The Theory That Would Not Die

by Sharon Bertsch McGrayne

  Cornfield typically rose at 5 a.m. to write and make paper-and-pencil calculations. He came up with clever approximations and computational tricks, much as Laplace had done. He visualized particularly difficult distributional functions by carving them out of a bar of soap. To collaborate with biochemists, he studied basic biology. And although he was a brilliant speaker, he never prepared a talk until the night before. He procrastinated even the day before he was scheduled to speak at 8:30 a.m. at Yale University about the contentious Salk poliomyelitis vaccine tests. “Don’t worry, Max,” he told a friend. “God will provide.”

  Cornfield was a voracious reader but did not own a TV and was blissfully unaware of popular culture. Once a biostatistician who dated Hollywood stars begged him to speed up a morning meeting: “I’ve got to get through because at 12 o’clock I have a luncheon date with Kim Novak.” Puzzled, Cornfield asked, “Kim Novak? Who’s he?”8 At the time she was Columbia Pictures’ answer to Marilyn Monroe.

  Another watershed medical study also occupied Cornfield’s attention during the 1950s. Death rates from cardiovascular disease had been rising steadily in the United States since 1900. Heart disease had been the nation’s leading cause of death since 1921, and strokes the third leading cause since 1938. Yet researchers at the midpoint of the twentieth century were as ignorant of the causes of heart disease and stroke as they had been of lung cancer.

  Understanding the causes of deaths from cardiovascular disease would require following a population for many years. A prospective study, however, was more feasible than with lung cancer because heart problems were far more common. In 1948 Cornfield helped design the Framingham Heart Study, which has since followed the health of three generations of Framingham, Massachusetts, residents.

  As one of the first important studies based on Framingham, Cornfield followed 1,329 adult male residents for a decade. Between 1948 and 1958, 92 of the group experienced myocardial infarction or angina pectoris.

  Longitudinal studies like Framingham are designed to investigate a large variety of variables, both singly and jointly, on the risk of developing a disease. Traditionally, epidemiologists studied their data by inspecting—“contemplating” was Cornfield’s word—the resulting multiple cross-classification arrays. Three risk factors, each considered at low, medium, and high levels, would produce a tidy 3x3 table of cells, but as the number of variables increased and they were considered singly and jointly, the number of cells to be contemplated quickly became impracticable. A cross-classification study with 10 risk factors at low, medium, and high levels would produce 59,049 cells. To get even 10 patients per cell, the study would need a cohort of 600,000 people, more than the population of Framingham.

  Cornfield realized he needed a “more searching form of analysis than simple inspection.”9 He would have to develop a mathematical model for summarizing the observations. He chose Bayes’ rule and used cardiovascular death rates as the prior. Framingham gave him data about two groups of people, those who had died of heart disease and those who had not. Within each group he had information about seven risk factors. Calculating Bayes’ rule, he got a posterior probability in the form of a logistic regression function, which he then used to identify the four most important risk factors for cardiovascular disease. In addition to age itself, they were cholesterol, cigarette smoking, heart abnormalities, and blood pressure.

  Bayes allowed Cornfield to reframe Framingham’s data in terms of the probability that people with particular characteristics would get heart disease. There was no critical level for cholesterol or blood pressure below which people were safe and above which they got the disease. And patients cursed with both high cholesterol and high blood pressure were 23% more at risk for heart attacks than those with low cholesterol and blood pressure rates.

  Cornfield’s identification in 1962 of the most critical risk factors for cardiovascular disease produced one of the most important public health achievements of the twentieth century: a dramatic drop in death rates from cardiovascular disease. Between 1960 and 1996 they fell 60%, preventing 621,000 fatalities. His report also showed researchers how to use Bayes’ rule to analyze several risk factors at a time; his multiple logistic risk function has been called one of epidemiology’s greatest methodologies.

  To measure the efficacy of a particular therapy, Cornfield used an early multicenter trial at NIH to introduce another Bayesian concept—Harold Jeffreys’s relative betting odds. Now known as the Bayes’ Factor, it is the probability of the observed data under one hypothesis divided by its probability under another hypothesis.

