CHAPTER IV-3 MATHEMATICS AND STATISTICIANS "It's a gift to be simple..." Old Shaker hymn, quoted by Chatterjee and Price, 1991, frontispiece). The headnote above is quoted in a statistics text. But in mathematical statistics, simplicity is given only lip service. Its breach is honored and rewarded with publication and professional status in the statistics (as in other academeic) profession. The real professional gifts are rigor, elegance, sophistication. Indeed, a propensity for simplicity often is not a gift professionally, but a curse. The resampling approach to statistical practice is an important contemporary example. By almost every conceivable test, resampling is simpler than conventional methods. Yet for decades statisticians would have nothing to do with resampling, and even at present they do not embrace it as the tool of first resort for everyday problems. A ILLUSTRATION TO FOCUS THE DISCUSSION Consider as an example a study that offers "A New Confidence Interval Method..." (Peskun, 1993). It works with the following example: 36 of 72 (.5) taxis surveyed in Pittsburgh had visible seatbelts, whereas 77 of 129 taxis in Chicago (.597) had visible seatbelts. The assigned task (whether or not it would be more sensible to think in terms of a test of a hypothesis is left aside) is to derive a confidence interval for the difference of .097 between the proportions. Peskun's new method provides a 95 percent confidence interval of -.237 to .047, to be compared with three other methods he cites whose results are, respectively, -.240 to .046, -.251 to .057, and -.248 to .054. The density of the four pages of mathematics that enter into Peskun's derivation must be seen to be believed. In contrast, consider how resampling handles the problem: 1. Construct urns with proportions like those observed int the two cities. 2. Draw samples with replacement from each urn of same size as actual samples. 3. Compute the observed difference between the experimental trial results. 4. Repeat steps (2-3), graph the results, and count off the confidence interval. For those with a taste for efficiency, the following RESAMPLING STATS computer program mimics the by-hand steps given above: URN 36#1 36#0 pitt Pittsburgh 36 seatbelts, 36 no seatbelts URN 77#1 52#0 chic Chicago 77 seatbelts, 52 no seatbelts REPEAT 15000 SAMPLE 72 pitt pitt$ Draw 72 taxis with Pittsburgh probability SAMPLE 129 chic chic$ Draw 129 taxis with Chicago probability MEAN pitt$ p Compute experimental proportion in Pittsburgh MEAN chic$ c Compute experimental proportion in Chicago SUBTRACT p c d Find difference in experimental proportions SCORE d z Record difference END HISTOGRAM z Show histogram of 15,000 trials PERCENTILE z (2.5 97.5) k Confidence interval from histogram PRINT k k = -0.23934 0.043605 1000+ + * + * * F + * * * r + * * * e 750+ ***** q + ********* u + ********* e + *********** n + *************** c 500+ **************** y + **************** + ******************* * + ******************* * + ********************* Z 250+ ************************ + ************************** + **************************** + ******************************* + *************************************** 0+-------------------------------------------------------------------- |^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^ -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 Vector z The result of 15,000 simulation trials requiring 5 minutes on a desk computer- a confidence interval from -.239 to .044 - agrees with Peskun's interval (-.237 to .047) but is even narrower (which may or may not constitute greater precision). But much more important than the apparent accuracy (assumed from the close agreement), the resampling treatment is transparent to the extent that any user understands exactly what s/he is doing, in contrast to the equational mystery of Peskun's Normal- approximation method which inevitably involves such quantities as pi and e, as a result of using Stirling's formula. The four methods Peskun cites, as well as the resampling method, agree to the second decimal point. It is hard to imagine any application in which the user would care about differences at that level of accuracy, because such differences are likely to be dwarfed by the variation in other aspects of the empirical research study and its data collection and analysis; even indicating that degree of accuracy may well mislead a reader . These are some of the ways in which resampling is simpler than the conventional method: 1. Simpler to understand. Even a freshman or sophomore in an introductory statistics class - indeed, seventh graders - can understand the resampling manipulation shown above. How many who are potential users of this device and not professional mathematical statisticianscan understand the mathematics in the ASA article? (Leave aside that the underlying logic of confidence intervals itself is unintelligible or meaningless to some statisticians of the stature of Leonard Savage and Ronald Fisher.) 