For more information contact Resampling Stats, 612 N. Jackson St., Arlington, VA, 22201 PROBABILITY AND STATISTICS THE RESAMPLING WAY Stats for Poets, Politicians - and Statisticians Julian L. Simon and Peter Bruce Probability theory and its offspring, inferential statistics, constitute perhaps the most frustrating branch of human knowledge. Right from its beginnings in the seventeenth century, the great mathematical discoverers knew that the probabilistic way of thinking -- which we'll call "prob-stats" for short -- offers enormous power to improve our decisions and the quality of our lives. Prob-stats can aid a jury's deliberation about whether to find guilty a person charged with murder...reveal if a new drug boosts survival from a cancer...help steer a spacecraft to Saturn...inform the manager when to take a pitcher out of the baseball game...aid a wildcatter to calculate how much to invest in an oil well...and a zillion other good things, too. Yet until very recently, when the resampling method came along, scholars were unable to convert this powerful body of theory into a tool that laypersons could and would use freely in daily work and personal life. Instead, only professional statisticians feel themselves in comfortable command of the prob- stats way of thinking. And the most frequent application is by social and medical scientists, who know that prob-stats is indispensable to their work yet too often fear and misuse it. Prob-stats continues to be the bane of students, most of whom consider the statistics course a painful rite of passage -- like fraternity paddling -- on the way to an academic degree. Even among those who study it, most close the book at the end of the semester and happily put prob-stats out of their minds forever. The statistical community has made valiant attempts to ameliorate this sad situation. Great statisticians have struggled to find interesting and understandable ways to teach prob-stats. Learned committees and professional associations have wrung their hands in despair, and spent millions of dollars to create television series and text books. Chance magazine imaginatively demonstrates and explains the exciting uses and benefits of prob-stats. Despite successes, these campaigns to promote prob-stats have largely failed. The enterprise smashes up against an impenetrable wall - the body of complex algebra and tables that only a rare expert understands right down to the foundations. Almost no one can write the formula for the "Normal" distribution that is at the heart of most statistical tests. Even fewer understand its meaning; yet without such understanding, there can be only rote learning. Almost every student of probability and statistics simply memorizes the rules. Most users of prob-stats select their methods blindly, understanding little or nothing of the basis for choosing one method rather than another. This often leads to wildly inappropriate practices, and contributes to the damnation of statistics. Indeed, in the last decade or so, the discipline's graybeards have decided that prob-stats is just too tough a nut to crack, and have concluded that students should be taught mainly descriptive statistics - tables and graphs - rather than how to draw inferences probabilistically, which is really the heart of statistics. The new resampling method, in combination with the personal computer, promises to change all this. Resampling may finally realize the great potential of statistics and probability. Resampling estimates probabilities by numerical experiments instead of with formulae - by flipping coins or picking numbers from a hat, or with the same operations simulated on a computer. And the computer language-program RESAMPLING STATS performs these operations in a transparently clear and simple fashion. The best mathematicians now accept resampling theoretically. And controlled studies show that people ranging from engineers and scientists down to seventh graders quickly handle more problems correctly than with conventional methods. Furthermore, in contrast to the older conventional statistics, which is a painful and humiliating experience for most students at all levels, the published studies show that students enjoy resampling statistics. THE REAPPEARANCE OF RESAMPLING IN THE HISTORY OF STATISTICS Resampling returns to a very old tradition. In ancient times, mathematics in general, and statistics in particular, developed from the needs of governments and rich men to count their armies and flocks, and to enumerate the taxpayers and their possessions. Up until the beginning of the twentieth century, the term "statistic" meant "state-istics", the number of something the "state" was interested in -- soldiers, births, or what-have-you. Even today, the term "statistic" usually means the quantity of something, such as the important statistics for the United States in the Statistical Abstract of the United States. These numbers are now known as "descriptive statistics," in contrast to "inferential statistics" which is the science that tells us how reliable is a set of descriptive statistics. Another stream of thought appeared by way of gambling in France in the 17th century. Throughout history people had learned about the odds in gambling games by experimental trial- and-error experience. To find the chance of a given hand occurring in a card game, a person would deal out a great many hands and count the proportion of times that the hand in question occurred. That was the resampling method, plain and simple. Then in the year 1654, the French nobleman Chevalier de Mere asked the great mathematician and philosopher Blaise Pascal to help him deduce what the odds ought to be in some gambling games. Pascal, the famous Pierre Fermat, and others went on from there to develop analytic probability theory, and Jacob Bernouilli and Abraham DeMoivre initiated the formal theory of statistics. The experimental method disappeared into mathematical obscurity except for its use when a problem was too difficult to be answered theoretically, as happened from time to time in the development of statistical tests -- for example, the development of the famous t-test by "Student", the pen-name of William S. Gossett -- and the World War II "Monte Carlo" simulations for complex military "operations research" problems such as how best to search for submarines with airplanes. Later