Chance News 58
Quotations
Berkeley law professor Kevin Quinn is working on a "statistical time machine" to compare Supreme Court justices' positions across historical time periods. He emailed Carl Bialik ("Statistical Time Travel Helps Answer What-Ifs", The Wall Street Journal, November 12, 2009) the following quotation:
The famous statistician George Box once wrote that "all models are wrong, but some are useful."
Submitted by Margaret Cibes
Note: I encourage you to read the article by Carl Bialik (The numbers Guy) referred to above. JLS
Everyone believes in the normal law, the experimenters because they imagine that it is a mathematical theorem, and the mathematicians because they think it is an experimental fact.
in Henri Poincaré's Calcul de probabilités, 1896
Submitted by Steve Simon
When times are good in financial markets, we’re willing to convince ourselves that they’re good for a reason. …. “When the trend is sideways to down, they think the machine is broken,” says [technical analyst] Robert Prechter. “Jeez, it can’t be us.” ....
Prechter readily admits that he’s far from infallible. The standard, he says he wants to be held to is similar to that of a hitter in baseball, in which batting .300 makes one a star and.400 an immortal.
Submitted by Margaret Cibes
From principles is derived probability, but truth or certainty is obtained only from facts.
Submitted by Laurie Snell
Forsooth
These Forsooths are from the December 2009 RSSNews.
When to undertake surveys
The ideal time for monitoring walking activity is when flows are highest, That is usually in June, and is linked to good weather and longer hours of daylight. However, because most walk journeys are for utility reasons, the number of walk journeys per month does not vary greatly - unlike cycling. School holidays influence walking patterns and the purpose of a trip is often time dependent.
It is uncertain to what extent the weather influences the amount of walking activity overall. It is likely that leisure walking is more strongly affected by weather conditions than walking for utility purposes.
October 2009
40% rise in swine flu deaths in 48 hours as two more die The number of swine flu deaths in Scotland has soared by 40% in just 48 hours, after the Scottish Government Confirmed last night that a further two people died after contracting the virus.
The patients, a 48-year-old man from Greater Glasgow and Clyde and an 81- year-old Fife woman, were both carrying the H1N1 strain
Their deaths take the total swine flu fatalities to 14, marking a sudden increase in the number of deaths since Glasgow mother Jacqui Fletcher became the UK's first swine flue victim in June.
In a Wall Street Journal article, "These Hobbyists Add to Calculators, Multiplying Their Fun", November 17, 2009, Dionne Searcey reports:
After two months of trying to crack the code - a process that involved factoring two huge prime numbers - Mr. Moody says he succeeded in July.
Submitted by Margaret Cibes
In a Psychology Today article, “Interacting with women makes men stupid”, May 18, 2009, Scott Barry Kaufman reports about a Dutch research paper, “Interacting with women can impair men’s cognitive functioning” (Journal of Experimental Social Psychology, May 2009).
The article’s author reports:
[M]ale participants tended to perform worse on a cognitive task …following the mixed-sex interaction compared to the same-sex interaction. …. Also, this effect was even stronger when the male participant reported higher attraction to the opposite-sex person they [sic] were interacting with. ….
It should be noted that there was evidence that women's cognitive performance did tend to decline after mixed-sex interactions if they reported having a relatively strong goal to impress the opposite-sex other.
Another study result is reported:
Part of boys' valuable cognitive resources may be spent on impressing their female class members.
Submitted by Margaret Cibes
In an Emax Health article, “Cell Phone Ringtones Can Impair Retention”, June 3, 2009, Kathleen Blanchard reports about an LSU study of the effect of cell phone ringtones on short-term information retention.
[P]eople exposed to cell phone ringtones had lower scores on tests after hearing ringtones in the classroom. ….
[The researcher] said the familiar LSU fight song… ”slowed down their decision-making performance for a longer time than even a standard ringtone."
See also “Cell phone ringtones can pose major distraction, impair recall”, by Gerry Everding, Washington University at St. Louis, May 28, 2009.
Submitted by Margaret Cibes
From “Why Changing the CEO May Not Change the Company”, by Jason Zweig, The Wall Street Journal, December 4, 2009
If you took the CEOs with the best track records and brought them in to run the businesses with the worst performance, how often would those companies become more profitable? According to [an MIT] economist …, who has studied the effects of hundreds of management changes, the answer is roughly 60%. That isn't much better than the flip of a coin. ….
