An Accidental Statistician

by George E P Box

  My office was downstairs in Patterson Hall, which was where most of the graduate students were. Some of their lectures were somewhat obscure, so they would come to me for help and we became friends. Halfway through my year in Raleigh, I had to move and I asked Sid Wiener, one of the graduate students, if he could recommend a mover. (Sid always wore a cigar and was from Brooklyn.) He said, “Ya don't want to go to a movah, Professah. We'll move ya, we'll move ya.” And he organized a crew of fellow students and rented a truck, and all went well until we got to the piano. Unfortunately my new living quarters were upstairs and the stairs contained a right-hand turn. Try as they would, the students could not get my piano up the stairs and around the corner. Their gallant attempts reminded me of the famous picture of the three marines with the American flag, and they were reluctant to admit, especially to a visiting Englishman, that it couldn't be done. In the end, we took the piano to my office in the basement of Patterson Hall at the university.

  I remembered that when my father had sometimes felt a bit down, he'd say, “Let's go and have a tune up!” He particularly liked to play marches. I did too. So one particular day without thinking too much about it, and forgetting I was in the South, I played “Marching through Georgia.” A janitor quickly opened my door and said firmly, “We don't play that song down here.”

  It soon became clear that to get around the campus at Raleigh, I needed some kind of conveyance. When I suggested buying a bicycle, the graduate students shook their heads and told me that, like everyone else in the United States, what I needed was a car (Figure 5.3). Among them was a student from Egypt called Alex Kahlil. I told Alex I was planning to buy a second-hand car. He said that he would help me but that he was supposed to take an exam that afternoon. He suggested that if I called his professor and told him I needed Alex's help, perhaps the professor would postpone the exam and Alex could help me buy the car. The exam was postponed, and under Alex's direction, we found a car that we thought might be okay at, let's call him, Dealer A. Alex asked to take it for a trial run; however, on the way back he took it to Dealer B, who also wanted to sell us a used car. But Alex said, “We've got this car to try from Dealer A. Would you check it out for us?” Dealer B did so and told us a number of things that might be wrong with it. Then we went back to Dealer A, and we told him about these possible defects and suggested that he might lower the price, and so on and so on. I think three dealers were eventually involved in this way. After an afternoon of haggling, we got a pretty good car at a bargain price.

  Figure 5.3 Jessie and our first English car (a Hillman).

  The graduate students at Raleigh took some of their courses at Chapel Hill, and they took it in turns driving there in one of the university cars. A story that went the rounds was about the time when Alex drove the car. His driving was unusual in that under his guidance, the car tended to oscillate somewhat from side to side. After a bit, a police car started to follow; I think the policeman might have suspected, incorrectly, that the driver was drunk. He told Alex to stop and asked for his driver's license, and sniffed around a bit, but everything seemed to be in order. He said to Alex, “The reason I stopped you was that you were veering from side to side.” Alex said, “I'm sorry, officer, but as you can see, I'm not a native of this country. I'm Egyptian.” The police officer said, “Yes.” Alex said, “Well in Egypt, of course, you learn to drive a camel, and sitting on the back of a camel you have to allow for how they sway from side to side.” The policeman glared at him, but Alex kept a straight face, so finally the officer said, “Does anyone else in this car have a driver's license?” Someone else showed his license, and the officer said, “You don't come from Egypt, do you? Okay, then change seats with him.”

  About 37 years later, I was in Egypt at a meeting of the International Statistical Institute listening to a lecture in a darkened room and someone came and sat beside me and whispered, “Hello, George.” It was Alex. He told me he had given up statistics and was growing oranges!

  In North Carolina, I took some driving lessons from a woman instructor who told me I must be “slow with the hands and quick with the feet.” Later I had to have a licensed driver with me, and Stu Hunter bravely undertook the task. I think Stu became rather fed up with this, because before very long, he suggested that I take the driver's test. At that time in North Carolina, the police did the testing. I wasn't much of a driver, and on the appointed day, I was very nervous and did everything wrong. Stu witnessed my performance during the test and expressed his surprise when the policeman started to fill out a form saying I'd passed. Somewhat defensively the officer said to Stu, “Well he didn't actually break the law.”

  On another occasion, when I was still a novice driver, I foolishly shot out onto a main road without stopping. A motorist who had the right of way on the main road narrowly missed me and came to a halt on the opposite side of the road. He rolled down his window. I was expecting what would have happened in England at this point: a lot of bad language. But all the driver said was, “Well, hi!”

  Later, I got a flat tire one Sunday morning. A car came along with a man and his family, who were in their best clothes, and I suspect they were on their way to church. The car stopped, and the man asked what seemed to be the trouble. I was embarrassed to tell him I didn't know how to change a tire, and I asked if he could perhaps tell me how to do it. He said, “I'll change the tire for you. You watch me and you'll know what to do next time.” He got down on his hands and knees in his Sunday suit and changed my tire. I was so overwhelmed by his kindness that I didn't know how to thank him. He said, “Well you know we're all here to help each other,” and he drove away. On my return to England, when I sometimes encountered what I thought were unjust criticisms of Americans, I told these stories.