  When Cornfield worked with researchers who used mice to screen for anticancer drugs, the rigidity of frequentist methods struck him like a blow from behind. According to their rules, even if their initial test results disproved their hypothesis, they had to take six more observations before stopping the experiment. Frequentist methods also banned switching a patient to better treatment before a clinical trial was finished. Frequentist experimenters could not monitor interim results during the clinical trial, examine treatment effects on subgroups of patients, or follow leads from the data with further unplanned analyses. When Cornfield discovered that Bayesian methods would let him reject some hypotheses after only two solidly adverse observations, he was converted. He had started out using Bayes’ theorem as an enabling tool to solve a particular problem, the way it had been used for cryptography, submarine hunting, and artillery fire during the Second World War. But now he was moving gradually toward making Bayes’ theorem the foundation of a broad philosophy for handling information and uncertainties. As he began thinking about Bayes as a philosophy rather than just a tool, he became part of the profound conversion that Jeffreys, Savage, Lindley, and others were also going through in the 1950s and 1960s. While Fisher considered a hypothesis significant if it was unlikely to have occurred by chance, Cornfield declared airily, “If maintenance of [Fisher’s] significance level interferes with the release of interim results, all I can say is so much the worse for the significance level.”10

  Interestingly, most of NIH’s other statisticians failed to follow their leader into Bayesian fields. Cornfield published scientifically important papers about Bayesian inference in mainstream statistics journals. Nevertheless, when he included Bayesian methods in some of the trials he worked on, their main conclusions were based on frequentism. It would be another 30 years before NIH started using Bayes for clinical trials. Savage thought that many researchers were content to reap the benefits of Bayes’ theorem without embracing the method.

  Cornfield declared cheerily, however, “Bayes’ theorem has come back from the cemetery to which it has been consigned.”11

  In 1967 Cornfield retired from NIH and moved later to George Washington University, where he chaired the statistics department and developed Bayes’ rule into a full-scale logical mathematical approach. In one paper he proved to the satisfaction of many Bayesians that, according to frequency rules, any statistical procedure that does not stem from a prior can be improved upon.

  Despite his Bayesian conversion, Cornfield was in great demand as a consultant. He advised the U.S. Army on experimental design; the investigating committee critiquing the best-selling Kinsey Report on female sexuality; the U.S. Department of Justice on sampling voting records to reveal bias against black voters; and the State of Pennsylvania after the Three Mile Island nuclear power plant incident.

  In 1974 the Bayesian biostatistician with a bachelor’s degree in history became president of the American Statistical Association. In his presidential address, the man who used humor and good cheer to reassure physicians about randomized trials, who gave epidemiologists some of their most important methodologies, and who established causes of both lung cancer and heart attacks, asked, “Why should any person of spirit, of ambition, of high intellectual standards, take any pride or receive any real stimulation and satisfaction from serving an auxiliary role [as a statistician] on someone else’s problem?” Smiling at his own question, Cor
nfield continued, “No one has ever claimed that statistics was the queen of the sciences. . . . The best alternative that has occurred to me is ‘bedfellow.’ Statistics—bedfellow of the sciences—may not be the banner under which we would choose to march in the next academic procession, but it is as close to the mark as I can come.”12

  When Cornfield was diagnosed with pancreatic cancer in 1979, he knew as well as anyone at NIH the disease’s appalling six-month survival rate. Nonetheless, he was determined to continue living to the fullest. Despite serious postoperative complications, his humor remained intact. A friend told him, “Jerry, I’m so glad to see you.” Smiling, Cornfield replied, “That’s nothing compared to how happy I am to be able to see you.”13 As he was dying he said to his two daughters, “You spend your whole life practicing your humor for the times when you really need it.”14

  9.

  there’s always a first time

  Bayes’ military successes were still Cold War secrets when Jimmie Savage visited the glamorous new RAND Corporation in the summer of 1957 and encouraged two young men to calculate a life-and-death problem: the probability that a thermonuclear bomb might explode by mistake.