The real force in corporate performance isn't the boss, but regression to the mean: Periods of good returns are highly likely to be followed by poor results, and vice versa.
Submitted by Margaret Cibes
“It’s only fifty-fifty you’ll get him back if you pay it,” he said factually …. I tried to smile. “Two to one, not bad odds at the track.”
by Shari Shattuck, Pocket Books, 2006
Submitted by Margaret Cibes
The value of negative data
Little Benefit Seen, So Far, in Electronic Patient Records, Steve Lohr, The New York Times, November 15, 2009.
Study Raises Questions About Cholesterol Drug’s Benefit, Natasha Singer, The New York Times, November 15, 2009.
Seeking a Shorter Path to New Drugs, Steve Lohr, The New York Times, November 15, 2009.
Negative data, data that disproves a commonly held belief about the superiority of a particular medical treatment, is especially valuable from an economic perspective, but doesn't get the respect it deserves.
Providing high tech electronic health records should lead to better care, but apparently it doesn't.
The nation is set to begin an ambitious program, backed by $19 billion in government incentives, to accelerate the adoption of computerized patient records in doctors’ offices and hospitals, replacing ink and paper. There is wide agreement that the conversion will bring better care and lower costs, saving the American health care system up to $100 billion a year by some estimates. But a new study comparing 3,000 hospitals at various stages in the adoption of computerized health records has found little difference in the cost and quality of care.
Previous studies had used a selected subset of health care practices.
The study is an unusual effort to measure the impact of electronic health records nationally. Most of the evidence for gains from the technology, Dr. Jha said, has come from looking at an elite group of large, high-performing health providers that have spent years adapting their practices to the technology. The group usually includes Kaiser Permanente, the Mayo Clinic, the Cleveland Clinic and Intermountain Healthcare, among others.
In another study, an expensive cholesterol lowering drug was found to perform less well than a simple inexpensive alternative.
For patients taking a statin to control high cholesterol, adding an old standby drug, niacin, was superior in reducing buildup in the carotid artery to adding Zetia, a newer drug that reduces bad cholesterol, according to a new study. The results of the study, published in The New England Journal of Medicine, were presented here Sunday night at an annual meeting of the American Heart Association.
The study was small (208 patients) and used a surrogate outcome, arterial wall thickness. The findings pitted raising good cholesterol against lowering bad cholesterol, and found that raising good cholesterol was better.
Over the course of the 14-month study, the bad cholesterol of the patients on Zetia decreased by 19.2 percent, but the patients’ arterial wall thickness stayed the same, the study said. In the niacin group, good cholesterol increased by 18.4 percent and the carotid wall thickness decreased.
But the use of arterial wall thickness also led to criticism by Dr. Peter S. Kim, the president of Merck Research Laboratories who said that
a drug’s ability to improve artery-wall thickness has not been proved to automatically correlate with a reduction in heart attacks.
The efficacy of Zetia has also been established on the basis of a surrogate outcome, reduction in levels of bad cholesterol.
Zetia, he said, lowers bad cholesterol and lowering bad cholesterol is a known good. The study results “should be compared to the overwhelming body of evidence that lowering LDL cholesterol is an important thing to do to improve cardiovascular health,” Dr. Kim said.
Others, however, felt that this study showed problems with a heavily marketed drug.
Some cardiologists here hailed the study as an indication that the popularity of Zetia and Vytorin, which had combined sales last year of about $4.6 billion, has far outstripped their evidence of a concrete benefit on heart health.
The final article noted the huge expense associated with drug development. Why does it cost $800 million to bring the average drug to market?
Most of the cost in drug development is the price of failure, said Mervyn Turner, the chief strategy officer at the drug giant Merck. This linear, trial-and-error method is no longer a sustainable model for big pharmaceutical companies. “We invest far too long in bad ideas,” Dr. Turner said in a phone interview. “It is really important to stop that at an earlier stage in the cycle.”
One of the suggestions to reduce drug development cost is to publicize early failures.
One idea is for drug makers to share information about compounds they have tried and shelved, for reasons like toxicity or inefficacy. Although many companies have committed to publishing the results of clinical trials, whether or not they succeed, drug makers don’t typically publish information about projects that fail at an earlier stage. A result is that companies waste many millions going down experimental paths that their competitors have already found to be dead ends.