  Some years later, I needed to change a tire on my car. My son Harry, who was then five years old, enthusiastically helped me change the tire, and when it was finished, he said, “Now let's change the other three.”

  My year in Raleigh gave me a view of American social habits that were unfamiliar to me. Initially, my only obligation was to help Stu Hunter get his Ph.D., and during the year, I did the same thing for another student, Sigurd Anderson. Sig had recently married Sally, and Stu was married to Tady, and we had a fine time together. It started off with what my American friends called a “Round Robin Dinner” in which a number of graduate students took part. The idea was that the first course was prepared and served at somebody's house, the next course at someone else's house, and so on. It turned out that Jessie and I were responsible for the pre-dinner drinks. We had no experience in this. We knew that people in America drank cocktails because we'd seen it in the movies, but we knew nothing about mixing drinks except that they had to be very strong. What we served must have been close to neat gin, I think, because one of the ladies passed out before the first course.

  Toward the end of my stay in Raleigh, Miss Cox told me we should take four or five weeks off to see something of the United States. We began by attending a conference in Canada, where a number of statisticians took the Thousand Islands boat trip. We needed gas at one point and stopped on the U.S. side. Immediately someone from U.S. immigration jumped on board and started asking us where we were from. Sigurd Andersen replied, “Denmark.” I said, “England,” and so it went. Everyone on the boat was from a different country. Finally, when Wassily Hoeffding said, in his very thick accent, “Russ-i-a,” the man shouted, “Okay, that's it! Get this boat out of here!” So went the Cold War. (I can't remember what we did for gas.)

  I remembered that I had two cousins living somewhere near Chicago whom I had never met. They were the son and daughter of my Uncle Pelham. I had no idea of how I might find them so I wrote to the mayor of Chicago. Very shortly I received a reply as follows:

  When Jessie and I drove from North Carolina to Chicago, my driving was far from perfect, especially in congested cities. I ended up in the Chicago Loop at rush hour. I wanted to turn right, but whenever the light changed, a mass
of people started crossing and I didn't see how I could turn without running them all over. Finally a policeman put his head in my window and said, “Say, buster, don't we have any colors you like?”

  My cousin Evelyn was married to a lawyer who was on the Chicago city council. He was very kind and took Jessie and me to a council meeting to see what it was like. My other cousin, Lester, worked in the car industry about 20 miles north of Chicago. He had a very lively family, and while we were visiting him, we went out to go dancing at about 11 o'clock in the evening, which was rather late for us.

  While in Chicago we visited an English friend, Professor Brownlee, at the University of Chicago. He was tremendously enthusiastic about the beauties of the United States, and when we said we weren't sure where to go on our trip, he invited us over to a spectacular slide show in his office. He had hundreds of slides that he had taken of his many travels, and he gave us an illustrated commentary with a map. He was particularly intrigued, and so were we, by the meandering of the Colorado River, the Goosenecks of the San Juan River in Utah, the Black Canyon of Gunnison National Park in Colorado, the Painted Desert in Northern Arizona, and of course, the Grand Canyon. We decided to follow the route that he sketched out.

  After we left the traffic of Chicago, our journey to the West progressed smoothly until we were about 200 miles from the Grand Canyon. There, the car that Alex had so methodically helped me choose broke down. It took two days to fix, and the fix, unfortunately, was short-lived, for as we neared the Canyon, it broke down again. We hung about while it was being repaired and began to wonder whether this part of the trip was worth the bother. There was a cowboy-looking guy leaning on a post nearby, so I asked him, “This Grand Canyon—what's it like?” He thought for a minute, and then he said, “Well, I'll tell ya, it's just about the biggest hole in the ground you ever did see!” We went, and it was. It was also one of the most beautiful and awe inspiring sights we had ever seen.

  On the trip we drove through country where there were many Native Americans living in tepees. Our friend had told us that if we wanted a chance to talk with them, we must bring a gift—an apple would do. So we stopped at a tepee and someone came out, took our apple, and then disappeared into the tepee never to reemerge.

  Our journey required that we travel a long way through the desert, so we had to have a carboy full of water before we started. So much of the West was remarkable to us, but one of the most interesting sites was an Anasazi Indian village at Mesa Verde where dwellings had been cut out of the side of a cliff. These had been abandoned many centuries ago, and there were signs that the inhabitants had left in a hurry. The climate was so arid that an old woman had been found in a remarkable state of preservation after 500 years. When we went to an evening campfire show put on by the Park Service, we learned more about the culture of the Native Americans who had lived in the area. Descendents of the Anasazi danced for us and told us about the system of lookouts that had warned their people of danger—most likely to come from their enemies, the Utes. They demonstrated the call that told the cliff dwellers that “the Utes are coming!” although they allowed that this was sometimes used just to keep people on their toes.

  Jessie and I very much enjoyed our trip around the United States. One day, when we were on a narrow road in the mountains, we met a massive flock of sheep that formed an impassable barrier across the road, so while I drove the car slowly forward, Jessie sat on top of the car, banging on the metal hood and shouting loudly at the sheep so that they made way for us. The Native American sheepherder looked at me appreciatively and said, “You have good woman there!”

