Stats Made Easy

While waiting around for my daughter to register for her first semester of college at the University of Wisconsin at Eau Claire, I read this sign in their physics department. It took me a moment to process the directions at the bottom, but then it hit me.

My bookseller friend Rich emailed recently about a find he made:
>In your physics days, did you ever encounter the famous Feynmann Lectures in book form (three volume set)? Feynmann was a respected renegade, if there IS such a thing. But he was good enough to be appointed to the Challenger’s explosion evaluation. Interestingly, before the SSTs flew, he predicted a 2% failure rate — and he’s been right on.< That reminded me of one of my favorite quotes by this renowned that I bring up when discussing the dangers of deleting experimental outliers:
“The first principle is that you must not fool yourself — and you are the easiest person to fool.”
It comes from a Feynmann’s talk on Cargo Cult Science.

PS. While searching the internet on the topic of bias, I came across an interesting website that provides “an outlet for experiments that do not reach the traditional significance levels (p < .05)”! It’s called the Journal of Articles in Support of the Null Hypothesis. The journal’s purpose is to reduce the ‘file drawer problem,’ that is the tendency for unpublished results to get buried in researchers’ file cabinets.*

*To learn about this propensity to publish only the positive, read this study by Jeffrey D. Scargle of the Space Science Division National Aeronautics and Space Administration, Ames Research Center.

Earlier this month I was listening to a local talk radio show when a caller provided this “end of the world” development: To be more kindly and gentle to students stressed out by math tests, teachers now refer to these as “celebrations of knowledge”!

At the time I heard of this outrageous slackening of educational standards, I was in the midst of reading a book that my buddy Rich sent me that’s titled “The World of W. Edwards Deming.” (If you are a bibliophile like me, check out his eclectic mix of uncommon books offered for sale via Ebay). My entry into the world of quality assurance was catalyzed by the electrifying documentary “If Japan can… Why can’t we?” by NBC in 1980 featuring Deming and his use of statistical methods.

One of my favorite stories in the book on Deming, which was written by his long-time personal assistant Cecelia S. Kilian, involves another pioneer in the field of statistics, a fellow named Harold Dodge. Deming worked with Dodge during WWII to develop statistical standards on a wartime emergency basis. Over a decade before that, Deming had an internship with Bell Telephone Laboratories – Dodge’s employer. During these times the statisticians working under Dodge played a neat trick on him as he tried to get a feeling for the cord length on a newly-developed handset: They clipped off millimeter or two every day. Deming recalls seeing Dodge stoop to an astoundingly uncomfortable level to take a phone call. Evidently it’s not hard to fool an absent-minded statistical genius!

Getting back to Deming’s views on education, I really do wonder how he would feel about the practice of testing as an incentive for students to develop a profound knowledge of their subject. In his book “Out of the Crisis” (1982, MIT) he said “I have seen a teacher hold a hundred fifty students spellbound, teaching what is wrong.” He credits Sir Ronald Fisher as his inspiration for learning statistics, despite being a “poor teacher on every count”! Deming made no secret of his dislike for grading, rating, and testing in industrial settings. Therefore I suppose he would approve of the new, more positive approach of celebrating knowledge, rather than making students take final exams.

“When teachers are forced to teach to the test, students get bored and genuine education ceases, no matter what the test scores may say… The examination as a test of the past is of no value for increased learning ability. Like all external motivators, it can produce a short term effect, but examinations for the purpose of grading the past do not hook a student on learning for life.”
— Myron Tribus (from his essay Quality in Education According to the Teachings of Deming and Feuerstein )

“I had many years that I was not so successful as a ballplayer, as it is a game of skill.”
— Casey Stengel (from testimony before United States Senate Anti-Trust and Monopoly Hearing, 1958)

Last week the University of Minnesota School of Statistics sponsored a talk titled “In-season prediction of batting averages: A field test of empirical Bayes and Bayes methodologies,” presented by Lawrence D. Brown from the Statistics Department of Wharton School at the University of Pennsylvania. My colleague Pat Whitcomb attended and told me about a few findings by Brown that baseball fanatics like me would find a bit surprising:
“The simplest prediction method that uses the first half [of a season’s] batting average …performs worse than simply ignoring individual performances and using the overall mean of batting averages as the predictor for all players.”*

Evidently these professional players perform at such a consistent level that the ones hitting at a higher than average rate up until the mid-season break tend to regress back to the mean the rest of the way, and vice-versa.