Submitted by Steve Simon
Questions
1. Some people are trying to put a "spin" on the positive effects of electronic medical records at leading health care institutions and the lack of effect in a nationwide survey, as indicating that the electronic medical record works, but it takes time and effort. Do you agree or disagree?
2. Zetia was approved by the FDA on the basis of a surrogate outcome, reduction in bad cholesterol, rather than in an outcome like decreased mortality or reduction in the number of heart attacks. Should the FDA require a new drug to show effectiveness on a direct measure instead of a surrogate measure?
3. What are the barriers to drug companies sharing information about early failures in the drug development process?
Art for the birds
“A bird’s eye view of art”, Science News, June 30, 2009
According to this very brief article, Japanese Professor Shigeru Watanabe has published a study in Springer’s Animal Cognition, which concludes:
Pigeons could be art critics yet, ... like humans, pigeons can be trained to tell the difference between "good" and "bad" paintings.
Curious readers without access to this journal might be interested in an earlier, full-text online, Watanabe paper, “Pigeons’ Discrimination of Paintings by Monet and Picasso” (Journal of the Experimental Analysis of Behavior, March 1995).
Abstract: Pigeons successfully learned to discriminate color slides of paintings by Monet and Picasso. Following this training, they discriminated novel paintings by Monet and Picasso that had never been presented during the discrimination training. Furthermore, they showed generalization from Monet's to Cezanne's and Renoir's paintings or from Picasso's to Braque's and Matisse's paintings. These results suggest that pigeons' behavior can be controlled by complex visual stimuli in ways that suggest categorization. Upside-down images of Monet's paintings disrupted the discrimination, whereas inverted images of Picasso's did not. This result may indicate that the pigeons' behavior was controlled by objects depicted in impressionists'paintings but was not controlled by objects in cubists' paintings.
This paper describes in great detail (methodology, statistical test results) several controlled experiments on “eight experimentally naïve pigeons."
Submitted by Margaret Cibes
Diversification of stock portfolios
“More Stocks May Not Make a Portfolio Safer”
by Jason Zweig, The Wall Street Journal, November 26, 2009
Conventional wisdom among financial planners is that investing in 10 up to 30 or 40 stocks provides adequate diversification for risk reduction. This "wisdom" is apparently backed up by studies.
But this research on diversification was based on the average results of a large number of portfolios randomly generated by computer.
When LSU business professor Don Chance had his students build a portfolio of 30 stocks, one at a time, the results confirmed the conventional wisdom in that, after the first 20 stocks, portfolio risk, as measured by fluctuation in price, had been reduced by about 40% from that of the first stock alone.
However, when Chance analyzed his individual students’ portfolios, he found that increasing the portfolio size from the first stock to the 30th resulted in 11% of the portfolios having more fluctuation than their first choice and 23% having more fluctuation than their first 5 choices.
The lesson: For any given investor, the averages mightn't apply.
Chance found that his students had started their portfolios with a few brand-name companies with which they were familiar and soon ran out of familiar company names, subsequently picking stocks with much lower capitalization and thus more risk.
One financial planner commented:
Humans can't think randomly …. Once people think of Exxon Mobil, they're a lot more likely to think of Chevron or another oil stock. For a lot of investors, diversification is like doing a word-association game.
Another planner commented:
People who regard themselves as risk-averse will assemble portfolios of highly similar stocks that all seem to be "safe." The result, paradoxically, is a risky portfolio with every egg in one basket.
Chance also found that 13% of computer-generated 20-stock portfolios were riskier than one-stock portfolios.
See “Experimental Evidence on Portfolio Size and Diversification: Your Mileage May Vary” to download a copy of the report.
Submitted by Margaret Cibes
The Monty Hall Problem
Oxford Univerity Press, 2009
Jason Douglace Rosenhouse
Jason graduated from Dartmouth in 2000 and is currently Associate Professor of Mathematics, at James Madison University, Harrisonburg Virginia. His research has been in number theory but here he has written a book on the Monty Hall problem.
Chapter 1 is called Ancestral Monty and includes among others a discussion of the Three Prisoners Paradox, Lets Make a Deal, and the Birth of the Monty Hall Problem. the Marylyn Vo Savant story that we all know about .