  RAND was the quintessential Cold War think tank. Gen. Curtis E. LeMay, the commander of the Strategic Air Command (SAC), had helped start it in Santa Monica, California, 10 years earlier as “a gimmick” to cajole top scientists into applying operations research to long-range air warfare.1 But RAND, an acronym for Research ANd Development, considered itself a “university without students” and its 1,000-odd employees “defense intellectuals.” Their mission was to use mathematics, statistics, and computers to solve military problems, pioneer decision making under conditions of uncertainty, and save the United States from Soviet attack. The U.S. Air Force, which funded RAND, gave its researchers carte blanche to choose the problems they wanted to investigate. But since President Eisenhower’s “New Look” military policy depended on early nuclear bombing (“massive retaliation”) as the cheapest way to respond to a Soviet attack, RAND’s top issues were nuclear strategy, surviving nuclear attack, and response options. Because SAC bombers had a monopoly on transporting America’s nuclear arsenal and General LeMay sat at the pinnacle of the world’s military might, RAND’s voice was often influential.

  By the time Savage visited Santa Monica that summer RAND reports had already challenged some of SAC’s sacred cows. To drop nuclear weapons on Soviet targets, macho air force pilots wanted to fly the new B-52 Strato-fortress jets; RAND recommended fleets of cheaper conventional planes. RAND had also described SAC’s overseas bases for manned bombers as sitting ducks for Soviet attack. A year after Savage’s visit, RAND would challenge Cold War dogma by arguing that victors typically fare better with negotiated settlements than with unconditional surrenders. RAND would even urge counterbalancing LeMay’s B-52s with the navy’s submarine-based missiles. In retaliation, SAC would almost break off relations with RAND on several occasions between Savage’s visit in 1957 and 1961.

  Circulating gregariously among RAND researchers that summer, Savage met Fred Charles Iklé, a young Swiss-born demographer who had studied the sociological effects of nuclear bombing on urban populations. At 33, Iklé was seven years younger than Savage and had earned a Ph.D. in 1950 from the University of Chicago, where Savage was teaching. Seeking a wide-open field that no one else at RAND was studying, Iklé chose nuclear catastrophes that an Anglo-American nuclear arsenal would not deter: those caused by accident or by someone mentally deranged. Referring to massive retaliation a few years later, Iklé would declare, “Our method of preventing nuclear war rests on a form of warfare universally condemned since the Dark Ages—the mass killing of hostages.”2 With SAC poised to expand its bomb-bearing flights, Iklé and Savage talked about assessing their impact on nuclear accidents. Eventually the issue came around to the question, what was the probability of an accidental H-bomb explosion?

  After a summer of conversations, Savage was preparing to return to academia when Albert Madansky, a 23-year-old Ph.D. graduate of Savage’s statistics department, arrived at RAND. Madansky had financed his graduate studies by working part-time for Arthur Bailey, the Bayesian theorist in the insurance industry. Until Bailey’s death, he had considered an actuarial career. Savage, who had published his book Foundations of Statistics but had not yet embraced Bayes’ rule, talked over the H-bomb problem with Madansky without considering it in Bayesian terms. As he left Santa Monica, he handed over the H-bomb study to Madansky but left the young man to do as he saw fit. Madansky would come up with a Bayesian approach on his own.

  Because RAND’s report on the project would eventually be classified, Madansky could not talk about his work for 41 years. But when Savage returned to Chicago he lectured openly about the fundamental statistical issues involved. Bayes’ rule was emerging in fits and starts from the secrecy of the Second World War and the Cold War.

  The H-bomb problem facing Madansky was politically and statistically tricky. No atomic or hydrogen weapon had ever exploded accidentally. In the 12 years since the United States had dropped atomic bombs on Japan in August 1945, nuclear bombs had been detonated, but always deliberately, as part of a weapons test. Barring accidents, the nation’s leaders believed that their stocks of nuclear weapons deterred all chance of thermonuclear war—and that accidents could not occur in the future because none had occurred in the past. Nonetheless, the question remained: could the impossible happen?