2. Simpler to perform. The resampling treatment requires far fewer manipulations than does any conventional treatment, unless the conventional user simply pushes the button on a computer and calls for a complex black-box program. 3. Logically simpler. The resampling treatment requires fewer assumptions. 4. Conceptually simpler. The resampling approach is theoretically simpler because it does not require a count of the size of a sample space and of one or more partitions of it, which is the nub of the problem in any probabilistic calculation, and therefore underlies every probabilistic statistic somewhere in its derivation. *** Why does everyone not handle basic problems the resampling way? Mathematical statisticians have been antagonistic to resampling - and perhaps more generally, to simplicity - for two main reasons, one intellectual-esthetic and the other nitty- gritty issues of self-interest. The latter issue is discussed in the context of teaching statistics in Chapter 00. Simulation fulfills your need when you aim to get a specific answer to a specific question; this is the situation in scientific research and in decision-making in business and elsewhere. Analytic methods provide general answers to classes of questions; this is the aim of mathematicians in their mathematical work. The pedagogical problem is that teaching is in the hands of mathematicians who impute their needs to the students who have entirely different needs, and hence answer questions that are not being asked, with explanations that are not understood; hence the teaching does not even have general cultural value to the students let alone meet their intellectual and practical needs. The end result is that the students are turned away from the subject and lose the chance to gain this valuable knowledge, while the mathematical teachers are frustrated and disgusted by the apparent stupidity of the students. ESTHETICS, MATHEMATICS, AND STATISTICS A mathematician, like a painter or a poet, is a maker of patterns...The mathematician's patterns, like the painter's or the poet's, must be beautiful (C. H. Hardy, A Mathematician's Apology, pp. 84-85, italics in original) Tobias Dantzig pinpointed the key element of thought better, the attitude of the user of statistics]: Between the philosopher's [and the scientist's] attitude toward the issue of reality and that of the mathematician there is this essential difference: for the philosopher [or scientific researcher] the issue is paramount; the mathematician's love for reality is purely platonic. The mathematician is only too willing to admit that he is dealing exclusively with acts of the mind. To be sure, he is aware that the ingenious artifices which form his stock in trade had their genesis in the sense impressions which he identifies with crude reality, and he is not surprised to find that at times these artifices fit quite neatly the reality in which they were born. But this neatness the mathematician refuses to recognize as a criterion of his achievement: the value of the beings which spring from his creative imagination shall not be measured by the scope of their application to physical reality. No! Mathematical achievement shall be measured by standards which are peculiar to mathematics. These standards are independent of the crude reality of our senses. They are: freedom from logical contradictions, the generality of the laws governing the created form, the kinship which exists between this new form and those that have preceded it. The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. (1954, p. 235) Greek philosophy, we must remember, was essentially aristocratic. The methods of the artisan, ingenious and elegant though they may appear, were regarded as vulgar and banal, and general contempt attached to all those who used their knowledge for gainful ends. (There is the story of the young nobleman who enrolled in the academy of Euclid. After a few days, he was so struck with the abstract nature of the subject that he inquired of the master of what practical use his speculations were. Whereupon the master called a slave and commanded: "Give this youth a chalcus, so that he may derive gain from his knowledge.") (Dantzig, p. 117) The activity and knowledge of mathematics have always had a special appeal to many - a fascination that can be most absorbing, and even pass over into religious belief and practice (as seen in such religious disciplines involving mathematics as the numerology of kabbala in mystic Judaism). We also see this fascination in mathematical puzzles, as Martin Gardner notes; There is a fascination