on, these two streams of thought -- descriptive statistics and probability theory -- joined together. Users of descriptive statistics wondered about the accuracy of the data originating from both sample surveys and experiments. Therefore, statisticians applied the theory of probability to assessing the accuracy of data and created the theory of inferential statistics. HOW RESAMPLING DEVELOPED Too much book-learning, too little understanding. The students had swallowed but not digested a bundle of statistical ideas which now misled them, taught by professors who valued fancy mathematics even if useless or wrong. It was the spring of 1967 at the University of Illinois, in my (Simon's) course in research methods with four graduate students working toward the PhD degree. I required each student to start and finish an empirical research project as a class project. Now the students were presenting their work in class. Each used wildly wrong statistical tests to analyze their data. "Why do you use the technique of cluster analysis?" I asked Moe Taher (a pseudonym). "I want to be up-to-date," said Taher. "How much statistics have you studied?" I asked. "Two undergraduate and three graduate courses," Taher answered proudly. It was the usual statistical disaster. A simple count of the positive and negative cases in Taher's sample was enough to reveal a clear-cut conclusion. The fancy method Taher used was window-dressing, and inappropriate to boot. It was the same sad story with the other three students. All had had several courses in statistics. But when the time came to apply even the simplest statistical ideas and tests in their research projects, they were lost. Their courses had plainly failed to equip them with basic usable statistical tools. So I wondered: How could we teach the students to distill the meaning from their data? Simple statistical methods suffice in most cases. But by chasing after the latest sophisticated fashions the students overlook these simple methods, and instead use unsound methods. I remembered trying to teach a friend a concept in elementary statistics by illustrating it with some coin flips. Given that the students' data had a random element, could not the data and the events that underlie the data be "modeled" with coins or cards or random numbers, doing away with any need for complicated formulas? Next class I shelved the scheduled topics, and tried out some problems using the resampling method (though that label had not yet been invented). First the students estimated the chance of getting two pairs in a poker hand by dealing out hands. Then I asked them the likelihood of getting three girls in a four- child family. After they recognized that they did not know the correct formula, I demanded an answer anyway. After suggesting some interesting other ideas -- we'll come to them later -- one of the students eventually suggested flipping coins. With that the class was off to the races. Soon the students were inventing ingenious ways to get answers -- and sound answers -- to even subtle questions in probability and statistics by flipping coins and using random numbers. The very next two-hour seminar the students re-discovered an advanced technique originally invented by the great statistician Ronald A. Fisher. The outcome of these experiments was resampling. Even before this time, though I had not known of it, resampling had been suggested for one particular case in inferential statistics in technical articles in the Annals of Mathematical Statistics by Meyer Dwass in 1957, and in the Journal of the American Statistical Association by J. H. Chung and D. A. S. Fraser in 1958. They preceded me in applying sampling methods to the problem of deciding whether two sample means differ from each other, basing the procedure on Fisher's famous "randomization" test. The new idea I contributed was handling all (or at least most) problems by resampling. And to that end, I taught a systematic procedure for carrying out resampling procedures and illustrated it for a variety of problems, while also teaching conventional methods in parallel. Then it was natural to wonder: Could even children learn this powerful way of dealing with the world's uncertainty? Max Beberman, the guru of the "new math" who then headed the mathematics department in the University of Illinois High School, quickly agreed that the method had promise, and suggested teaching the method to a class of volunteer juniors and seniors. The kids had a ball. In six class hours they were able to discover solutions and generate correct numerical answers for the entire range of problems ordinarily found in a semester-long university introductory statistics class. Furthermore, the students loved the work. A semester-long university class in statistics, with resampling and the conventional method taught side by side, came next. But students complained that dealing cards, flipping coins, and consulting tables of random numbers gets tiresome. So in 1973 I developed a computer language that would do with the computer what one's hands do with cards or dice. The RESAMPLING STATS program, which handles all problems in statistics and probability with only about twenty commands that mimic operations with cards, dice, or random numbers, is a simple language requiring no computer experience. Even 7th graders quickly understand and use it, though it is powerful enough for scientific and industrial use. (It does, however, provide a painless introduction to computers, which is a valuable educational bonus.) A major sub-part of the general resampling method - the "bootstrap", which was independently developed by Bradley Efron in the 1970s - has now swept the field of statistics to an extraordinary extent. The New York Times had this to say: "A new technique that involves powerful computer calculations is greatly enhancing the statistical analysis of problems in virtually all fields of science. The method, which is now surging into practical use after a decade of refinement, allows statisticians to determine more accurately the reliability of data analysis in subjects ranging from politics to medicine to particle physics... "`There's no question but that it's very, very important' said Frederick Mosteller, a statistician at Harvard University...Jerome H. Friedman, a Stanford statistician who has used the new method, called it `the most