Of course, by looking at many years of past performance, one would gain some predictive powers. For example, in 1978, more than ten years into his Hall of Fame (HOF) career, Rod Carew batted .333 for the Minnesota Twins. He made it to the Major Leagues only a few years ahead of fellow Twin Rick Dempsey, who hit at an average of .259 in 1978. Carew finished up his 19-year playing career with a lifetime batting average (BA) of .328, whereas Dempsey hung on for an astounding 24 years with a BA of only .233! It would not require a sabermetrician to predict over any reasonable time frame a higher BA for a HOF ballplayer like Carew versus a dog (but lovable, durable and reliable defensively at catcher) such as Dempsey.

Brown also verifies this ‘no brainer’ for baseball fans: “The naıve prediction that uses the first month’s average to predict later performance is especially poor.” Dempsey demonstrated the converse of this caveat by batting .385 (5 for 13) for his Baltimore Oriole team in the 1983 World Series to earn the Most Valuable Player (MVP) award!

Statistical anomalies like this naturally occur due to the nature of such binomial events, where only two outcomes are possible: When a batter comes to the plate, he either gets a hit, or he does not (foregoing any credit for a walk or sacrifice). It is very tricky to characterize binomial events when very few occur, such as in any given Series of 4 to 7 games. However, as a rule-of-thumb the statistical umpires say that if np>10 (for example over 50 at-bats for a fellow hitting at an rate of 0.200), the normal approximation can be used for binomial distributions and the variance becomes approximately p(1-p)/n.** From this equation one can see that the bigger the n, that is – at-bats, the less the fraction (batting average) varies.

PS. I leave you with this paradoxical question: Is it possible for one player to hit for a higher batting average than another player during a given year, and to do so again the next year, but to have a lower BA when the two years are combined?

*Annals of Applied Statistics, Volume 2, Number 1 (2008), 113-152

**This Wikipedia entry on the binomial distribution says that “this approximation is a huge time-saver (exact calculations with large n are very onerous); historically, it was the first use of the normal distribution, introduced in Abraham de Moivre’s book The Doctrine of Chances in 1733.”

Evidently due to its concentration algorithmic ‘philic’ minds, Stat-Ease gets review copies of new technical tomes from the Society for Industrial and Applied Mathematics (SIAM). I once fancied myself as a ‘mathelete,’ but I learned differently after moving up from being the big fish in my small pond at high school to a miniscule minnow at a major university in the Midwestern USA. My mistake was skipping into an advanced calculus class populated by some of the country’s top talent – National Merit scholars like me. Very quickly I realized that my math skills only put me on the very bottom rung and that only by the very tip of one fingernail. What saved me was begging for mercy by the teacher who, luckily, was sick and tired of the smart-mouths in the class who really got it and made sure to flout their chops in math. Thus, when the newest SIAM publication arrives, I always look it over in wonder before quickly passing it along to our master’s statisticians and algorithmic programmers, who may understand its true value.

For example, the book this week is Functions of Matrices, Theory and Computation by Nicholas J. Higham, which “emphasizes Schur decomposition, block Parlett recurrence, and judicious use of Padé approximants.” That blew me away immediately, but I rifled through the pages anyways and found a few pages of interest on the history of matrix functions, which really are useful in our business of experiment design and statistical analysis. (Thank goodness for the power of computers to do the calculations!) Higham credits English-born James Joseph Sylvester as the inventor of the matrix (not to be confused with the famous movie trilogy!). Sylvester emigrated to the USA where he founded the American Journal of Mathematics in 1878, the self-proclaimed “oldest mathematics journal in the Western Hemisphere in continuous publication.”

What amazes me is that anyone can read such esoteric materials, but it’s good they do, because great advances are made possible by developments in math. For example, Higham points out that the first practical application of matrices led to the elimination of unwanted flutter in aircraft wings. (Galloping flutter, or wake vortex flutter, caused the spectacular failure of the Tacoma Narrows Bridge in 1940.*) This work was done by the Aerodynamics Department of England’s National Physical Laboratory (NPL) in the 1930s. In parallel, not far away in the UK, Ronald Fisher, the founder of modern-day applied statistics, was developing the core catalog of experimental design matrices that still remain in use today.

“Here I stand because of you, Mr. Anderson. Because of you, I’m no longer an Agent of this system. Because of you, I’ve changed. I’m unplugged.”
– Agent Smith (played by Hugo Weaving ) from The Matrix Reloaded (2003)

PS. Neither the quote nor the picture really have much to do with matrices, but they provide me some amusement. For example, I saw the second movie of the Matrix trilogy with my brother Paul, an techie type like me. We annoyed the exiting theater patrons greatly by regurgitating Agent Smith’s lines about “Mr. Anderson” this and “Mr. Anderson” that – all with gagging glee.