about recreational mathematics that can, for some persons, become a kind of drug. Vladimir Nabokov's great chess novel, The Defense, is about such a man. He permitted chess (one form of mathematical play) to dominate his mind so completely that he finally lost contact with the real world and ended his miserable life-game with what chess proble- mists call a suimate or self-mate. He jumped out of a window. It is consistent with the steady disintegra- tion of Nabokov's chess master that as a boy he had been a poor student, even in mathematics, at the same time that he had been "extraordinarily engrossed in a collection of problems entitled Merry Mathematics, in the fantastical misbehavior of numbers and the wayward frolic of geometric lines, in everything that the schoolbook lacked." The moral is: Enjoy mathematical play, if you have the mind and taste for it, but don't enjoy it too much. Let it provide occasional holidays. Let it stimulate your interest in serious science and mathematics. But keep it under firm control (1966, p. 9). An interesting sidelight here concerns the number pi (and also e) which frequently appear in connection with probability and statistics, and especially in the Normal Appoximation. Many people have been impressed by that. For example, the Nobel- prize-winning physicist Eugene Wigner wrote thusly about pi and e (see introduction to Chapter 00 [Normal distribution], as well as other complex numbers: Certainly, nothing in our experience suggests the introduction of these quantities. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, some some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius, (Wigner, p. 224). Resampling - aside from the "real mathematics"<1> that constitutes the state-of-the-art investigations of resampling techniques - offends mathematical statisticans because it does not meet the main criteria of what has always been considered truth in mathematics. Monte Carlo simulations of all kinds butt up against the fundamental attitude of the mathematics profession toward non-proof-based methods. As S. Stigler put the matter in a related connection: "Within the context of post-Newtonian scientific thought, the only acceptable grounds for the choice of an error distribution were to show that the curve could be mathematically derived from an acceptable set of first princi- ples" (1986, p. 110, italics added). This may be related to Mosteller's comment somewhere that the bootstrap (and presumably all resampling) is "anti-intuitive" [? see history file], whereas to the layperson resampling certainly is much more intuitive than is the formulaic method. Statisticians also often reject simple methods because it seems prudent to do so. ...profitable, practical operations-research work is often arrived at by so simple, even crude, an analysis that very often its author himself would hesitate to write a paper about it for publication in a learned journal. And if he did it might well be rejected for its simplicity and lack of mathematical sophistica- tion. ...learned periodicals and journals value mathematical refinement and academicism for their own sake..." (Singh, 1972, p. 17) Mathematical physicist John Barrow invented a revealing scenario about proof-based mathematics. He imagined what might happen if we were to receive a response from Martians to an Earth-transmitted extra-terrestrial messages. Those messages depend heavily upon mathematics, on the assumption that that will be the easiest for the Martians to decode. Barrow writes first about the excitement: There is great excitement at NASA today. Years of patient listening have finally borne fruit. Contact has been found. Soon the initial euphoria turns to ecstasy as computer scientists discover that they are eavesdropping not upon random chit-chat but a systemat- ic broadcast of some advanced civilisation's mathemati- cal information bank. The first files to be decoded list all the contents of the detailed archives to come. Terrestrial mathematicians are staggered: at first they see listings of results that they know, then hundreds of new ones including all the great unsolved problems of human mathematics.... Soon, the computer files of the extraterrestrials' mathematical textbooks begin to arrive on earth for decoding and are translated and compiled into English to await study by the most distinguished representa- tives of the International Mathematical Congress. Mathematicians and journalists all over the world wait expectantly for the first reactions to this treasure chest of ideas. Then he writes about the next peculiar reaction: But odd things happened: the mathematicians' first press conference was postponed, then it was cancelled without explanation. Disappointed participants were seen leaving, expressionless, making no comment; the whole atmosphere of euphoria seemed to have evaporated. After some days still no official statement had been made but rumours had begun to circulate around the mathematical world. The extraterrestrials' mathematics was not like ours at all. In fact, it was horrible. They saw mathematics as another brance of science in which all the facts were established by observation or experiment.