important new idea in statistics in the last 20 years, and probably the last 50'. He added, `Eventually, it will take over the field, I think.'" (Nov. 8, 1988, C1, C6) Resampling is best understood by seeing it being learned. The instructor walks into a new class and immediately asks, say, "What are the chances if I have four children that three of those children will be girls?" Someone says "Put a bunch of kids into a hat and pick out four at a time". Teach says, "Sounds fine in theory, but it might be a bit difficult to actually carry out...How about some other suggestions?" Someone else says, "Have four kids and see what you get." Teach says, "Sounds good. But let's say you have four children once. Is that going to be enough to give you a decent answer?" So they discuss how big a sample is needed, which brings out the important principle of variability in the samples you draw. Teach then praises this idea because it focuses attention on learning from experiment, one of the key methods of science. S/he points out, however, that it could take a while to have a hundred families, plus some energy and money, so it doesn't seem to be practical at the moment. Teach asks for another suggestion. Someone suggests taking a survey of families with four children. Teach praises this idea, too, because it focuses on getting an answer by going out and looking at the world. But what if a faster answer is needed? Someone else wonders if it is possible to "do something that is like having kids. Put an equal number of red and black balls in a pot, and pull four of them out. That would be like a family." This kicks off discussion about how many balls are needed, and how they should be drawn, which brings out some of the main concepts in probability - sampling with or without replacement, independence, and the like. Then somebody wonders whether the chance of having a girl the first time you have a child is the same as the chance of a red ball from an urn with even numbers of red and black balls, an important question indeed. This leads to discussion of whether 50-50 is a good approximation. This brings up the question of the purpose of the estimate, and the instructor suggests that a clothing manufacturer wants to know how many sets of matched girls dresses to make. Coins seem easier to use than balls, all agree. And Teach commissions one student to orchestrate the rest of the class in a coin-flipping exercise. Then the question arises: Is one sample of (say) thirty coin-flip "families" enough? The exercise is repeated several times, and the class is impressed with the variability from one sample of thirty to the next. Once again the focus is upon variability, perhaps the most important idea inherent in prob-stats. Or another example: The instructor asks, "What are the chances that basketball player Magic Johnson, who averages 47 percent success in shooting, will miss 8 of his next 10 shots?" The class shouts out joking suggestions such as "Go watch Magic," and "Try it yourself on the court." Teach responds, "Excellent ideas, good scientific thinking, but not feasible now. What else could we do?" Soon someone - say, Adam - suggests flipping a coin. This leads to instructive discussion about whether the 50-50 coin is a good approximation, and whether ten coins flipped once give the same answer as one coin flipped ten times. Eventually all agree that trying it both ways is the best way to answer the question. Teach then invites Adam up front to run the simulation. Adam directs each of the 30 students to flip ten coins and count the resulting heads. Teach writes the results on the board. The proportion of the thirty trials in which there are 8 or more misses is then counted. The students discuss this result in the light of the variability from one sample of ten shots to another - an important and now-obvious idea. And Teach points out how the same procedure is at the heart of industrial quality control. After a while someone complains that flipping coins and dealing cards is wearisome. Aha! Now Teach breaks out the computer and suggests doing the task faster, more accurately, and more pleasurably with the following computer instructions: REPEAT 100 obtain a hundred simulation trials GENERATE 10 1,1OO A generate 10 numbers randomly between 1 and 100 COUNT A 1,53 B count the number of misses in the trial (Magic's shooting average is 47% hits, 53% misses.) SCORE B Z record the result of the trial END end the repeat loop for a single trial COUNT Z 8,10 K count the number of trials with 8 or more misses The histogram (Figure 1) shows the results of 100 trials, and Figure 2 shows the results of 1000 trials. The amount of variability obviously diminishes as the as the number of trials increases, an important lesson. Figures 1 and 2 Then Teach asks: "If you see Magic Johnson miss 8 of 10 shots after he has returned from an injury, should you think that he is in a shooting slump?" Now the probability problem has become a problem in statistical inference -- testing the hypothesis that Magic is in a slump. And with proper interpretation the same computer program yields the appropriate answer -- about 6.5 percent of the time Magic will miss 8 or more shots out of 10, even if he is not in a slump. So don't take him out or stop him from shooting. Understanding this sort of variability over time is the key to Japanese quality control. Now the instructor changes the question again and asks: "If you observe a player - call him Houdini - succeed with 47 of 100 shots, how likely is it that if you were to observe the same player take a great number of shots - a thousand or ten thousand - his long-run average would turn out to be 53 per cent or higher?" A sample of 47 baskets out of 100 shots could come from players of quite different "true" shooting percentages. Resampling can help us make transparent several different approaches to this problem in "inverse probability". Clearly, we need to have some idea of how much variation there is in samples from shooters like Houdini. If we have no other information, we might reasonably proceed as if the 47/100 sample is our best estimate of Houdini's "true" shooting percentage. We could and take repeated samples from a 47% shooting machine to estimate how great the variation is from shooters with long-run averages in that vicinity, from which we could estimate the likelihood that the "true" average is 53%. (This is the well-known "confidence interval" approach. In truth, the logic is a bit murky, but that is seldom a handicap in daily practice. ) Alternatively, we might be interested in a particular shooting percentage - say, Houdini's lifetime average before a shoulder injury. In such a case, we might want to know whether the 47 for 100 is just spell of below-average shooting, or an indication that the injury has affected his play. In this situation, we could repeatedly sample from a 53% shooting machine to see how likely a 47/100 sample is. Using this "hypothesis- testing" approach, if we find that the 47/100 is very unusual, we concluded that the injury is hampering Houdini; if not, not. Consider still another possibility: If Houdini is a rookie with no history in the league, we might want to apply additional knowledge about how often 53% shooters are encountered in the league. Here we might bring in information about the "distribution" of the averages of other players, or of other rookies, to see how likely a 47/100 sample is in light of such a distribution - a "Bayesian" approach to the matter. The resampling approach to problems like this one helps clarify the problem. Because there are no formulae to fall back upon, you are forced to think hard about how best to proceed. Foregoing these crutches may make the problem at hand seem confusing and difficult, which is sometimes distressing. But in the long run it is also the better way, because it forces you to come to terms with the subtle nature of such problems rather than sweeping these subtle difficulties under the carpet. You will then be in a better position to choose a step-by-step logical procedure which fits the circumstances. To repeat, in the absence of black-box computer programs and cryptic tables, the resampling approach forces you to directly address the problem at hand. Then, instead of asking "Which formula should I use?" students ask such questions as "Why is something `significant' if it occurs 4% of the time by chance, yet not `significant' if a random process produces it 8% of the time?" MAKING THE PROCEDURE MORE PRECISE Let's get a bit more precise and systematic. Let us define resampling to include problems in inferential statistics as well as problems in probability, with this "operational definition": With the entire set of data you have in hand, or with the given data-generating mechanism (such as a die) that is a model of the process you wish to understand, produce new samples of simulated data, and examine the results of those samples. That's it in a nutshell. In some cases, it may also be appropriate to amplify this procedure with additional assumptions that you deem appropriate. Problems in pure probability may at first seem different in nature than problems in statistical inference. But the same logic as stated in the definition above applies to both varieties of problems. The only difference is that in probability problems the "model" is known in advance -- say, the model implicit in a deck of cards plus a game's rules for dealing and counting the results -- rather than the model being assumed to be best estimated by the observed data, as in resampling statistics. Many problems in probability -- all of them, we conjecture - have a corresponding flip-side "shadow" or "dual" problem in statistics, and vice versa; the basketball case above is an example. THE GENERAL PROCEDURE The steps in solving the particular problems above have been chosen to fit the specific facts. We can also describe the steps in a more general fashion. The generalized procedure simulates what we are doing when we estimate a probability using resampling problem-solving operations. Step A. Construct a simulated "universe" of random numbers or cards or dice or another randomizing mechanism whose composition is similar to the universe whose behavior we wish to describe and investigate. The term "universe" refers to the system that is relevant for a single simple event. For example: a) A coin with two sides, or two sets of random numbers "1- 5" and "6-0", simulates the system that produces a single male or female birth, when we are estimating the probability of three girls in the first four children. Notice that in this universe the probability of a girl remains the same from trial event to trial event -- that is, the trials are independent -- demonstrating a universe from which we sample without replacement. b) An urn containing a hundred balls, 47 red and 53 black, simulates the system that produces 47 baskets out of 100 shots. Hard thinking is required in order to determine the appropriate "real" universe whose properties interest you. Step(s) B. Specify the procedure that produces a pseudo- sample which simulates the real-life sample in which we are interested. That is, you must specify the procedural rules by which the sample is drawn from the simulated universe. These rules must correspond to the behavior of the real universe in which you are interested. To put it another way, the simulation procedure must produce simple experimental events with the same probabilities that the simple events have in the real world. For example: a) In the case of three daughters in four children, you can draw a card and then replace it if you are using a deck of red and black cards. Or if you are using a random-numbers table, the random numbers automatically simulate replacement. Just as the chances of having a boy or a girl do not change depending on the sex of the preceding child, so we want to ensure through replacement that the chances do not change each time we choose from the deck of cards. b) In the case of Magic Johnson's shooting, the procedure is to consider the numbers 1-47 as "baskets", and 48-100 as "misses". Recording the outcome of the sampling must be indicated as part of this step, e.g. "record `yes' if girl or basket, `no' if a boy or a miss. Step(s) C. If several simple events must be combined into a composite event, and if the composite event was not described in the procedure in step B, describe it now. For example: a) For the three girls in four children, the procedure for each simple event of a single birth was described in step B. Now we must specify repeating the simple event four times, and determine whether the outcome is or is not three girls. b) In the case of Magic Johnson's ten shots, we must draw ten