Chapter 2 is called Classical Monte. And described as follows
Version One: You are shown three identical doors. Behind one of them is a car. The other two conceal goats. You are asked to choose, but not open, one of the doors. After doing so, Monty, who knows where the car is, opens one of the two remaining doors. He always opens a door he knows to be incorrect, and randomly chooses which door to open when he has more than one option (Which happens on those occasions when your initial choice conceals the car). After opening an incorrect door, Monty gives you the choice of either switching to the unopened door or sticking with your original choice. You then receive what is in the door that you choose. What should you do?
Chapter 3 is called Bayesian Monty
Version Two: As before, Monty shows you three identical doors. One contains a car, the other two contain goats. You choose one of the doors but do not open it. This time, however, Monty does not know the location of the car. He randomly chooses one of the two doors different from your selection and opens it. The door turns out to conceal a goat. He now gives you the options either of sticking with your original door or switching to the other one. What should you do?
Jason shows how to solve these two versions explaining the mathematics used in the two solutions.
Chapter 4 is called Progressive Monty described as follows.
This time we assume there are n identical doors, where n is an integer satisfying n >=3. One door conceals a car, the other n-1 conceal goats. You choose one of the doors at random but do not open it. Monty then opens a door he knows to conceal a goat, always choosing randomly always choosing randomly among the available doors. At this point he gives you the choice of sticking with your original door or switching to one of the remaining doors.
You make your decision. Monty now eliminates another goat-concealing door (at random) and once more gives you the choice either of sticking or switching. This process continues until only two doors remain in play. What strategy should you follow to maximize your chances of winning?
At the end of these chapters Jason writes:
We have trodden a long and winding road to reach this point. We have navigated the rapids of the classical problem and some of the most natural variations. We confronted the full horror of the progressive version and emerged stronger for the experience. Along the way we have illuminated much of the world of probability theory and its offshoots. Yet for all of that, there remain certain variations on and aspects of the Monty Hall problem that have not fit comfortably into the preceding chapters. Our purpose now is to tie up some of these loose ends.
While the number 4 version is a horror the solution is amazingly simple. The optional strategy is to stick with your original door until only 2 doors remain and then switch, Your probability of winning for any n >= 4 is (n-1)/n.
The proof is discussed in this book and also in
Optimal strategies for progressive Monty Hall Problems
The Mathematical Gazette
Vol 93 No 528, Page 410
November 2009
Steven K. Lucas, Jason Rosenhouse
You can find an earlier version by putting the title in google.
Submitted by Laurie Snell
Simpson’s Paradox in the news
“When Combined Data Reveal the Flaw of Averages”
by Cari Tuna, The Wall Street Journal, December 2, 2009
The subtitle of this article is “In a statistical Anomaly Dubbed Simpson’s Paradox, Aggregated Numbers Obscure Trends in Job Market, Medicine and Baseball.”
The author reports about an anomaly which results from comparing unemployment rates for two periods, the early 1980s recession period and the current 2009 period. While unemployment rates are now lower for the population of all adult Americans, the rates for some subgroups of the population are higher.
So how can the overall unemployment rate be lower today but higher among each group? The anomaly is an example of Simpson's Paradox -- a common but misleading statistical phenomenon rooted in the differing sizes of subgroups. Put simply, Simpson's Paradox reveals that aggregated data can appear to reverse important trends in the numbers being combined.
The author discusses the well known example of Berkeley’s 1973 graduate admissions data, a 1986 study of kidney-stone treatments, and baseball statistics.
[Harvard’s statistics chair] says he thinks many people who wield similarly misleading data do so unintentionally. "When you find data that go with your theory, then you don't dig deeper."
Bloggers [1] mentioned additional examples of Simpson’s Paradox, on topics such as mean SAT scores compared over time, and infant mortality rates compared among different countries.
On the ISOSTAT listserv, readers were referred to Andrew Gelman’s comments on this article in [2], posted on his website, Statistical Modeling, Causal Inference, and Social Science, December 3, 2009.
Submitted by Margaret Cibes
STATS editor starts Forbes column
A recent STATS email announced that its editor, Trevor Butterworth, has debuted a weekly column, "Medialand," on Forbes.com. The focus appears to be on critiquing the coverage of scientific/statistical news by media folks.
