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Dicing with Death: Chance, Risk and Health

Stephen Senn
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Steven R. Dunbar
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The safety and efficacy of therapeutic drugs and medical procedures is front-page news. Beyond the pharmaceutical issues, the real question is who knew what and when, and just as important, how sure were they of the information? What is enough knowledge to keep or pull drugs from the market, balancing lives and health against millions of dollars in research, development and marketing?

The operative words in those sentences are "knew" and "knowledge". This book is about the "science of knowing". Really knowing involves counting and measuring, so statistics is the science of quantitative reasoning. "'This is plausible, how might it be wrong?' is the statistician's catch-phrase." (page 4) This science has more in common with epistemology, than it does with accounting. "Statistics, like Bill Shankly's football is not just a matter of life and death, 'Son, it's much more important than that'". (page 2)

Stephen Senn has written an amusing and readable book intended to inform quantitatively and scientifically literate readers (not necessarily statisticians and mathematicians) about a number of statistical methods and concepts. He mostly treats the topics of classical statistics, with confidence intervals, significance tests, correlations, risk analysis and randomized trials taking the central stage. The discussion is supported by excursions into probability theory and its foundations to explain the meaning of the statistics. Each topic is discussed in the context of a specific episode from statistical history, usually medically related, interleaved with surprisingly colorful thumbnail biographies of statisticians, mathematicians, physicians and public health officers. The area of statistics is huge and Senn acknowledges that he has not tried to cover everything, especially the statistics of genetics and decision sciences.

Chapter 1 is a review of elementary statistics, beginning with a familiar puzzle involving the family with two children, one of which is a son. What is the probability that the other child is a girl? This is the starting point for a discussion of the statistics of sex ratios at birth, and a statistical, not mathematical, answer to the question. Interestingly enough, the evidence suggests that the probability is nearer to 0.7 than to the theoretical 2/3. This leads to a discussion of maximum-likelihood estimation and p values. The chapter also discusses Simpson's paradox, and apparent paradoxes due to regression to the mean. The lessons are that "making sense of data can be difficult and one must take care... if one wishes to come to a reasonable conclusion." (page 24)

Chapter 2 contains thumbnail biographical sketches of Dr. John Arbuthnot, whom Senn calls the father of the significance test. Also briefly mentioned are Daniel Bernoulli and his illustrious family, Rev. Thomas Bayes, Laplace, Sir R. A. Fisher, Student (also known as William Gossett) and Karl Pearson. The theme of the chapter is significance tests.

Chapter 3 covers the origins and practice of the randomized clinical trial. The chapter begins with a short biography of medical statistician Austin Bradford Hill. Bradford Hill was the first to apply double-blind randomized pairs study to the effect of the antibiotic streptomycin on tuberculosis. R. A. Fisher had pioneered the randomized trial applied to agricultural experiments. The most interesting part of the chapter is an extended discussion of the ethics of the randomized clinical trial. If the experimenter knows that a new drug may have a helpful effect on a disease, is it ethical to knowingly withhold the treatment from a group of affected patients? Senn concludes that although the randomized clinical trial may seem morally repugnant, it is difficult to produce ethically superior alternatives. What is significant is that Senn concludes this on the basis of quantitative arguments, rather than moral principles.

Chapter 4 is a starred chapter, one of a few such designated sections and chapters. Senn has marked more mathematical sections with a star so they can be safely avoided by the non-mathematical reader. However the starred chapters and sections will not be discouraging to a mathematics or statistics major who knows algebra and matrix multiplication. The chapter covers the philosophical foundations of probability and statistical theory. It discusses Laplace's law of succession, Bayesian approaches, subjectivist probability, and the scientific falsification approach of Karl Popper. Thrown in for good measure are mini-biographies of a number of well-known and not-so-well-known scientists, philosophers, mathematicians and statisticians.

Chapter 5 is one of the most interesting to me, as it concerned the intersection of politics with the misuse of statistics. The issue was a 2-level, 2-factor study conducted by the NIH on the effects of aspirin and beta-carotene on cardiovascular disease and cancer in 22,071 U.S. male physicians. In 1992, the Congressional Women's Caucus called for Congress to pass legislation that required the director of NIH to carry out studies "in a manner sufficient to provide for a valid analysis of whether the variables being studied in the trial affect women." Senn demonstrates how the variance of a random sample varies as the inverse square of the sample size. Therefore, to obtain the same sample variance and similar confidence in the statistical results would require 4 times the sample size and presumably 4 times the study cost. The chapter is an excellent model of mixing a powerfully motivating real-life example with a fairly abstract result in probability and statistics to produce a lesson with real-life consequences.