  We carried memories of this trip when we returned to England, this time on the French ship, the Île de France. On this occasion, we had a pleasant trip. I remember there were a number of French Air Force cadets returning home. One group amused itself by dropping puff balls on the dancers, testing their bombing skills by aiming for the rather expansive cleavages of the ladies below:

  Upon our return to England, we bought a new car—a Hillman—but we never forgot the one that Alex helped us buy.

  1 Then called North Carolina State College, part of the University of North Carolina system, which included the campus in Chapel Hill.

  2 G.E.P. Box, “The Exploration and Exploitation of Response Surfaces: Some General Considerations and Examples,” Biometrics, Vol. 10, No. 1, 1954, pp. 16–60.

  “Ah! Then yours wasn't a really good school”

  Chapter Six

  Princeton

  While I was at the University of North Carolina, I was invited to give seminars at various institutions. One of these was Princeton where I met John W. Tukey, an extremely able mathematician and statistician. In addition to his work at Princeton, John had an important job at Bell Labs where he practiced and encouraged the use of statistics. John and I respected one another, but we didn't always see eye to eye.

  Some of my early research at ICI had concerned “tests of statistical significance.” That an effect is “statistically significant” means that it is unlikely to be a result of chance. You might, for example, be testing the efficacy of a new drug and you might want to check that the difference in efficacy between this and the standard drug was not just a result of experimental error.1 Clearly questions of statistical significance must be considered because without them the scientist can, on the one hand, be “chasing red herrings” or, on the other, be missing important small differences.

  Now there are a variety of tests of significance to choose from and they all involve assumptions. In particular, we may assume that we know the form of the probability distribution of the data (the probability distribution of the “noise”). Some tests assume that this distribution is of a particular kind called the “normal” distribution and this is in fact a distribution that quite often can approximate reality. The difficulty is that although while some important statistical tests are insensitive to large departures from important assumptions like normality, others are not. In 1953, I decided to call tests that are insensitive to a particular assumption “robust” tests, and this has become a generally accepted term. It is important to understand that it is the nature of the test as well as the nature of the assumptions that affect robustness. For example, if you ask the question, “Is A bigger than B?” an assumption of normality of the distribution might not be important, but if the question is, “Is A more variable than B?” it could matter a lot. One of the problems I worked on, therefore, was how to produce tests that are robust to assumptions. A robust statistical procedure, although not necessarily “optimal” for any particular set of experimental conditions, would work well in practice over a wide, relevant range of conditions likely to be met in practice.

  Among his many abilities, John Tukey had a flare for coming up with ad hoc statistics that were “robust” to particular contingencies. Bob Hogg, and many other statisticians, did likewise, so there was a plethora of robust statistics. I preferred to achieve robustness using a Bayesian approach by introducing a suitable model that allowed efficiently for specific likely contingencies. These differences of approach led to my writing the song, “Let's Go Robust,” based on Cole Porter's “Let's do It”:

  John does it

  Hogg does it

  Every statistician that's in vogue does it

  Let's do it

  Let's go robust

  Outliers

  Bad data,

  Totally excluded by this estimator

  Let's do it

  Let's go robust

  I got this

  De-scending Psi function

  That is free from all flaws

  Moreover my function

  Is much better than yours

  A students

  C students

  Now all get robustified to B students

  Let's do it

  Let's go robust

  We had our

  Nice observationsx

  Circumcised from both ends2

  Removing the data

 
; On which our theory depends

  We tackle

  Difficult cases

  We can hit an outlier at fifty paces

  Join the movement

  Let's go robust

  Box only

  Does it with models

  In all kinds of different ways

  Claiming he learned it

  From some preacher called Bayes.

  In fame and fortune

  You can be sharing

  If you find a way to do it that's a wee bit daring

  Strike out now

  And go robust

  Looking at the problem in the light of today's technology, you can see that the basic problems of tests of significance and robustness were solved in principle at very early stages of Fisher's thinking. From the first Fisher insisted on randomization to avoid biased conclusions, but he also pointed out that randomization itself provided nonparametric significance tests. One calculated what had happened with all the values of the relevant criterion that could have happened if the randomization process had occurred differently. These tests were not used at the time because of the lack of computing power, but are now possible with the advent of fast computers.

  After I returned to England from North Carolina, I kept getting calls from John. He wanted me to leave ICI and come to Princeton to be the director of the Statistical Techniques Research Group (STRG) that was being set up there. Finally, in 1956, Jessie and I agreed to go along with our newly adopted son, Simon. John had arranged housing for us, a log cabin in the woods. When we arrived at the house for the first time, there was a man there who said, rather bluntly, “I'd like to see the money.” After we had been in Princeton for a few months, we were dining with the Hunters when Jessie announced that she had befriended “a wonderful kitty” at the cabin. As she continued her description of this animal, Stu and his wife realized that the “kitty” was in fact a skunk (a creature that doesn't exist in England). After living at the cabin for a year, we moved to university housing where the wildlife was more familiar.