An Accidental Statistician

by George E P Box

  On one occasion during the meeting, an idea occurred to me that no one else seemed to have thought about. So even though I was only a student, I put up my hand, was acknowledged by the chairman, and made my point in about three minutes with the help of a piece of chalk and a blackboard. Immediately after the meeting was over, a stranger came over to me and said, “I'm George Barnard. What are you doing this evening?” When I replied, “Nothing,” he said, “Well let's go and get a bottle of wine and have dinner together” (Figure 2.5).

  Figure 2.5 George Barnard.

  George, it turned out, was a professor at Imperial College. From that date forward, he became my friend and mentor and there began a close association that lasted until the end of George's life. For example, it wasn't long after I met George that I became confused about the theory of least squares, an important statistical technique for estimating unknown constants. I was dissatisfied with a “proof” I had been taught at University College. It seemed to me that it was not rigorous and only covered a special case. I asked George, and he said, “I'll show you how Gauss did it.” He wrote three lines of matrix algebra and that was it. It was beautiful and completely general.10 In a subsequent class, there was a complex problem in an exam that took almost all the students a long time to do, but using the Gauss proof that George had shown me, I completed the problem in five minutes.

  It took me 18 months to get my undergraduate degree in statistics with first-class honors. I spent the rest of the three years for which I was supported doing graduate work. I think there were seven students in the class the first year of instruction. Of course much had happened to all of us during World War II. Unfortunately, we had an instructor who didn't seem to notice this, and persisted in treating us like school children. She would point a finger at one of us and demand, for example, that he or she “Define an expectation.” This nonsense was put a stop to by a quiet student called Westgarth. He was an ex-major in the tank corps who had been a commander in the desert in North Africa fighting against General Rommel. She pointed to him and said, “What is a random variable?” Eyeing the instructor fixedly, Westgarth slowly put his feet on the table, leaned back, and said, “I haven't the slightest idea.” That seemed to cure her.

  Thankfully, the poor classroom technique of this particular instructor was the exception. I was most fortunate during this period to have E.S. Pearson as my professor and H.O. Hartley as my advisor. My Ph.D. thesis was titled “A General Distribution Theory for a Class of Likelihood Criteria.”

  1 The war at sea was a different story. It began with the torpedoing of the British liner, SS Athenia, soon after Britain and France declared war on Germany. We suffered huge losses of our merchant ships. We had to import a great deal of our food, and the enemy's intention was to starve us out.

  2 Noncommisioned officer.

  3 The huts were wide enough to accommodate two beds end to end with a walking space in between. You could make them as long as you liked by adding more sections. The stoves were upright with a pipe that went through the roof.

  4 Known in the U.S. Army as a SNAFU.

  5 Weekly meat rations were limited to the value of one shilling six pence—or 6p in today's terms.

  6 Instructions for American Servicemen in Britain 1942, United States War Department, 1942.

  7 Two of the books were Fisher's Statistical Methods for Experimenters and his Design of Experiments. The others were about applications by followers of Fisher. I remember Statistical Methods in Forestry and Range Management, which had an excellent account of the “method of Least Squares.” Another was about using statistical design to improve teaching methods.

  8 H. Cullumbine and G.E.P. Box, “Treatment of Lewisite Shock with Sodium Salt Solutions,” British Medical Journal, April 20, 1946, pp. 607–608; G.E.P. Box and H. Cullumbine, “The Relationship between Survival Time and Dosage with Certain Toxic Agents,” British Journal of Pharmacology, Vol. 2, 1947, pp. 27–37.

  9 The reason that the scientist at the station knew Fisher was that Fisher did not get on very well with Karl Pearson and his people at University College, where they both had laboratories, and rather than eat with them, he crossed the street and had lunch at University College Hospital.

  10 The Gauss proof showed that the squared differences between the least-square estimates and any other set of linear estimates was always greater than zero!

  “Can you answer useful questions?”

  Chapter Three

  ICI and the Statistical Methods Panel

  While I was a student at the university, my three-month summers were spent working for Imperial Chemical Industries (ICI) where I was later employed. The company was huge and was divided up into divisions located at different places in Great Britain. The divisions included dyestuffs, paints, textiles, pharmaceuticals, heavy chemicals, explosives, and so on.

  The first year I was an assistant to Mr. L. R. Connor at ICI's London headquarters, where as a vacation student, I earned four pounds a week. Mr. Connor was a lawyer and was a very precise gentleman. After I had been with him for about a month, he suddenly asked me whether ICI had been paying my four pounds a week. I said that no, I hadn't been paid yet, but I wasn't depending on it to survive. Mr. Connor, said, rather gravely, “That's all very well, but suppose you were to sue ICI?” Somehow I didn't think I'd get very far suing one of the largest companies in post-war Britain.