A click-able list of previous Butterworth columns [3] includes:
“Can Plastic Change Your Sex?,” November 19, 2009
“Tricked by Treats,” October 29, 2009
Submitted by Margaret Cibes
Statistics make you stingy
Nicholas Kristof and Sheryl WuDunn have recently published "Half the Sky: Turning Oppression into Opportunity for Women Worldwide," a book about the global oppression of women, focusing on the sex slave trade, honor killings, rape as a wartime tactic, lack of maternal health care. Though they often cite statistics about the magnitude of these problems, they prefer to offer individual stories.
About halfway through the book (page 99), they explain why.
Frankly, we hesitate to pile on the data, since even when numbers are persuasive, they are not galvanizing. A growing collection of psychological studies show that statistics have a dulling effect, while it is individual stories that move people to act. In one experiment, research subjects were divided into several groups, and each person was asked to donate $5 to alleviate hunger abroad. One group was told the money would go to Rokia, a seven-year-old girl in Mali. Another group was told that the money would go to address malnutrition among 21 million Africans. The third group was told that the donations would go to Rokia, as in the first group, but this time her own hunger was presented as part of a background tapestry of global hunger, with some statistics thrown in. People were much more willing to donate to Rokia than to 21 million hungry people and even a mention of the larger problem made people less inclined to help her.
In another experiment, people were asked to donate to a $300,000 fund to fight cancer. One group was told that the money would be used to save the life of one child, while another group was told it would save the lives of eight children. People contributed almost twice as much to save one child as to save eight. Social psychologists argue that all this reflects the way our consciences and ethical systems are based on individual stories and are distinct from the part of our brains concerned with logic and rationality. Indeed, when subjects in experiments are first asked to solve math problems, thus putting in play the parts of the brain that govern logic, afterward they are less generous to the needy.
These thoughts bring to mind another famous (infamous?) quote by Joseph Stalin:
A single death is a tragedy. A million deaths is a statistic.
Questions
1. Later in the book (page 141), Kristoff and WuDunn cite the need for "relentless empiricism" in developing policies for the prevention of AIDS. Is this a contradiction?
2. Would you suspect that statisticians, as a group, are less generous than people in other professions?
Submitted by Steve Simon
Mammogram Math
New York Times, December 11 2009
John Allen Paulos
The U.S. Preventive Services Task Force (USPSTF) published a paper "Screening for Breast Cancer" in the Annals of Internal Medicine on November 17 2009 and recommended:
(1) The USPSTF recommends against routine screening mammography in women aged 40 to 49 years. The decision to start regular, biennial screening mammography before the age of 50 years should be an individual one and take patient context into account, including the patient's values regarding specific benefits and harms.
(2) The USPSTF recommends biennial screening mammography for women aged 50 to 74 years.
(3) The USPSTF concludes that the current evidence is insufficient to assess the additional benefits and harms of screening mammography in women 75 years or older
These recommendations were changed on the December 15 2009 version to:
(1) The USPSTF recommends biennial screening mammography for women aged 50 to 74 years.
(2) The decision to start regular, biennial screening mammography before the age of 50 years should be an individual one and take patient context into account, including the patient's values regarding specific benefits and harms.
(3) The USPSTF concludes that the current evidence is insufficient to assess the additional benefits and harms of screening mammography in women 75 years or older. Since John's article was published on December 11 he must have used the first recommendations.
John writes:
Much of our discomfort with the panel's findings stems from a basic intuition: since earlier and more frequent screening increases the likelihood of detecting a possible fatal cancer, it is always desirable. But is this really so?
To understand why this is, John uses some expressions and words that we might not be familiar with. Here are two of these: Reductio ad absurdum is a mode of argumentation that seeks to establish a contention by deriving an absurdity from its denial, thus arguing that a thesis must be accepted because its rejection would be untenable. It is a style of reasoning that has been employed throughout the history of mathematics and philosophy from classical antiquity onwards.
In medicine, a disease is asymptomatic if a patient carries a disease or infection but experiences no symptoms. A condition might be asymptomatic if it fails to show the noticeable symptoms with which it is usually associated.
Then John writes:
Applying it (reductio ad absurdum) to the contention that more screening is always better leads us to note that if screening catches the breast cancers of some asymptomatic women in their 40's then it would also catch those of some asymptomatic women in their 30's. But why stop there, why not monthly mammograms beginning at age 15? The answer, of course is that they would cause more harm than good.
The rest of the article is devoted to the false positive paradox and a plea for the public to better understand probability and mathematics.