(Barrow, 1992, pp. 178-179) The key is the disappointment. Terrestrial mathematicians are not excited by a method that simply offers answers or solutions. The method must also meet aesthetic tests to be acceptable. It is here that resampling fails. The connection between the esthetic of mathematics, and the rejection of simplicity, can be seen in the comment of a referee on one of my papers: "Most of us [presumably, statisticians] would be concerned about the implication that users should always [I do not say "always"] use self-invented ultra-simple techniques" (italics added). Here we see in starkest terms the issue that separates the mainstream thinking in the statistics profession and the approach suggested here -- whether simplicity is better. I believe that simplicity -- and re-creation of the technique from first principles (rather than taking a technique off the shelf) is part-and-parcel of simplicity -- reduces the chance of using an unsound technique. But to that referee and the others s/he refers to, simplicity is unaesthetic, and perhaps threatening. The attitude is that with which G. Stigler characterizes science, "The ...work should be pursued with non- vulgar instruments" (1973, quoted by Fisher, 1986, p. 78). Dantzig contrasts the outlook of the mathematician with the person who has practical interests: Greek philosophy, we must remember, was essentially aristocratic. The methods of the artisan, ingenious and elegant though they may appear, were regarded as vulgar and banal, and general contempt attached to all those who used their knowledge for gainful ends. (There is the story of the young nobleman who enrolled in the academy of Euclid. After a few days, he was so struck with the abstract nature of the subject that he inquired of the master of what practical use his speculations were. Whereupon the master called a slave and commanded: "Give this youth a chalcus, so that he may derive gain from his knowledge.") (Dantzig, p. 117) The irrational quantities can be expressed through rational approximations to any desired degree of accuracy... Such methods enable one to "trap" the irrational number between two sequences of rational numbers, of which the first is consistently "less" than the irrational, and the second consistently "greater." And, what is more, the interval between these rational approximations may be rendered as small as one desires. Well, what further can be desired? The physicist, the engineer, the practical man generally are fully satisfied. What the physicist requires of his calculating methods is a degree of refinement which will permit him to take full advantage of the growing precision of his measuring devices. The fact that certain magnitudes, like **2, **, or e, are not expressible mathematically by means of rational numbers will not cause him to lose any sleep, as long as mathematics is furnishing him with rational approximations for such magnitudes to any accuracy he desires. The position of the mathematician with respect to this problem is different. (Dantzig, p. 107) There also was the problem at first of resampling have been suggested from outside of statistics. Even the greatest of scientists sometimes look down their noses at upstarts without the proper credentials. Consider this lovely quote by James Clerk Maxwell (by way of Conant, 1965, pp. 39, 40) about Alexander Graham Bell: When about two years ago news came from the other side of the Atlantic that a method had been invented of transmitting by means of electricity the articulate sounds of the human voice so as to be heard hundreds of miles away from the speaker, those of us who had reason to believe that the report had some foundation in fact began to exercise our imagination, picturing some triumph of constructive skill--something as far surpassing Sir William Thomson's siphon recorder in delicacy and intricacy as that is beyond a common bell pull. When at last this little instrument appeared, consisting, as it does, of parts every one of which is familiar to us and capable of being put together by an amateur, the disappointment arising from its humble appearance was only partially relieved on finding that it was really able to talk... Professor Graham Bell, the inventor of the telephone, is not an electrician who has found out how to make a tin plate speak, but a speaker who, to gain his private ends, has become an electrician. Two more quotations from commentators on science are relevant to the subject of the initial rejection and now the slow and grudging acceptance of resampling by the mathematical- statistical profession: 1) "It is the essence of a profession that the skills required therein are not possessed by those without" (Goodwin, 1973, cited by Fisher, p. 80). 