The picture exhibits a physical matrix – the screen window. I just inserted all the screens earlier this week when it seemed as if Spring had finally arrived in Minnesota. However, we citizens of this northern State were chagrined to see a coating of snow yesterday morning – over a foot in some parts. 🙁

*If you’d like to set up an experiment on flutter that requires only a hair blower and some other materials that can be procured from your local hardware store, see this posting on Aeroelasticity Phenomenon by Wright State’s College of Engineering and Computer Science (Dayton, Ohio).

That’s what meteorologist Edward Lorenz postulated in his 1972 paper on predictability of weather. Lorenz, who died last week at the age of 90, used this example to illustrate his “chaos theory,” which linked small changes in a system to large, unforeseen consequences. For more background on the life and accomplishments of the 1991 Kyoto prize winner for earth and planetary sciences, see this article by Thomas Maugh.

I am certain I heard of chaos theory well before the movie Jurassic Park, but who can forget the pessimistic views the scientific character Doctor Ian Malcolm, who cited Lorenz’s thories to predict the subsequent catastrophe of dinosauric proportions. This is a recurring theme of Jurassic Park author, Michael Crichton: Any complex system will inevitably break down due to the natural state of disorder, or entropy.

I fear that I shall always remain unclear on distinctions a fine as this – chaos vs entropy. Perhaps things may come into focus after I read this article on “Chaos, Complexity, and Entropy”— a physics talk for non-physicists by Michel Baranger of the Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics at Massachusetts Institute of Technology.

It seems to me that Lorenz in his chaos theory considered Earth’s meteorology as a system that often becomes so tightly wound that it comes right to the brink of breaking down — so close that the tiniest disturbance, such as that caused by a benign Brazilian butterfly, can create a terrible upset. Being a chemical engineer, what comes to mind for me is a supersaturated solution of a salt that solidifies around the tiniest seed.

I only hope that I do not get twisted up in Earth’s chaotic meteorology — a very real possibility here at the northern end of the USA’s tornado alley. Maybe a minnow in the Amazon is wiggling a fin at this very moment! I’d better bunker down in the basement…

I collected three articles for my blog this week that all involve the decision to go left or right.

Last Sunday’s Parade magazine reported that Tom Dowdy, an engineer for UPS delivery, estimates a savings of 3 million gallons of gas per year by biasing delivery routes to right, rather than left, turns. The reduction in idling time reduced UPS truck emissions by 32,000 metric tons – the equivalent air pollution of 5300 cars.

This week’s “Ask Marilyn” column in Parade features an observation by Bob English of Lakeland, Florida, who avoided a head on collision thanks to time seemingly slowing down. Marilyn calls this phenomenon “extreme concentration” – a positive reaction to incredible stress. This happened to me some years ago. On a peaceful weekend morning with ideal driving conditions I took my daughter and niece up the Saint Croix Valley for a visit with my mother. Halfway there the one car we encountered on the 15 mile country route veered into directly at us. To me it felt like time stood still as I realized that we’d hit head on in just a second. I remember seeing that I had only a narrow shoulder on the right and realizing that we’d roll if I went any further that direction. Then I clearly recall looking beyond the oncoming driver, who must have dozed off on this sunny morning. There were no other cars coming down the road. I then decided to go around to the left of the opposing automobile – a very radical move. What I did not consider was the other driver waking up and moving out of my lane back to the correct side of the road. I made the move successfully in any case. However, as I learned later from a defensive driving course, the correct maneuver is to go right not matter what – even it means you will crash into a ditch – better that then a head on collision.

The last of the three articles I collected this week is by Shankar Vedantam of the Washington Post. He discusses the natural “action bias” of people who would do best by doing nothing. This causes investors to hold stocks as they peak and sell them after a big fall in price – not an optimal strategy! In another example of action bias, economist Ofer Axar compiled statistics on soccer goalies defending a penalty kick. He concluded that they would stop the most goals by standing still. However, over 90 percent of kicks were defended by diving left or right.

So, the next time you feel pressured into a decision one way or the other, consider the option of not doing anything just yet. However, if something bad will happen for sure by sitting still, I hope that you will benefit from a spell of extreme concentration and not the other typical reaction of people under extreme stress – a paralyzing ‘freeze.’