Chance News 84

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"There's a saying in the San Francisco Bay Area: There are lies, damned lies, and next bus."

reported in Get onboard: It's time to stop hating the bus, Talk of the Nation, NPR, 29 March 2012

Submitted by Jerry Grossman

"Skepticism enjoins scientists — in fact all of us — to suspend belief until strong evidence is forthcoming, but this tentativeness is no match for the certainty of ideologues and seems to suggest to many the absurd idea that all opinions are equally valid. ...What else explains the seemingly equal weight accorded to the statements of entertainers and biological researchers on childhood vaccines?..."

--John Allen Paulos, in Why don’t Americans elect scientists?, New York Times, 13 February 2012

Submitted by Bill Peterson

From Significance magazine, February 2012:

“Diaconis and Mosteller … introduced an adage that they called the law of truly large numbers: with a large enough sample, almost anything outrageous will happen.”

“What a coincidence? It’s not as unlikely as you think”

“[After a drug trial] no one in the world wants to know what the chance is that, for this experimental group, A was better than B …. We know exactly what the chance is, because the event has already happened. …. The probability that A was better, given this evidence, is either 1 or 0. …. What the drug company actually wants to know is, given the evidence from the trial, and possibly given other evidence about the two drugs, what are the chances that a greater proportion of future patients will get better if they take A instead of B? …. [After a hypothesis test] the civilian assumes that his original question has been answered and that the probability that A is better is high. But he should not believe this, because of course it might not be true. The p-value can be small, as small as you like, and the probability that A is better could still be low.”

“Why do statisticians answer silly questions that no one ever asks”

Submitted by Margaret Cibes

"If you want to show that things are better, study the mortality rate, not the survival rate."

--Aaron Carroll, in Maybe we can have a better press corps!, Accidental Economist blog, 11 April 2012

Carroll is criticizing this article comparing cost-effectiveness of cancer treatments in the US and Europe. (The notion that screening saves lives because it improves the metric known as the 5-year survival rate is a difficult one to counter in the mind of the lay public.)

Submitted by Paul Alper


“Let's start with T. Rowe Price's U.S. stock fund lineup. I have plugged in 15 of its largest actively managed U.S. equity funds. Let's start at the top with T. Rowe Price Blue Chip Growth. [Note] that T. Rowe Price Growth Stock is a tight fit with a 1.00 correlation--the highest it can get. So, we know that owning those two large-growth funds is rather redundant. [Note] that Small-Cap Value has the lowest correlation at 0.91--thus, it's a good choice for diversification purposes.”

“Fun With Mutual Correlation Matrices”, Morningstar, April 2, 2012

Submitted by Margaret Cibes

“This past week, Obama sports a 53-44 fav[orable] / unfav[orable] rating among women, but 43-54 among men. That's a whopping 20-point gender gap. .... In 2008, President Barack Obama won women 56-43, while narrowly edging out John McCain among men 49-48. That 12-point gender gap appeared massive at the time, but it appears that we're headed toward an even bigger margin in 2012.”

“Republicans face massive gender deficit”, Daily KOS, March 23, 2012

Submitted by Margaret Cibes

From Significance magazine, February 2012:

“It is ridiculous to believe that the legal profession has to get involved with the complex parts of statistics. The statistics of gambling uses the frequency approach to statistics, and that is straightforward. The reasoning involved with Bayes is more complex and we cannot expect juries to accept it.”

Letter to the editor

“Strangely, DNA experts are required to give probabilities for their evidence of matching; fingerprint expert[s] are forbidden to.”

“Fingerprints at the crime-scene"

“We cannot increase the probability of winning a [lottery] prize as this is fixed. However, we can increase the amount we can expect to win if we do strike [it] lucky …. [C]hoosing popular combinations of numbers decreases the expected value of your ticket as you have a higher probability of having to share your prize if you win. …. [W]hen you select your numbers, you may as well try to choose a less popular combination that might increase your expected value. Unfortunately, it is not possible to determine what exactly the unpopular combinations are: there is not enough data.”