  At the time of my first summer, the Statistical Methods Panel, which ICI had established to coordinate the statistical work of its various divisions, had just finished writing the book, Statistical Methods in Research and Production, often called “Little Davies” after it's editor, O. L. Davies. It had been written by scientists at ICI for internal use. Someone thought that it would be a good idea to ask Lord McGowan, the company's CEO, to write a few words for the preface. Lord McGowan surprised everyone when he said that in the aftermath of WWII, it was our duty to help not just ICI but the whole of British industry. So he decided that the book should be published externally and made available to the general public. There was a bit of a flap on, however, because the authors of the book couldn't very well say, “It was good enough for ICI but not good enough for the general public.” I was asked to read through the draft and comment. My suggestions were well received, and because of this I was made a joint author of the book, and later a member of the Statistical Methods Panel.

  My second year as a vacation student was spent at the Dyestuffs Division in Blackley, near Manchester. The people there offered to make me a salaried employee during my third year as a student, with the understanding that I would join them afterward. The salary they offered was considerably more than the government grant that I had been receiving, so I agreed.

  Soon after I joined the Statistical Methods Panel, we had a meeting in London. On the first morning, Harold Kenney, the chairman of the panel, told me that very unexpectedly, our Managing Director would be attending the meeting. There was nothing on our agenda that was likely to be of interest to this VIP, so at about 8:15 a.m., with the meeting due to start at 9:00, Harold came to me, the most junior member of the panel, and asked whether I would make a presentation. Harold knew that I was working on what came to be called, “Response Surface Methods,” a way to run experiments to determine conditions that maximized the yield of a chemical process. I was put in a separate room to think, in the few minutes available, how to explain this clearly to this very intelligent but very nontechnical person. I hastily made some notes and hoped for the best. To my surprise, our visitor became quite interested in what I said and asked lots of very sensible questions. So it was a success, and Harold never forgot it. After this I could do no wrong. The other members of the panel were all very senior to me, but during meetings, Harold would always interject, “I'd like to know what George thinks about this.”

  The Statistical Methods Panel met several times a year, and because the various parts of the company were spread all over the country, we needed to find a place conducive to creativit
y that was readily accessible to all of us by train. We found such a place at Keswick, a beautiful location in the Lake District of northern England, and we met there for periods of three to six days.

  Some of the happiest years of my life were the eight that I worked at ICI. Among the things ICI made were synthetic dyestuffs, textiles, and waterproofing and mothproofing agents. Expert teams of highly skilled chemists and engineers developed and improved the complicated processes associated with these products. I quickly got myself involved with them and was able to increase the efficiency of their experiments, both on the full scale and in the lab. Typically a 1% increase in yield could produce huge profits. To help them design effective experiments, I had to know details of the processes and testing methods, so I found myself climbing up and down ladders, talking and arguing every day with technical staff and process workers, and teaching them a little about statistical design and analysis. The woman who brought tea to our desks each morning and afternoon soon tired of taking away my untouched cup, and complained to my secretary, “'E's never 'ere.”

  In 1955, a young man named Norman Draper worked with me at ICI as a summer student. He recalls riding his motorbike to the job, which paid the paltry sum of five pounds a week. He expected that the job would entail endless data entry. Instead I sent him all over to talk to scientists, get answers to questions, and discuss problems. Norm followed that summer by enrolling in the Ph.D. program at North Carolina, where he studied with Chandra Bose. In 1960, after he graduated, he came to Madison to work at the Math Research Center, and later the Statistics Department.

  I turned out a great deal of work for the company, and as a result, my boss said if I wanted to go to meetings or to lectures or anything like that, I should just go. I need not ask permission from anyone. So I went to the afternoon lectures of Professor M. S. (Maurice) Bartlett who was then at Manchester University. Professor Bartlett taught a number of courses, including one on a branch of mathematics called the theory of games and ethical decisions. A number of student athletes enrolled in this class with the mistaken idea that it would increase their prowess in their particular sport. I took his course on multivariate analysis, and his lectures were extremely clear and inspirational, particularly since he used n-dimensional geometry to illustrate the mathematics. The discussions that took place during afternoon tea were most informative as well.

  Using these ideas, George Tiao and I subsequently wrote a paper that included the key idea of finding linear combinations of nonstationary time series that are stationary, that is, the idea later called cointegration.1

  The work that I did at ICI was to help chemists and chemical engineers design and analyze experiments to improve chemical processes; some of the work was done on the laboratory scale, some in the pilot plant, and some on a full scale. Experimentation on the process itself was expensive and difficult. Lab experiments were easy, but the drawback was that you always needed to make an educated guess as to how these small-scale results might apply on the full scale, and we knew that sometimes our guesses could be wide of the mark. As a supplement to this work, therefore, in 1954 I devised a technique called “evolutionary operation,” which I communicated to the Board of Directors in a short memo.

  The objection to experimentation on the full scale is that combinations of variables would have to be tried that might disrupt the normal operation of the process and produce unsalable products. Evolutionary operation was run on the full scale but did not have this disadvantage because it used Darwin's concept of evolution and natural selection. The changes from the best known process conditions were very small, but they were repeated many times over. The idea is illustrated here for the simple case of just two reaction conditions of temperature and concentration and one response variable percentage yield.