2) "Is not every new discovery a slur upon the sagacity of those who overlooked it?" (Jewkes, 1991, p. 11). <2> The main points of divergence between the statistics profession and me, then, are three-fold: 1) I am a Martian, in Barrow's phrase, providing simple new methods but not rigorous proofs; proofs are of the highest value to the profession, while the simplicity is anathema. 2) As mathematicians, statisticians aim at generality (embodied in formulae), whereas I aim at specific solutions (embodied in numerical answers and the procedures to obtain them). 3) I seek methods that are so simple and transparent that those who are not mathematical statisticians can use them correctly and with full understanding, but the statistics profession honors methods that require the intercession of a professional statistician. THE VALUE OF DEXTERITY WITH MATHEMATICAL REASONING This section picks a fight with the most self-assured and prestigious sub-collection of these people - those who pride themselves on being clever with mathematical thinking and puzzle- solving.<3> The only conceivable counter-force that might support the argument being mde here includes those who think themselves clever yet know that that are not good at mathematical closed- system thinking. But this group tends to be intimidated in these matters by the mathematically-clever types, and therefore are not likely to come out in support. Let's be clear about the supposed equation in mathematicians' minds of cleverness and mathematical thinking. For example, a famous book of mathematical puzzlesstarts its first page as follows: "To see how good your brain is..." (Kordemsky, 1972, p. 1). And in Francis Galton's early discussion of intelligence, he wrote: "There can hardly be a surer evidence of the enormous difference between the intellectual capacity of men, than the prodigious differences in the numbers of marks [the grades] obtained by those who gain mathematical honours at Cambridge" (1869, quoted in Herrnstein and Boring, 1965, p. 416). A professorial colleague of the author's teaches elementary statistics (his department limits him to that course) and is not fully witted by any practical test I know of. His behavior borders on the bizarre in business and personal matters. He apparently has never produced a useful piece of research work in his several decades of university employment. He is not even amiable or amusing. By no sensible measure that I know of could his "intelligence" (whatever one means by that) be considered even borderline average, except for his capacity to manipulate mathematical symbols. Yet he contemptuously refers to his university students as stupid because they don't follow the formulas he writes on the blackboard. Not only do I not know of any empirical evidence to believe that those who think in mathematical terms (whatever that means) are generally better thinkers than those who don't, but I can't find much reason to believe that it ought to be so. "When was the latest year that is the same upside down?" is Kordemsky's (1972, puzzle 39, p. 15). So you figure out that 1961 is the answer, either by some shortcut logical process or by trying individual years going backwards. So what? What does finding the answer prove? Perhaps there is some imagination shown in working out a system for finding the number, but is this "better" in any way than figuring out a system to water the garden more efficiently? Or to set up a new club for neighborhood children? In another puzzle (Kordemsky, p. 4), a troop of soldiers stands by a river with no boat or bridge. Two boys are nearby in a rowboat that is big enough for the two of them, but only for one soldier. How can you move the soldiers across the river? The book's solution has to do with moving combinations of boys and soldiers. But what about making a rope and pulling the boat back and forth? The mathematical approach shuts out such ideas that go outside the facts that have been given. Such closed- system thinking certainly can be useful, but it certainly is not the whole of good thinking, and it may well direct people away from creative open-system thinking. Another flaw with treating mathematical puzzles as a test of the goodness of one's brain is that solving many math puzzles is simply a matter of knowing mathematical rules. For example, the solution of one well-known puzzle hinges on remembering that one is not allowed to divide by an algebraic expression that equals zero. This has