“Playing the lottery with a little bit of stats know-how…”

Submitted by Margaret Cibes

Chocolate hype

“Association Between More Frequent Chocolate Consumption and Lower Body Mass Index”
Archives of Internal Medicine, March 26, 2012

This report (full text not yet available online) was published as a “research letter.” It involved about 1000 Southern Californians, who were surveyed about their eating and drinking habits and whose BMIs were computed. Funded by the National Institutes of Health, the study found that, among this group, people who ate chocolate more frequently had lower BMIs than less frequent chocolate eaters. The authors, in the “research letter” at least, apparently did not claim any cause-and-effect result and indicated that a controlled study would be needed before jumping to any such conclusion.

The Knight Science Journalism Tracker (subtitled “Peer review within science journalism”) presented a nice critique of press accounts of the “research letter” in “Eat Chocolate. Lose Weight. Yeah, right.”.

Before I became aware of the Knight project, I had tracked down a number of press accounts of the study and was encouraged to find it appropriately described in the articles, if not in the headlines, as non-definitive. (This is not to say that all reports were accurate or complete with respect to other aspects of the study.) Comparing media reports to original study reports might be a good exercise for a stats class, when a similarly enticing topic presents itself in the news, and when the original study report, or even an abstract, is available for comparison.

(a) The New York Times[1]: “Dietary studies can be unreliable, ... and it is difficult to pinpoint cause and effect.”
(b) BBC News[2]: “But the findings only suggest a link - not proof that one factor causes the other.”
(c) TODAY [3]: “The researchers only found an association, not a direct cause-effect link.”
(d) The Times of India[4]: “[T]he reasons behind this link between chocolate consumption and weight loss remain unclear.”
(e) Reuters[5]: “Researchers said the findings … don't prove that adding a candy bar to your daily diet will help you shed pounds.”
(f) Poughkeepsie Journal[6]: “The study is limited. It was observational, ... rather than a controlled trial ....”
(g) The Wall Street Journal[7]: “[B]efore people hoping to lose weight indulge in an extra scoop of chocolate fudge swirl, the researchers caution that the study doesn't prove a link between frequent chocolate munching and weight loss…..”


1. According to Knight, the original purpose of the project was to study “non-cardiac effects on statin drugs,” not chocolate consumption. Should this information have been reported in the articles? Why or why not?
2. Do you think that the size of the chocolate-consuming group was the same as the size of the entire group of respondents? Why would you want to know the “sample” size behind the chocolate results?
3. How would you monitor a controlled study involving a group of people who were instructed to eat chocolate occasionally and another group who were instructed to eat it more frequently?

Submitted by Margaret Cibes

"Hangman" and conditional probability

A better strategy for hangman
by Nick Berry, lifehacker blog, 5 April 2012

If we order the 26 letters by their frequency of occurrence in English, we get:


But does it follow that this is the right order for guessing your first letter in a game of Hangman? A better strategy, Berry argues, is to condition on the length of the word, information that we have at the start of the game. He develops the following table:

Number of letters Optimal calling order
1 A I
2 A O E I U M B H
3 A E O I U Y H B C K
4 A E O I U Y S B F
5 S E A O I U Y H
6 E A I O U S Y
7 E I A O U S
8 E I A O U
9 E I A O U
10 E I A O U
11 E I A O D
12 E I A O F
13 I E O A
14 I E O
15 I E A
16 I E H
17 I E R
18 I E A
19 I E A
20 I E

He has some interesting conclusions based on the above conditional probabilities:

  • The most challenging (least deterministically obvious) words to guess are three letter words. It can take up to ten guesses before getting a letter on the board!
  • With fewer than three letters, it gets easier (there are fewer possible words), and with more than three letters it becomes less likely there will be any words that you cannot find a letter for quickly.
  • For five letter words, the best first guess is the letter S. This is the only time a consonant is the most likely first guess letter.
  • For four letter words, the first non-vowel guess is an S, followed by B and then F (remember, these are only called if all preceding letters have failed to hit).
  • No row contains more than ten guesses, and since a Hangman game takes eleven fails to lose, it is impossible to come up with any English letter word that will fail at Hangman without a single letter appearing on the board (assuming the optimal search strategy above is followed).
  • A should only be your first guess if the word length is four or fewer letters. If five letters, go for S first. Between six and twelve letters try E and above that you should call I.

How will this work in practice? Berry notes that "Battle plans are excellent up until the first shot is fired!" Indeed, as Andrew Gelman point out on his blog post, Hangman tips (4 April 2012), if if you knew your opponent was guessing according to these probabilities, you could counter by adjusting your word selections to shift the distribution.