The issues discussed here have occurred in other Chance News's. We encourage you to read the following
Chance News 12 Screening
Chance News 8 Mammograms Validated as Key in Cancer Fight
Chance News 14 Calculated Risks Revisited
A more recent analysis of the current screening studies can be found in the latest issue of Plus Magazine. Here David Spiegelhalter has an article Understanding uncertainty: Breast screening, a statistical controversy. Here you will find among other things an interesting discussion of the USPSTF study including the data from this study. Plus Magazine articles are written in a way to be used in a mathematics courses and this is a fine example of this.
by Laurie Snell
For a Forsooth about the false positive paradox, readers may be interested in the article “Industry Official Faults Radiologists In Mammogram Debate”, Hartford Courant, December 31, 2009, in which "a high-ranking official at National Imaging … reached a different conclusion: Poorly performing radiologists are the root of the problem."
Submitted by Margaret Cibes
N.F.L. Suspends Its Study on Concussions
New York Times, December 19,2009
Alan Schwarz
The following quotations are from Linda T. Sanchez, Democrat of California, who complained (1) that the person doing the study of concussions of players was employed by the NFL and therefore suspect. Further, (2) she criticized the basis of comparison.
Hey, why don't we let tobacco companies determine whether smoking is bad for your health or not?
That is sort of like comparing two-pack-a-day smokers with one-pack-a-day smokers to see what the differences are instead of two-pack-a-day smokers with the general population to see whether there is an increased risk of the activity that they are participating in to their health.
Submitted by Paul Alper
George Clooney and the golden ratio
“Soy Consumption Appears to Help Breast-Cancer Survivors”
by Jeremy Singer-Vine, The Wall Street Joural, December 15, 2009
The article consisted of brief summaries of some current research results.
Here is an excerpt from a section on blood pressure:
Healthy people's blood pressure adheres to the "golden ratio," according to research described in a letter to the British Medical Journal. That ratio [is] approximately equal to 1 to 1.618033987 …. The researchers analyzed 20 years of blood-pressure data on 166,377 Austrian subjects. They found that the average ratio between systolic and diastolic blood pressure—that is, between the peak and trough—was 1.6235 overall. But it was 1.7459 for subjects who died during the study, and, remarkably, 1.6180 in patients who survived all 20 years.
Caveat: The researchers conceded that their findings had little practical use for doctors.
For the full text of the reporter’s source, see “George Clooney, the cauliflower, the cardiologist, and phi, the golden ratio”.
Here is another excerpt from a section on testosterone and bargaining:
High testosterone appears to increase fair play among women, according to a study in Nature. …. Remarkably, women who thought they were given testosterone were less generous than women who thought they received placebo, though the difference was not statistically significant. ….
Caveat: The experiment provided no positive proof of the researchers' social-status hypothesis, but it did appear to disprove competing theories.
For an abstract of the reporter’s source, see "Prejudice and truth about the effect of testosterone on human bargaining behavior”.
Submitted by Margaret Cibes
Sanitizing hands
“Kills 99.9% of Germs – Under Some Lab Conditions”
by Carl Bialik (The Numbers Guy), The Wall Street Journal, December 16, 2009
Bialik discusses concerns related to the high effectiveness rates touted by health product companies, such as the claim of 99.9% protection from germs by hand-sanitizing liquids.
One skin care manager emailed Bialik, “The 99.99% message is more powerful among consumers than 'antibacterial' or 'germ kill' alone.” According to Bialik, the chief scientist and founder of a hand-sanitizer maker “markets his products with a 99.99% germ-mortality rate, in part because his competitors do.”
One problem with relying on these claims is that manufacturers have to demonstrate only that the product is effective on a “representative sample of microbes,” without disclosing which bugs the product kills.
Another problem involves the standard test, which is run on 60 slides that have been inoculated with a specific bug, and requires that bugs have been killed on at least 59 slides on which the sanitizer has sat for 10 minutes each. If a test falls short, manufacturers are permitted to re-do the test. Apparently, in real life, people do not leave the sanitizing products on surfaces that long.
A blogger [4] commented, “Since dirty surfaces may well contain over a billion bacteria, 99.9% effectiveness translates to [1 million] bacteria still surviving. What good does that do?”
Another blogger [5] stated, “98.6% of people believe anything presented in numbers.”
Submitted by Margaret Cibes