  According to more than a century of conventional statistics, the impossible could not be calculated. Jakob Bernoulli had decreed in 1713 that highly improbable events do not happen. David Hume agreed, arguing that because the sun had risen thousands of times in the past it would continue to do so. It was Thomas Bayes’ friend and editor, Richard Price, who took the contrary view that the highly improbable could still occur. During the nineteenth and early twentieth centuries, Antoine-Augustin Cournot concluded that the probability of a physically impossible event was infinitely small and thus the event would never occur. Andrei Kolmogorov slightly refined “never” by saying that if the probability of an event is very small, we can be practically certain that the event will not happen in the very next trial.

  Fisher was no help either. He argued that probability is a simple relative frequency in an infinitely large population; until a nuclear bomb accident occurred, he had no way of judging its future probability. Mercifully, Madansky did not have an infinitely large population of H-bomb accidents, and further experimentation was out of the question. Fisher’s approach left him with the banal observation that zero accidents had occurred and that the probability of a future accident was also zero.

  Madansky concluded, “As long as you are set that the probability is going to be zero, then nothing’s going to change your mind. If you have decided that the sun rises each morning because it has always done so in the past, nothing is going to change your mind except one morning when the sun fails to appear.”3

  He did not buy the argument that an accident could never occur simply because none had happened in the past. First, the political and military establishment’s assumption that a well-stocked American arsenal of nuclear weapons would deter war rested on increasingly shaky grounds. In the six years between 1949 and 1955 the Soviets had exploded their first atomic bomb, the United States had detonated the world’s first hydrogen bomb, and Britain had tested both an atomic and a hydrogen bomb. The USSR had launched the first man-made satellite into orbit around Earth in 1957. Meanwhile, the United States trained countries in the North Atlantic Treaty Organization (NATO) to fire nuclear weapons and supplied Britain, Italy, and Turkey with nuclear missiles. By the time the Anglo-American Agreement for Cooperation on the Uses of Atomic Energy for Mutual Defense Purposes was signed in 1958, all hope of preventing the spread of nuclear weapons had evaporated. France tested its first atomic bomb in 1960.

  In addition to the rapid expansion of nuclear weapons, Madansky had 16 top-secret reasons for doubting that the proba
bility of a future accident was zero. A classified list detailed 16 of “the more dramatic incidents” involving nuclear weapons between 1950 and 1958.4 They included accidental drops, jettisons, aircraft crashes, and testing errors. Incidents had occurred off British Columbia and in California, New Mexico, Ohio, Florida, Georgia, South Carolina, and overseas. RAND’s list omitted accidents that had not attracted public attention.

  An atomic or hydrogen bomb consists of a small capsule, or “pit,” of uranium or plutonium inside a case covered with powerful conventional explosives. Only if these high explosives blow up at the same instant can the uranium or plutonium capsule be sufficiently compressed on all sides to trigger a nuclear blast. In a few cases these conventional explosives had detonated, generally on impact in a plane crash. Because the capsule of nuclear material had not been installed inside the weapons, however, there had been no nuclear accidents. That fact had convinced SAC that its procedures were sound and that no nuclear accident would occur.

  Still, scores of people had died when the high explosives in unarmed nuclear weapons blew up. The year 1950 was a banner year for accidents. On April 11, 1950, 13 people died near Kirtland Air Force Base, outside Albuquerque, New Mexico, when a B-29 crashed into a mountain; flames from the high explosives were visible for 15 miles. On July 13, 16 were killed when a B-50 nose-dived near Lebanon, Ohio. Nineteen people, including Gen. Robert F. Travis, died when a B-29 with mechanical problems crash-landed in California on August 5, injuring 60 people in a nearby trailer camp. Also that year, two bombs without nuclear capsules were jettisoned and abandoned in deep water in the Pacific Ocean off British Columbia and in an unnamed ocean outside the United States.