1. The sequence


comes from conceptually thinking of a very large bucket which contains all the words in the English language used in a very large book so that common words such as "and," '"or," "but," "down," "I," "me," "birth" and so on appear very frequently; words such as 'hyperventilate" or "colonoscopy" appear less frequently. However, if one thinks of a large bucket which contains all the words in the English language but each word appears only once, then the frequency of occurrence according to Berry is


Now, if you are told the specific length of a word, we can ignore the words of different length in the very large bucket and this leads to the above table.

2. The sequence


is beloved by the cognicenti because of its connection to the printing trade.

Submitted by Paul Alper

Show-and-tell opportunities

Significance, February 2012

Timandra Harkness is a writer and comedian, who is currently on tour in the UK and Australia with her show, “Your Days Are Numbered: the maths of death”. She notes that she’s the “only comedian currently touring the UK who has an article in [a] Journal of the Royal Statistical Society.” The article, “Seduced by stats?”, appears in the February 2012 issue of Significance, a joint production of RSS and ASA.

A Cambridge University professor has created a website, Cambridge Coincidences Collection, where visitors can record and/or read stories about coincidences which they have experienced. He expects to analyze the possible scientific explanations for them and/or the mathematical chances of their occurrence.

Submitted by Margaret Cibes

Red meat and mortality risk

Risks: More red meat, more mortality
by Nicholas Bakalar, New York Times, 12 March 2012

The risk numbers
by Carl Bialik, Wall Street Journal, Numbers Guy blog, 23 March 2012

By now we have all heard that we should be eating less red meat. A new study published in the Archives of Internal Medicine (available by subscription here) finds that consuming red meat is associated with mortality risk from cancer and heart disease. As described in the NYT article

People who ate more red meat were less physically active and more likely to smoke and had a higher body mass index, researchers found. Still, after controlling for those and other variables, they found that each daily increase of three ounces of red meat was associated with a 12 percent greater risk of dying over all, including a 16 percent greater risk of cardiovascular death and a 10 percent greater risk of cancer death.

In his Numbers Guy post, Carl Bialik calls for a closer examination of "12 percent greater risk" figure, noting that it corresponds to a relative risk, without providing information about absolute risk. He notes that people tend to overreact to information presented in this way. He references the British pharmaceutical industry’s Code of Practice, where, in a section on "Misleading Information, Claims and Comparisons," (p. 16) we read:

Reference to absolute risk and relative risk. Referring only to relative risk, especially with regard to risk reduction, can make a medicine appear more effective than it actually is. In order to assess the clinical impact of an outcome, the reader also needs to know the absolute risk involved. In that regard relative risk should never be referred to without also referring to the absolute risk. Absolute risk can be referred to in isolation.

In this YouTube video, Gerd Gigerenzer describes a famous example of the consequences of not understanding the distinction between relative and absolute risk. In 1995, a British study found a 100 percent increase risk of thrombosis for women using third-generation oral contraceptive pills. Widespread media reporting of this figure resulted in public alarm, and many women stopped using the pill. In the year following the report, there were some 13,000 additional abortions in the UK, reversing what had been a downward trend. However, Gigerenzer explains, what the studies actually found was that for every 7000 women who took the pill of the second-generation pills, 1 had a thrombosis, compared to 2 per 7000 women who took the third-generation pill. So the relative risk did indeed increase by 100%, but the absolute risk had increased only by 1 in 7000.


Of course, we all have a 100% overall chance of dying eventually. What does the NYT mean by "each daily increase of three ounces of red meat was associated with a 12 percent greater risk of dying over all"?

Submitted by Bill Peterson

The Supreme spittoons

Former Oklahoma AG shares insight into U.S. Supreme Court
by Dan Bewley, News On 6 (Tulsa, OK), 28 March 2012 This backgrounder piece on the experience of arguing before the Supreme Court concludes with the following:

Two other Supreme Court traditions that are interesting, attorneys who argue before the Supreme Court are given a white goose-quill pen to take home as a souvenir.

And just below the bench that the justices sit at, are ceramic spittoons as a remnant of the court's early history, only now they're used as waste baskets.

Paul Alper wrote to point out the irony of the spittoons, in light of the famous statistical controversies surrounding tobacco products.