nothing to do with having a brain with good general capabilities; it has to do only with knowing mathematics (and it might just as well be the rules of chess or of bingo). Amos Tversky once remarked that nothing is surer than that people err in their thinking. In a valuable body of research, cognitive psychologists point out the large defects in people's thinking about issues that require statistical understanding, and they document these effects in controlled experiments. For example, people do not recognize the effect of sample size upon the extent of variability (as was the case even with the great John Graunt; see Chapter 00), and they do not estimate posterior odds well when Bayesian thinking is called for, as well as misapplying a variety of heuristics. In response, the psychologists diagnose the problem as a lack of "intuition" deriving from a poor grasp of statistical theory, and with the aim of "improving the quality of thinking" (Kahneman and Tversky, 1982, p. 494) they suggest more and better statistical training. For example, Tversky and Kahneman note that when asked how many different different committees of k members can be formed from a group of n people, subjects will guess that there are more possible 2-person than 5-person in a group of 10 persons. They then say that "One way to answer this question without computation is to mentally construct..." (p. 12, italics added), and they go on to suggest a mode of imagining. Lewis Carroll produced an entire book full of "Pillow Problems" that he solved in bed without paper and pencil. He wrote that he did not take special pride in having solved them that way, but is that believable? Should we consider a person a better carpenter who chooses to build a house only with old-style hammer and saw rather than using the entire array of modern tools that are available? Or playing a piano concerto with one hand? A tour de force may be amusing and impressive, but it displays art and not productive power. Kahneman and Tversky show subjects the series 8 * 7 *...1, and 1 * 2 *...8 for five seconds and ask them to compute the answer, finding different answers for the two modes of presentation. But why not focus on the problems people make when they have plenty of time to solve them? (Perhaps there is a hint here of the idea that cleverness is associated with being "quick" or "fast" in one's thinking.) Certainly these authors would agree that the better scientist is not the one who thinks up conclusions or research ideas more rapidly, but rather the scientist who gets better conclusions and more important research ideas. Always our aim should be to make people more sound in their thinking, as tested by their effectiveness with regard to the world of objects, people, and events, rather than making them more clever (although admittedly, being seen as clever can enhance one's effectiveness with people.) Nisbett et. al., say that "reasoning is based on models", and recommend that people learn to "call to mind a statistical heuristic" (1982, p. 448). And as a way of teaching people that there will be more variability in the percentage of babies born in a small hospital than in a large hospital they suggest that "a correct answer can be elicited in a series of easy steps" of a Socratic nature (1982, p. 500). I submit that the essence of the problem is not the lack of well-trained intuition but lack of a more important mental tool: the habit of actually (not in the imagination) simulating the situation at hand so as to estimate the relevant probabilities. That is, we should advise people not to rely on mental operations, and not to try to improve their thinking with statistical theory directly. Instead, we should teach them the heuristic "Try it". Instead of asking people whether the percentage will vary more in large and small hospitals, ask them the probability that the proportion will be greater than (say) 60 percent with daily samples of 15 and 45 babies, and then teach them to simulate the situation either with coins or with a simple computer program. The ironic part is that even though the subjects do not need to understand the theory to arrive at a sound answer, they are likely to learn from the results the very theory that would help them deal with the problem correctly by mental computation alone. But learning the theory and improving one's intuition as a result of the doing simulations is just a bonus; the main goal - which should always be kept in mind - is arriving at sound answers to the questions that arise in a person's work and personal life. Let's be specific. One of the most-used problems in the study of cognitive psychology is K and T's taxi question. It goes like this: A cab was involved in a hit-and-run accident at night. Two cab companies, the Green and the Blue, operate in the city. You are given the following data: (a) 85% of the cabs in the city are Green and 15% are Blue. (b) a witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accidentally and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time. What is the probability that the cab involved in the accident was Blue rather than Green?