Chance News 47
It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge.
Pierre Simon Laplace
Théorie Analytique des Probabilités, 1812,
Their visual acuity is only fractionally – not geometrically – better than that of the common primates from which they were engineered.
When do you draw the line? When do you take action to avoid that logarithmic point where things take off exponentially?
Submitted by Margaret Cibes
Andrew Gelman has an interesting article regarding the statistics of elections. He starts with
“Presidential elections have been closer in the past few decades than they were for most of American history. Here's a list of all the U.S. presidential elections that were decided by less than 1% of the vote:
Funny, huh? Other close ones were 1844 (decided by 1.5% of the vote), 1876 (3%), 1916 (3%), 1976 (2%), 2004 (2.5%).
Four straight close elections in the 1870s-80s, five close elections since 1960, and almost none at any other time.”
Perhaps more interesting is his take on House and Senate elections:
From the graph, “the rate of close elections in the House has declined steadily over the century. If you count closeness in terms of absolute votes rather than percentages, then close elections become even rarer, due to the increasing population. In the first few decades of the twentieth century, there were typically over thirty House seats each election year that were decided by less than 1000 votes; in recent decades it's only been about five in each election year.”
Another way of putting it: “Consider that, in the past decade, there were 2,175 elections to the United States House of Representatives held on Election Days 2000, 2002, 2004, 2006 and 2008. Among these, there were 41 instances — about 1.9 percent — in which the Democratic and Republican candidates each received 49 percent to 51 percent of the vote (our calculations exclude votes cast for minor parties). In the 1990s, by contrast, there were 65 such close elections. And their number increases the further one goes back in time: 88 examples in the 1950s, 108 in the 1930s, 129 in the 1910s.”
1. A contention mentioned in the NYT article for this bifurcation of opinion is: “as the economy has become more virtual, individuals can now choose where to live on an ideological rather than an occupational basis: a liberal computer programmer in Texas can settle in blue Austin, and a conservative one in the ruby-red suburbs of Houston.” Argue for and against this assertion of coupling ideology and occupational mobility.
2. Gelman and Silver end their NYT article with “Elections like those in New York’s 20th district or in Minnesota, as contentious as they are, actually hark back to a less divisive era in American politics.” Explain the seeming paradox of a close race indicating less divisiveness.
3. As of this posting, the Minnesota senate seat is unfilled despite a manual recount, a canvassing board followed by a so-called election contest and awaits the results of the appeal to the Minnesota Supreme Court—possibly further. See When Does ‘Close Become Too-Close-to-Call? for an analysis of error rates and how likely it is that the real winner would be Norm Coleman instead of Al Franken who currently leads by 312 votes out of about 2.9 million cast. The Minnesota Supreme Court will hear oral arguments on June 1, 2009.
Submitted by Paul Alper
Bayes theorem in the news
In a letter to the editor of the Miller-McCune magazine, headed "For those of You Who Paid Attention in Statistics Class" , Professor Howard Wainer, of The Wharton School, writes about the "mathematical reality ... that so long as only a very small minority of people commit crimes and the criminal justice system is fair ... there will always be a very large proportion of innocent people convicted." He provides a hypothetical example, with calculations based on Bayes theorem.
Some interesting challenges to his assumptions, both from the editor and online bloggers, are included . Professor Wainer's response to an editor's comment includes the following medical example.
Each year in the U.S., 186,000 women are diagnosed, correctly, with breast cancer. Mammograms identify breast cancers correctly 85 percent of the time. But 33.5 million women each year have a mammogram and when there is no cancer it only identifies such with 90 percent accuracy. Thus if you have a mammogram and it results in a positive (you have cancer) result, the probability that you have cancer is: 186,000/(186,000+3.35 million) = 4%.
So if you have a mammogram and it says you are cancer free, believe it. If it says you have cancer, don't believe it.
The only way to fix this [is to] reduce the denominator. Women less than 50 (probably less than 60) without family history of cancer should not have mammograms.
1. Consider Professor Wainer's calculation of the probability (4%) of having cancer given a positive mammogram result. Do you agree with the numbers used?
2. Consider Professor Wainer's advice to a person who receives a diagnosis of being cancer free. What is the probability of being cancer free if one receives a negative mammogram result? Do you agree with his advice?
3. Comment on Professor Wainer's advice for "women less than 50."
Submitted by Margaret Cibes
Except for Voltaire who famously (albeit, possibly apocryphally) said, “Lord, protect me from my friends; I can take care of my enemies,” few doubt the benefits of having friends.
From Tara Parker-Pope we find some surprising side effects of friendship. She suggests looking at an Australian study which “found that older people with a large circle of friends were 22 percent less likely to die during the study period than those with fewer friends.” Further, “last year, Harvard researchers reported that strong social ties could promote brain health as we age.”
She also refers to a 2006 study of nearly 3000 nurses with breast cancer which “found that women without close friends were four times as likely to die from the disease as women with 10 or more friends. And notably, proximity and the amount of contact with a friend wasn’t associated with survival. Just having friends was protective.” She closes her article with a quote from the director of the center for gerontology at Virginia Tech: “People with stronger friendship networks feel like there is someone they can turn to. Friendship is an undervalued resource. The consistent message of these studies is that friends make your life better.”
1. Parker-Pope also mentioned researchers here who “studied 34 students at the University of Virginia, taking them to the base of a steep hill and fitting them with a weighted backpack. They were then asked to estimate the steepness of the hill. Some participants stood next to friends during the exercise, while others were alone. The students who stood with friends gave lower estimates of the steepness of the hill. And the longer the friends had known each other, the less steep the hill appeared.” In fact, three of the 34 were excluded because they were deemed outliers. The participants estimated the slant via three different methods as can be seen in the figure below:
The haptic measurement "required adjusting a tilt board with a palm rest to be parallel to the hill, importantly, without looking at ones hand." As can seen from the above figure, it appears to more accurate than either the verbal, merely a guess, or the visual which a (presumably crude) disk-like device acted as an aide.
The researchers performed a two-way ANOVA (sex and social support) separately for each of the three measuring methods. They reported the value of each F(1,27) to determine a p-value for each method to see if Friend compared to Alone is statistically significant. So, why the number 27? From merely looking at the figure, which of the three methods for determining slant would appear to be unrelated to friendship?
2. The above study took place in Virginia. In Plymouth, England the researchers did a similar slant study but this time instead of friendship directly, imagining of support was tested as can be seen from the following figure:
This study had 36 participants and similarly to the first study, they did a two-way ANOVA (sex and imagery of support) leading to F(2,30) for each slant measuring technique. So, why the 2 and the 30? From merely looking at the figure, which of the three methods for determining slant would appear to be unrelated to imagery of support?
3. In either study, visual or verbal on average markedly overstate the slant of the hill. What does that suggest about peoples ability to judge a task?
4. The researchers admit that for either study, "Participants in this study were not randomly assigned." Why would this pose a problem?
5. To give Voltaire his due, Parker-Pope points out that "A large 2007 study showed an increase of nearly 60 percent in the risk for obesity among people whose friends gained weight."
Submitted by Paul Alper