and the expected time until HTH occurs is 10.

Discussion

Suppose that Mary and John play a game in which Mary chooses HTH and John chooses HTT and the person whose pattern comes up first wins. Then this is a fair game even though the expected time for John's pattern coming is less than the expected time that Mary's pattern comes for the first time. How is this possible?

Additional reading

Mathematical games, Sci. Amer. 10, 120-125. Here you will find an elegant combinatorial solution to this coin tossing problem, due to John Conway. This article is also included in Gardner's book "Time Travel and Other Mathematical Bewilderments" and in some of his other books.

Introduction to Probability, Grinstead and Snell, pp 428,430, 432.

Supersizing

“XXXL”, by Elizabeth Kolbert, The New Yorker, July 20, 2009

This article starts out describing some results of the CDC’s National Health and Nutrition Examination Surveys, which have been carried out since the 1950s. According to the CDC’s surveys, the percentage of overweight American adults (“body-mass index greater than twenty-seven”) took a giant leap in the 1980s. CDC researchers published their results in JAMA in 1994: First survey - early 1960s - 24.3%; Second survey - early 1970s - 25%; Third survey - late 1970s - 25.4%; Fourth survey - 1980s - 33.3%.

Men are now on average seventeen pounds heavier than they were in the late seventies, and for women that figure is … nineteen pounds. The proportion of overweight children, age six to eleven, has more than doubled, while the proportion of overweight adolescents, age twelve to nineteen, has more than tripled. ….
“If this was about tuberculosis, it would be called an epidemic,” another researcher wrote in an editorial accompanying the report.

Reviewing five books about the recent surge in weight of many Americans (The End of Overeating, Fat Land, Mindless Eating, The Fat Studies Reader, Globesity), the article's author describes some hypotheses about our eating habits, as well as the results of some psychological experiments performed to try to identify possible psychological causes.

One story has Ray Kroc, of McDonald’s, questioning why they could sell more French-fries if they “supersized’ them.

Kroc pointed out that if people wanted more fries they could always order a second bag.

“But Ray,” [a McDonald’s Board member] is reputed to have said, “they don’t want to eat two bags – they don’t want to look like a glutton [sic, and possibly sick].” ….

The result is that as French-fry bags get bigger, so, too, do French-fry eaters.

A solution to the probability puzzle

In Chance News 50 we gave the following probability puzzle

Find three random variables X, Y, Z, each uniformly distributed on [0; 1], such that their sum is constant. Since each random variable has expectation 1/2, the sum must in fact be 3/2.

The following solution was provided by Roger Pinkham

If {z} denotes the fractional part of z, then supposing x uniform (0,1) put X={2x}, Y=1-{x+1/2}, and Z=1-x. X+Y+Z = 3/22

Congratulations Rodger

Laurie Snell

A bet with no losers?

“Using the Lottery Effect to Make People Save”
by Jason Zweig, The Wall Street Journal, July 18-19, 2009

The author discusses a Michigan pilot program to increase people’s savings rates.

[P]sychologists have long known that people tend to overestimate the odds of rare events. Applying that behavioral insight, [a] finance professor … has devised a clever program called "Save to Win." Launched earlier this year for members of eight credit unions in Michigan, it is a cross between a certificate of deposit and a raffle ticket. Members who put $25 or more into a Save to Win one-year CD are entered into a monthly "savings raffle" for prizes up to $400, plus one annual drawing for a $100,000 jackpot. Only Michigan residents are eligible to participate.

This CD pays between 1% and 1.5% annual interest and has attracted about $3.1 million in new deposits. One new customer described her experience:

"The teller said somebody else she told about it won, … so I said, 'Well, you must be good luck then.' I thought it was a good idea, because earning interest means you win anyway. So I put down the minimum, $25." This past week, [the customer] won $400. She plowed the $400 back into her Save to Win account, getting a second shot at winning the $100,000 grand prize.

A credit union president stated, “You are sort of betting, but there's no losing."

Bloggers [12] commented:

(a) "Apply that same logic to the $250 SS stimulus check. Instead of sending $250 to everyone (which is mostly used to pay bills), give $2500 to every tenth person. Much more spending and stimulus will result."

(b) "One thing at least is certain - the credit union wins by offering a lower rate."

More scientific fraud

[http://online.wsj.com/articleSB10001424052970203517304574306540899745318.html “Defining Data Down”]
Book review of Eugenie Samuel Reich’s Plastic Fantastic by John Derbyshire, The Wall Street Journal, July 24, 2009

The book describes a late 1990s-2004 science fraud by a postdoctoral German physicist at Bell Labs, who was “seeking to persuade organic materials, like plastics,” instead of inorganic substances, like silicon, “to exhibit behaviors useful in electronics.”

The prize-winning physicist’s results were widely published, in academic physics journals, as well as in more popular science magazines. While his work had its skeptics, it wasn’t until 2002 that fraud was suspected, after which a Bell Labs committee confirmed that lab results had been faked and data fabricated. The man lost his job and his Ph.D and returned to Germany.

A key moment in the denouement came in April 2002 when a Bell Labs researcher noticed that two of [the physicist’s] papers from two years before, one in ¬Science and one in Nature, had reported outputs that were identical—even down to the electrical “noise”—from two quite different devices. Two years, in two journals claiming a combined readership of nearly two million, and nobody noticed?

The physicist was skillful in maintaining support from his managers, possibly because he was “an amiable employee … and a cheap one, delivering striking results from a minimum of resources.” An inadequate peer-review process on the part of editors or fellow scientists appears also to have been at fault.

No one seems to have found a motive for the dishonesty. According to the article’s author, “Perhaps there is no better explanation, at last, than that he did it because he could.”

A blogger [13] commented:

“I have witnessed the bias in how proposals are often reviewed by "peers" - who at times seem more concerned with where the investigator was from as opposed to the quality of the ideas.”