numbers to make up a sample of shots, and examine whether there are 8 or more misses. Recording of "three or more girls" or "two or less girls", and "8 or more misses" or "7 or fewer", is part of this step. This record indicates the results of all the trials and is the basis for a tabulation of the final result. Step(s) D. Calculate from the tabulation of outcomes of the resampling trials. For example: the proportion of a) "yes" or "no", or b) "8 or more" or "7 or fewer", estimates the likelihood we wish to estimate in step C. There is indeed more than one way to skin a cat (ugh!). And there is always more than one way to correctly estimate a given probability. Therefore, when reading through the list of steps used to estimate a given probability, please keep in mind that a particular list is not sacred or unique; other sets of steps will also do the trick. Let's consider an extended example, my study in the 1960s of the price of liquor in the sixteen "monopoly" states (where the state government owns the retail liquor stores) compared to the twenty-six states in which retail liquor stores are privately owned. (Some states were omitted for technical reasons.) This problem in " statistical hypothesis testing" would conventionally be handled with Student's t-test, but with much less theoretical justification than the resampling method possesses here. These are the representative 1961 prices of a fifth of Seagram 7 Crown whiskey in the two sets of states: 16 monopoly states: $4.65, $4.55, $4.11, $4.15, $4.20, $4.55, $3.80, $4.00, $4.19, $4.75, $4.74, $4.50, $4.10, $4.00, $5.05, $4.20 26 private-ownership states: $4.82, $5.29, $4.89, $4.95, $4.55, $4.90, $5.25, $5.30, $4.29, $4.85, $4.54, $4.75, $4.85, $4.85, $4.50, $4.75, $4.79, $4.85, $4.79, $4.95, $4.95, $4.75, $5.20, $5.10, $4.80, $4.29. A social-scientific study properly begins with a general question about the nature of the social world such as: Does state monopoly affect prices? The scientist then must transform this question into a form that s/he can study scientifically. In this case, the question was translated into a comparison of these two sets of data for a single brand as collected from a trade publication. If the answer is not completely obvious from causal inspection of the data because of variation within the two samples - as in the case here, where the two samples overlap - the researcher may turn to inferential statistics help. The first step in using probability and statistics is to translate the scientific question into a statistical question. Once you know exactly which prob-stats question you want to ask - - that is, exactly which probability you want to determine -- the rest of the work is relatively easy. The stage at which you are most likely to make mistakes is in stating the question you want to answer in probabilistic terms. Though this step is difficult, it involves no mathematics. Rather, this step requires only hard, clear thinking. You cannot beg off by saying "I have no brain for math!" The need is for a brain to do clear thinking, rather than a brain for mathematics. People using conventional methods avoid this hard thinking by simply grabbing the formula for some test without understanding why they choose that test. But resampling pushes you to do this thinking explicitly. The scientific question here is whether the prices in the two sets of states are systematically different. In statistical terms, we wish to "test the hypothesis" that there is a "true" difference between the groups of states based on their mode of liquor distribution - that is, a difference that is not just the result of happenstance -- or whether the observed differences might well have occurred by chance. In other words, we are interested only in whether the two sub-groups of states are "truly" different in their liquor prices, or whether the difference we observe is likely to have been produced by chance variability. The resampling method proceeds as follows: We consider that the entire "universe" of possible prices consists of the set of events that have been observed, because that is all the information that is available about the universe. We therefore write each of the forty-two observed state prices on a separate card, and shuffle the cards together; the deck now simulates a situation in which each state has the same chance as any other state of being dealt into a given pile. We can now examine, on the "null hypothesis" assumption that the two groups of states do not really reflect different price-setting mechanisms but rather differ only by chance, how often that universe produces by chance groups with results as different as those we actually observed in 1961. (In this case, unlike many others, the states constitute the entire universe we are interested in, rather than being a sample taken from some larger universe, as is the case when one does a biological experiment or surveys a small sample draws from the entire U. S. population, say.) From the simulated universe we repeatedly deal groups of 16 and 26 cards without replacing the cards, to represent hypothetical monopoly-state and private-state samples.) We sample without replacement (and hence for convenience we need only look at the 16 state set, since, once it is set, the average of the remaining 26 is also fixed) because there are only 42 actual states for which data is available, and hence we are not making inferences to a larger, infinite universe. Instead, we have the entire universe at hand. The probability that prices in the monopoly states are "really" lower than those in the private-enterprise states may be estimated by the proportion of times that the sum (or average) of those randomly-chosen sixteen prices from the simulated universe is less than (or equal to) the sum (or average) of the actual sixteen prices. If we were often to obtain a difference between the randomly-chose groups equal to or greater than that actually observed in 1961, we would conclude that the observed difference could well be due to chance variation. This logic may not be immediately obvious to the newcomer to statistics. It is fairly subtle, and requires a bit of practice, even with the resampling method to bring it to the fore. But once you understand this way of thinking, you will have reached the heart of inferential statistics. The steps again: Step A. Write each of the 42 prices on a