(Tversky and Kahnemann, p. 157) Subjects mostly guess quite wrongly, the median answer being .8 whereas the correct answer is .41. A person who chooses to experiment rather than ratiocinate may proceed as follows: 1. Assign numbers 1-85 as Green taxis, 86-100 as Blue taxis. Because 80 percent of each group will be identified correctly and 20 percent incorrectly , assign numbers 1-68 as correctly identified Green taxis, 69-85 as incorrectly-identified Green taxis (as Blue), and numbers 86-97 and 98-100 as correctly- and incorrectly-identified Blue taxis respectively. 2. Choose x integers randomly between 1 and 100 and record them. 3. Count the number of integers between 69-85 and between 86 and 97. 4. Compute the ratio of the number of integers between 86 and 97 to the sum of those 69-85 and 86-97. One may ask why one should bother to experiment when one can go directly to step 4. But the fact is that people do not manage to go directly to step 4 because doing so requires an understanding that most people demonstrably do not possess. And that understanding is not necessary when one models the process directly and proceeds from step 1 to step 4. I do not argue that the latter process is better than former; I "merely" say that that it produces more correct answers. Such simulation is feasible in just about every situation. And it may not even take longer than ratiocination. Consider for another of the psychologists' pet problems (from Kahneman and Tversky), the variability in proportions of boys and girls born in hospitals of various sizes: A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day. As you know, about 50 percent of all babies are boys. However, the exact percentage varies from day to day. Sometimes it may be higher than 50 percent, sometimes lower. For a period of 1 year, each hospital recorded the days on which more than 60 percent of the babies born were boys. Which hospital do you think recorded more such days? Here is how it may be handled with the following computer program: REPEAT 1000 Begin the first of 1000 trials GENERATE 200 1, 2 a Create a sample of 200 boys or girls COUNT a 1 b Count the number of boys DIVIDE b 200 c Normalize result as percent boys SCORE b z Record the result of the trial END End the 1000 trials HISTOGRAM z Plot the results of the 1000 trials By changing the sample size of babies from 200 to 20 and 2000 - by simply substituting one of those numbers in the second and fourth numbers in the program above one can immediately the effect of changing the sample size on the variability. The cognitive psychologists ask: Why do people get these questions wrong? I answer on a different dimension than they do, saying "Because people do not experiment, but rather try to think unaided by concrete trials".[1] Nor is this change in dimension of the answer simply ducking the "real" question. Once I watched an Indian in a market of the city of Jubalpur carve bed legs on a lathe. The four legs for a single bed came out far from identical because the lathe worker worked by eye rather than using a patterned guide. It makes sense to say that the reason that the legs were uneven is that he did not use a guide, rather than saying that his eyes and hands are fallible. (When I suggested using a guide, the lathe worker said that it was a good idea. Maybe in the future. The shop was still new. I asked how new. Nine years old, I was told.) Why do people raticinate and not experiment? Because we teach them the former and not the latter, and because we put a high value - and the measure of "intelligence" - on the former rather than the latter. *** A comment of the philosopher Collingwood pertains to the mathematically clever: The tail-less dog praises tail- lessness. **FOOTNOTES** [1]: Nothing said here is intended to suggest that the cognitive psychologists' study of the causes of biases is not valuable. It certainly is important for us to understand biases if only to anticipate and counter them in non-quantitative situations where one cannot resort to experimentation. But I would suggest to the cognitive psychologists that it is worthwhile to teach people to seek concrete representations for abstract questions even in non-quantitative situations; the fathers-and-sons puzzle in Chapter 00 is a powerful example. Another example is estimating the area of a figure with the Archimedean method or by counting squares under the curve. Indeed, the cognitive psychologists have long shown the way in