  It is supposed in this example that the previously best known operating conditions for temperature and concentration were 300°C and 13%, respectively, labeled on point A in the left-hand diagram. In the evolutionary operation mode, the process was run for a suitable period at each of the different conditions labelled A–E. The conditions B–E are chosen to be only slightly different from A—not enough to cause problems. But the cycle was repeated a number of times and the results averaged. As soon as there was evidence that some variant was significantly better, this became the origin for a new cycle. In the illustration in the right-hand diagram, after nine cycles of the procedure, the data are starting to show that a slightly lower temperature with a slightly higher concentration would give an increased yield. The ICI board was reluctant to have this idea published but eventually allowed it, in 1957.2

  In 1998, the Institute of Electrical and Electronics Engineers (IEEE) produced an anthology of the papers and reports leading up to what became a method for computer calculation called evolutionary computation. One of the earliest of these was the one on evolutionary operation that I published. For this the IEEE awarded me the Evolutionary Computation Pioneer Award in 2000.

  At ICI, in addition to helping to improve chemical processes, a massive effort was needed to test the products thoroughly: dyestuffs, detergents' waterproofing agents, artificial leathers, and many others. How close to standard colors were our dyestuffs? How waterproof were fabrics treated with our products? How resistant to wear was our artificial leather? To obtain the answers to such questions, a sample of the material to be tested was compared with a standard that had to be matched or, in appropriate cases, exceeded. There were ingenious instruments and machines for carrying out the tests. Once more these provided a golden opportunity to employ Fisher's experimental designs. For example, in the Martindale wear-testing machine, four small pieces of fabric, one of which was the standard, were placed in holders and rubbed at a fixed pressure against abrasive emery paper. The loss in weight after a 1000 cycles of the machine was a measure of resistance to wear. Thus, three sample products could be compared with the fourth, which was the standard. There were obvious difficulties. For example, four holders were used. Were there differences due to holders? The holders were in four different positions on the machine. Were there differences due to positions? In each run of the machine, the emery paper abrasive needed to be changed. Were there differences due to different emery papers? How could we allow for such differences? The subsequent figure shows part of the replicated Hypergraeco Latin Square design with the numbers showing wear after four cycles of 1000 revolutions. This kind of design is due to Fisher, an elaboration of the Latin Square design mentioned earlier.

  Hyper-Graeco–Latin Square—Wear Testing3

  Using the design, it is possible to eliminate the effects of position, cycle, holder, and emery paper and thus to obtain very accurate comparisons with the standard. I very much enjoyed solving complex puzzles associated with such experimental designs.

  One thing I greatly missed after I left ICI were the stories and jokes that went around. The standard was high, and it seemed a new joke surfaced almost weekly. Harold Kenney, for example, was a source of good stories. Some of these were about a friend of his called Hetteridge, who had served in the trenches during the First World War and was one of the few people who seemed to have enjoyed it. He was quite blood thirsty. He had tried very hard to rejoin the Army at the beginning of the Second World War, but he was too old.

  After the German blitzkrieg that had defeated the French in a matter of weeks, the authorities in Britain were concerned about the possibility of an invasion that might perhaps use parachute troops. So the Home Guard [sometimes called “Dad's Army”] was formed. Able-bodied men who were too old to serve in the Army, Navy, and Air Force joined the Home Guard. At first they didn't have very much in the way of weapons, so they had drills with shovels and rakes temporarily substituted for guns. Hetteridge was a major in the Home Guard. He was extremely keen and became well known for the efficiency of his Home Guard unit.

  There happened to be a regiment of the regular Grenadier Guards stationed nearby, and their colonel asked Hetteridge over for dinner in the offic
ers' mess. Hetteridge arrived in his First World War uniform with two Lugars (German guns) stashed in his belt. The colonel met him at the door but pointed out that one did not take weapons into the officers' mess. So reluctantly Hetteridge hung up his belt with his guns in the hallway. The dining room had French windows overlooking a pristine lawn and flower gardens. The conversation at lunch was about the fall of France and the danger from parachutists. A lieutenant was speaking about this then novel form of warfare, and looking out onto the lawn, he said, “There could be a parachutist coming down on the lawn right there and what could we do about it?” Hetteredge felt inside his coat, produced another Lugar, and said, “I'd shoot the bastard.”

  Another Hetteridge story concerned the fact that among other things, he was a lay preacher. Unfortunately, one of Heterridge's characteristics inherited from WWI was his use of “bad language.” This kind of vocabulary does not translate easily from English to American and vice versa. For example, while the word “bugger” is fairly innocuous in American, in English, it is not. In particular the expression “bugger off,” meaning “go away!”, is not used in polite society. Now immediately after the fall of France, a large proportion of the British Army was rescued at Dunkirk and successfully brought back to England. The following Sunday, Hetteridge based his sermon from the parable of the seven lepers in which Jesus cured seven lepers of their leprosy, but only one returned to thank him. With great emotion, Hetteridge said, “We have just witnessed a miracle! The great majority of our army has been rescued in the face of the enemy. Yes, a miracle, and what are we doing? Are we on our knees thanking God? No! Things are just happening as usual! It's just like the story of the seven lepers—seven were cured of their leprosy and only one returned to thank Him. What did the other six do? Why! They just buggered off!”