card and shuffle. Steps B and C (combined in this case): Deal the cards into groups of 16 and 26 cards. Then calculate the mean price difference between the groups, and compare the experimental-trial difference to the observed mean difference of $4.84 - $4.35 = $.49; if it is as great or greater than $.49, write "yes", otherwise "no". Step D. Repeat step B-C a hundred or a thousand times. Calculate the proportion "yes", which estimates the probability we seek. The estimate -- not even once in 10,000 trials (see Figure 3) -- shows that it would be very unlikely that two groups with mean prices as different as were observed would happen by chance from the universe of 42 observed prices. So we "reject the null hypothesis" and instead accept the proposition that the type of liquor distribution system influences the prices that consumers pay. Figure 3 Under the supervision of Kenneth Travers and me at the University of Illinois during the early 1970s, PhD candidates Carolyn Shevokas and David Atkinson studied how well students learned resampling, working with experimental and control groups of junior college and four-year college students. Both found that with resampling methods -- even without the help of computer simulation -- students produce a larger number of correct answers to numerical problems than do students taught with conventional methods. Furthermore, attitude tests as well as teacher evaluations showed that students enjoy the subject much more, and are much more enthusiastic about it than conventional methods. It is an exciting experience to watch graduate engineers or high-school boys and girls as young as 7th grade re-invent from scratch the resampling substitutes for the conventional tests that drive college students into confusion and despair. Within six or nine hours of instruction students are generally able to handle problems usually dealt with only in advanced university courses. The computer-intensive resampling method also provides a painless and attractive introduction to the use of computers. And it can increase teacher productivity in the school and university systems while giving students real hands-on practice. Monte Carlo methods have long been used to teach conventional methods. Resampling has nothing to do with the teaching of conventional "parametric" methods, however. Rather, resampling is an entirely different method, and one of its strengths is that it does not depend upon the assumption that the data resemble the "Normal" distribution. Resampling is the method of choice for dealing with a wide variety of everyday statistical problems -- perhaps most of them. To repeat, the purpose of resampling is not to teach conventional statistics. Rather, resampling breaks completely with the conventional thinking that dominated the field until the past decade, rather than being a supplement to it or an aid to teaching it. For those in academia and business who may use statistics in their work but who will never study conventional analytic methods to the point of practical mastery -- that is, almost all -- resampling is a functional and easily-learned alternative. But resampling is not intended to displace analytic methods for those who would be mathematical statisticians. For them, resampling can help to understand analytic methods better. And it may be especially useful for the introduction to statistics of mathematically-disadvantaged students. (The method is in no way intellectually inferior to analytic methods, however; it is logically satisfactory as well as intuitively compelling.) Though we and the mathematical statisticians who have developed the bootstrap element in resampling (following Efron's work in the 1970s) have an identical intellectual foundation, they and we are pointed in different directions. They see their work as intended mainly for complex and difficult problems; we view resampling as a tool for all (or almost all) tasks in prob- stats. Our interest is in providing a powerful tool that researchers and decision-makers rather than statisticians can use with small chance of error and with total understanding of the process. Like all innovations, resampling has encountered massive resistance. The resistance has largely been conquered with respect to mathematical statistics and advanced applications. But instruction in the use of resampling at an introductory level, intended for simple as well as complex problems, still faces a mix of apathy and hostility. CONCLUSION Estimating probabilities with conventional mathematical methods is often so complex that the process scares many people. And properly so, because the difficulties lead to frequent errors. The statistical profession has long expressed grave concern about the widespread use of conventional tests whose foundations are poorly understood. The recent ready availability of statistical computer packages that can easily perform these tests with a single command, irrespective of whether the user understands what is going on or whether the test is appropriate, has exacerbated this problem. Probabilistic analysis is crucial, however. Judgments about whether to allow a new medicine on the market, or whether to re- adjust a screw machine, require more than eyeballing the data to assess chance variability. But until now, the practice and teaching of probabilistic statistics, with its abstruse structure of mathematical formulas cum tables of values based on restrictive assumptions concerning data distributions -- all of which separate the user from the actual data or physical process under consideration -- have not kept pace with recent developments in the practice and teaching of descriptive statistics. Beneath every formal statistical procedure there lies a physical process. Resampling methods allow one to work directly with the underlying physical model by simulating it. The term "resampling" refers to the use of the given data, or a data generating mechanism such as a die, to produce new samples, the results of which can then be examined. The resampling method enables people to obtain the benefits of statistics and probability theory without the shortcomings of conventional methods, because it is free of mathematical formulas and restrictive assumptions and is easy to understand and use, especially in conjunction with the computer language and program RESAMPLING STATS. chance 0-191 statwork December 