just such a fashion when they instruct people to estimate odds with various devices such as hypothetical bets, or by simply checking for consistency by estimating the negative of an event and calculating whether the two probabilities add to unity. (I have not, however, come across any research on improvement in skills from using such methods.) copied into refstats. take out this page for book, but keep for ref REFERENCES Barrow, John D., Pi in the Sky: Counting, Thinking and Being (New York: Oxford UP, 1992) Chatterjee, Samprit, and Bertram Price, Regression Analysis by Example (New York: John Wiley, 1991). Gardner, Martin, New Mathematical Diversions From Scientific American, ( New York: Simon & Schuster, 1966) Hardy, C. H., A Mathematician's Apology. Peskun, Peter H., "A New Confidence Interval Method Based on the Normal Approximation for the Difference of Two Binomial Probabilities", JASA, Vol 88, #422, pp. 656-660 Stigler, George J., "The Adoption of the Marginal Utility Theory", in R. D. Black, A. W. Coats and Craufurd D. W. Goodwin (eds), The Marginal Revolution in Economics: Interpretation and Evaluation (Durham: Duke U. Press, 1973) Stigler, Stephen M., The History of Statistics (Cambridge: Harvard U. Press, 1986). Bin Cum Center Freq Pct Pct -------------------------------------------- -0.38 2 0.0 0.0 -0.35 1 0.0 0.0 -0.34 7 0.0 0.1 -0.33 3 0.0 0.1 -0.32 7 0.0 0.1 -0.31 13 0.1 0.2 -0.3 8 0.1 0.3 -0.29 26 0.2 0.4 -0.28 29 0.2 0.6 -0.27 47 0.3 1.0 -0.26 55 0.4 1.3 -0.25 107 0.7 2.0 -0.24 134 0.9 2.9 -0.23 167 1.1 4.0 -0.22 195 1.3 5.3 -0.21 260 1.7 7.1 -0.2 269 1.8 8.9 -0.19 402 2.7 11.5 -0.18 386 2.6 14.1 -0.17 527 3.5 17.6 -0.16 528 3.5 21.2 -0.15 568 3.8 24.9 -0.14 721 4.8 29.7 -0.13 686 4.6 34.3 -0.12 840 5.6 39.9 -0.11 770 5.1 45.1 -0.1 910 6.1 51.1 -0.09 758 5.1 56.2 -0.08 929 6.2 62.4 -0.07 715 4.8 67.1 -0.06 711 4.7 71.9 -0.05 620 4.1 76.0 -0.04 593 4.0 80.0 -0.03 545 3.6 83.6 -0.02 477 3.2 86.8 -0.01 418 2.8 89.6 0 307 2.0 91.6 0.01 346 2.3 93.9 0.02 227 1.5 95.4 0.03 222 1.5 96.9 0.04 101 0.7 97.6 0.05 91 0.6 98.2 0.06 74 0.5 98.7 0.07 63 0.4 99.1 0.08 37 0.2 99.3 0.09 32 0.2 99.6 0.1 20 0.1 99.7 0.11 16 0.1 99.8 0.12 8 0.1 99.9 0.13 8 0.1 99.9 0.14 4 0.0 99.9 0.15 3 0.0 100.0 0.16 3 0.0 100.0 0.17 1 0.0 100.0 0.2 1 0.0 100.0 0.23 2 0.0 100.0 Note: Each bin covers all values within 0.005 of its center. ENDNOTES **ENDNOTES** <1>: This is the term that William Kruskal used in 1984 correspondence with me in distinguishing between a) the simple uses of resmampling and the bootstrap of the sort described here, and b) the sophisticated uses and explorations of the characteristics of these techniques that began with the work of Efron and colleagues in 1979. <2>: This discussion may reflect the fact that much of my professional life has been a tussle with the mathematical attitude - people I consider deductionists. In discussion of such issues as causality, duopoly theory, and of course resampling, they have always sought to arrive at conclusions by the logical analysis of closed systems, excluding rich variables which would make the analysis intractable mathematically, and also excluding any judgments which would make it impossible to close the system. My tendency always is the opposite. And I always begin with concrete examples, and work my way toward the abstract generality, whereas many deductionists praise the opposite route (even if they themselves often do not follow it.) In the same spirit, I tend to verify matters by using two or more methods to arrive at an answer, and if they agree, I need not worry about which is the better method. I find the deductionists often preferring to argue about which is the correct method. Perhaps my not being attracted to abstraction and deduction is due to my lack of capacity to manipulate equations, and this in turn seems due to my inability to remember the meanings of notation. This may be connected with my preference for working with concrete entitities. I do not share the ability of phyicists and others to forget the content of an equation and manipulate it symbolically to see where it arrives. (I am startled at how so many physicists have no sense of the "philosophical" content of Einstein's special relativity, but place great confidence in the results of their manipulations of its equations.) It makes sense to me, and seems confirmed by observation, that working with contentless symbols often leads to error. <3>: More generally, this section challenges the most powerful collection of persons in the world - those who think that they are clever. This set is also one of the largest collections of persons in the body of humanity, including almost everyone who went to college and a good many who never went to school at all. Most specifically, the argument in this section challenges