19, 1990 [BOX1] THE RESAMPLING STATS LANGUAGE AND PROGRAM COMPARED TO BASIC AND OTHER LANGUAGES The computer language and program RESAMPLING STATS enable the user to perform experimental trials in the simplest, as well as more complex, Monte Carlo simulations of problems in probability and statistics. Most of the twenty or so commands in RESAMPLING STATS mimic the operations one would make in conducting such trials with dice or cards; for example, SHUFFLE randomizes a set of elements. (The rest of the commands are such as IF, END, and PRINT.) This correspondence between the computer operations and the physical operations that one would undertake in a simulation with an urn or playing cards or whatever, which in turn correspond to the physical elements in the real situation being modeled, greatly helps the user to understand exactly what needs to be done with the computer to arrive at a sound answer. RESAMPLING STATS employs a fundamentally different logic than do standard programming languages such as BASIC and PASCAL. (APL is the only language with a similar logic.) Standard languages imitate mathematical operations by making a variable - a single number at a time - the unit that is worked with. In contrast, RESAMPLING STATS works with a collection of numbers - a vector. This enables each operation to be completed in one pass, whereas in other languages there must be repeat loops until each element in the vector is processed. Furthermore, whereas other languages name the variable, RESAMPLING STATS names locations, and moves otherwise-nameless collections from location to location. In computer logic this may not be a meaningful distinction. But it is as much a working distinction as between a) a set of instructions that tell how to process tourist group 37 - first show them where the bar is, have their suitcases put away, and get them onto the bus, and b) instructions that tell what to do with whomever is in hotel A on January 1 and move them to hotel B, then process whomever is in hotel B on January 2 and then move them to hotel C, and so on. RESAMPLING STATS programs are much shorter and clearer than BASIC programs. Typically, only about half as many instructions are needed. Here is an example of the same problem written in the two languages, selected for illustration because it is the very first problem in the 1987 book THE ART AND TECHNIQUES OF SIMULATION, by Mrudulla Gnanadesikan, Richard L. Scheaffer, and Jim Swift, prepared by the American Statistical Association for use in high schools. "Outcomes with a Fair Coin: What are the numbers of heads (or tails) you can expect to get if you flip a given number of coins?" Please notice that there is a statistics problem closely related to this probability problem, with the same program used to solve it. For example: "You have a device that produces (say) a sample of 15 successes in 20 attempts. How likely is it that the long-run ("true") rate for the device is 50% successes (or less)?" The BASIC program of Gnanadesikan et. al. is as follows: BASIC Program to Simulate Trials with Repeated Coin Tosses 80 INPUT "ENTER THE NUMBER OF KEY COMPONENTS";N 100 INPUT "ENTER THE NUMBER OF TRIALS";NT 120 DIM T$(NT,N),C(2*N) 140 FOR i = 1 TO NT 150 LET NH = 0 160 FOR J = 1 TO N 170 LET X = RND (1) 180 IF X < .5 THEN 220 190 T$ (I,J) = "H" 200 NH = NH + 1 210 GOTO 230 220 T$ (I,J) = "T" 230 IF J = N THEN 260 250 GOTO 270 270 NEXT J 280 C(NH + 1) = C(NH + 1) + 1 290 NEXT I 330 FOR K = 1 TO N + 1 350 NEXT K 360 END The BASIC program is written in general form and does not specify a particular number of coins and heads, as RESAMPLING STATS does. (It has been simplified by removing the many "print" statements.) Here is the RESAMPLING STATS program that does the same job, for a sample of five coins. REPEAT 100 Run a hundred simulation trials GENERATE 5 1,2 A Generate randomly a sample of five "1"s and "2"s COUNT A 1 B Count the number of "1"s in this trial sample SCORE B Z Record the results in vector Z END End the repeat loop HISTOGRAM Z Graph the results, and also produce a table of results with their relative and cumulative frequencies The results of this RESAMPLING STATS program are in Figure B1. Figure B1 Do you agree that the RESAMPLING STATS program is not only much shorter and easier to write, but also is much more obvious to your intuition? Even more important, the above program in RESAMPLING STATS language is written by the user, which leads to learning about both statistics and computers. In contrast, the BASIC program given by Gnanadesikan et. al. is pre-written by the authors, and all the user does is fill in the parameters. The students therefore do not learn what is necessary to develop an abstract model of the real-life situation, or write a computer program to simulate that model, both of which are crucial steps in the learning process. [END OF BOX] [BOX 2] THE PRO'S AND CON'S OF RESAMPLING 1) Does Resampling Produce Correct Estimates? If one does not make enough experimental trials with the resampling method, of course, the answer arrived at may not be sufficiently exact. For example, only ten experimental bridge hands might well produce far too high or too low an estimate of the probability of five or more spades. But a reasonably large number of experimental bridge hands should arrive at an answer which is close enough for any purpose. There are also some statistical situations in which resampling yields poorer estimates about the unknown population than does the conventional parametric method, usually "bootstrap" confidence-interval estimates made from small samples, especially yes-or-no data. But on the whole, resampling methods yield "unbiased" estimates, and not less often than do conventional methods. Perhaps most important, the user is more like to arrive at sound answers with resampling because s/he can understand what s/he is doing, instead of grabbing the wrong formula in error. 2. Do Students Learn to Reach Sound Answers? In the 1970s, Kenneth Travers, who was responsible for secondary mathematics at the College of Education at the University of Illinois, and Simon organized systematic controlled experimental tests of the method. Carolyn Shevokas's thesis studied junior college students who had little aptitude for mathematics. She taught the resampling approach to two groups of students (one with and one without computer), and taught the conventional approach to a "control" group. She then tested the groups on problems that could be done either analytically or by resampling. Students taught with the resampling method were able to solve more than twice as many problems correctly as students who were taught the conventional approach. David Atkinson taught the resampling approach and the conventional approach to matched classes in general mathematics at a small college. The students who learned the resampling method did better on the final exam with questions about general statistical understanding. They also did much better solving actual problems, producing 73 percent more correct answers than the conventionally-taught control group. These experiments are strong evidence that students who learn the resampling method are able to solve problems better than are conventionally taught students. 3) Can Resampling Be Learned Rapidly? Students as young as junior high school, taught by a variety of instructors, and in languages other than English, have in the matter of six short hours learned how to handle problems that students taught conventionally do not learn until advanced university courses. In Simon's first university class, only a small fraction of total class time -- perhaps an eighth -- was devoted to the resampling method as compared to seven-eighths spent on the conventional method. Yet, the tested students learned to solve problems more correctly, and solved more problems, with the resampling method than with the conventional method. This suggests that resampling is learned much faster than the conventional method. In the Shevokas and Atkinson experiments the same amount of time was devoted to both methods but the resampling method achieved better results. In those experiments learning with the resampling method is at least as fast as the conventional method, and probably considerably faster. 4. Is the Resampling Method Interesting and Enjoyable? Shevokas asked her groups of students for their opinions and attitudes about the section of the course devoted to statistics and probability. The attitudes of the students who learned the resampling method were far more positive -- they found the work much more interesting and enjoyable -- than the attitudes of the students taught with the standard method. And the attitudes of the resampling students toward mathematics in general improved during the weeks of instruction while the attitudes of the students taught conventionally changed for the worse. Shevokas summed up the students' reactions as follows: "Students in the experimental (resampling) classes were much more enthusiastic during class hours than those in the control group, they responded more, made more suggestions, and seemed to be much more involved". Gideon Keren taught the resampling approach for just six hours to 14- and 15-year old high school students in Jerusalem. The students knew that they would not be tested on this material. Yet Keren reported that the students were very much interested. Between the second and third class, two students asked to join the class even though it was their free period! And as the instructor, Keren enjoyed teaching this material because the students were enjoying themselves. Atkinson's resampling students had "more favorable opinions, and more favorable changes in opinions" about mathematics generally than the conventionally-taught students, according to an attitude questionnaire. And with respect to the study of statistics in particular, the resampling students had much more positive attitudes than did the conventionally-taught students. The experiments comparing the resampling method against conventional methods show that students enjoy learning statistics and probability this way. And they don't show the panic about this subject often shown by many others. This contrasts sharply with the less positive reactions of students learning by conventional methods, even when the same teachers teach both methods in the experiment. [END OF BOX] Additional Readings Edgington, Eugene S., Randomization Tests, Marcel Dekker, N. Y. , 1980 Efron, Bradley, and Diaconis, Persi; "Computer Intensive Methods in Statistic," Scientific American, May, 1983, pp. 116-130. Noreen, Eric W., Computer Intensive Methods for Testing Hypotheses, Wiley, 1989 Simon, Julian L., Basic Research Methods in Social Science, 1969, N. Y., Random House (3rd Edition in 1985 With Paul Burstein) Simon, Julian. L., Atkinson, David. T., and Shevokas, Carolyn. "Probability and Statistics: Experimental Results of a Radically Different Teaching Method," American Mathematical Monthly, v. 83, No. 9, Nov. 1976 Simon, Julian. L., Resampling: Probability and Statistics a Radically Different Way (unpublished manuscript available from the author) FIGURE FOR BOX (B1) 40+ + * + * F + * r + * e 30+ * q + * * u + * * e + * * n + * * c 20+ * * y + * * * + * * * * + * * * + * * * * Z 10+ * * * * 1 + * * * * + * * * * + * * * * * + * * * * * * 0+----------------------------------------------------- |^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^| 0 1 2 3 4 5 Number of Heads FIGURE 2 Magic Johnson, 47% shooter 100 trials 40+ + + F + r + e 30+ q + u + e + * n + * * c 20+ * * * y + * * * + * * * * * + * * * * + * * * * Z 10+ * * * * 2 + * * * * * * + * * * * * * + * * * * * * * + * * * * * * * * 0+------------------------------------------- |^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^| 2 4 6 8 10 Misses in 10 shots FIGURE 3 Magic Johnson, 47% shooter 1000 trials 400+ + + F + r + e 300+ q + u + e + n + * * c 200+ * * * y + * * * + * * * * + * * * * + * * * * Z 100+ * * * * * 3 + * * * * * + * * * * * * + * * * * * * + * * * * * * * * 0+----------------------------------------------------- |^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^| 0 2 4 6 8 10 Misses in 10 Shots FIGURE 3 Liquor Prices 10,000 trials 1000+ + + F + r + e 750+ q + u + **** e + **** * n + ******** c 500+ ********** y + ************ + ************* * + *************** + **************** Z 250+ ***************** 4 + ******************** + ********************* + ************************* + ******************************* 0+--------------------------------------------------------------- |^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^|^^^^^^^^^| -60 -40 -20 0 20 40 60 Cents

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