Chapter 6 starts with a discussion of journalists and their coverage of matters scientific. In particular, the issue is the fallacy of assuming that when one event happens after another, it necessarily follows that the first event caused the second. As an introduction, the chapter contains a short biography of Francis Galton. The statistical issue is one of the retrospective study, for which there is neither a randomized controlled study, nor could there be for ethical reasons. Galton, in addition to his founding of regression analysis, also conducted retrospective analyses, apparently aware of the issue of confounding variables inherent in such studies. The last half of the chapter is a history of the analysis of the effects of cigarette smoking on health, starting with Austin Bradford Hill and Richard Doll in 1950. Interestingly, the great statistician R. A. Fisher argued against the interpretation of Hill and Doll's analysis, insisting that hidden confounding factors may be responsible for the statistically poorer health of the smokers.

Chapter 7 is about the construction of life tables. It was new to me that Edmund Halley, the astronomer responsible for the calculation of the orbits of comets, was also the second to create a life table. John Arbuthnot, the father of the significance test, was the first, constructing a life table based on data for the town of Tunbridge Wells, where Thomas Bayes lived as a pastor. The topic of life tables leads to a discussion of the differences between the professional practices and outlooks of actuaries and medical statisticians and to hazard rates, or exposure. Ultimately, the chapter discusses the conclusions of statistician David Cox, who won the Kettering Prize for cancer research. He showed that the statistical inferences did not depend on the on the precise modeling of the background probability of dying, as long as it is hazard ratios the study is in interested in.

Newspapers report frequently about recalls of well-known drugs based on the results of multiple studies combined together. The newspapers rarely call such studies meta-analyses, or pooled data, but that is the topic of Chapter 8. I never understood how such studies were conducted until reading this chapter which briefly surveys the practice and gives a biography of Archie Cochrane, statistical developer of the method.

The modeling of epidemics with differential equations is now fairly standard, even in introductory courses on differential equations. As such, the names of Kermack and McKendrik are familiar, but again, I knew nothing of how they came to the modeling of diseases which is the content of Chapter 9. This chapter discusses stochastic processes and their statistics but goes a little light on the mathematical details, and concentrates on the probabilistic interpretations. There is also some discussion of the ethical implications of withholding children from inoculation for diseases, which can be justified individually by the threshold theorem, balancing the protection of the group against the protection of the individual against the slight chance of complications from the inoculation.

The probabilistic and statistical theme of Chapter 10 is the Poisson distribution, and the occurrence of rare events. The real concern of the chapter is the relationship of courts of law to statistics, using the claims against Dow Corning that silicone breast implants could cause rheumatoid arthritis. Senn presents the statistical conclusion, again based on pooled data, that a causal relationship between breast implants and connective tissue disease cannot be claimed. Nevertheless, "the judicial juggernaut had already developed its billion-dollar momentum, it wasn't going to be stopped by mere statistics." (page 210).

The book has many puns, especially in the Chapter and Section headings. There are plenty of references to pop culture, usually forming one of the puns in the headings, and these keep the book lively and light. The book is also witty and full of tart asides and tangential excursions. Reading the book is like attending the lectures of a curmudgeonly but beloved professor.

Who is the book intended for? The mathematical level is neither high nor demanding. I think the book is aimed at a well educated reader who had some mathematics at one point, and has the patience to work through a deep, but well-explained quantitative argument. I imagined a physician, an epidemiologist, a public health administrator, even a lawyer involved in litigation, who was curious about the foundations of the medical trials they now rely on, and may even participate in or conduct. This book would make good, and entertaining, background reading for someone now teaching statistics who did not have a full academic background in statistics. Each chapter would provide interesting anecdotes and classroom illustrations as well as biographical sketches on the people whose names are now attached to theorems and statistical tests. The book might also make interesting supplementary reading for students, particularly those who intend to pursue medical statistics, biostatistics, applied statistics or epidemiology.

I enjoyed this book even though I am not a statistician nor do I teach statistics. I learned some things, I picked up a few biographical morsels that I can use when I teach applied probability. The real enjoyment of this book though comes with seeing the impact that the mathematical sciences can have when the outcomes are literally life and death.

Steven R. Dunbar ( teaches at the University of Nebraska - Lincoln and is MAA Director of Mathematics Competitions.

Preface page ix
Permissions xii
1 Circling the square 1
2 The diceman cometh 26
3 Trials of life 50
4 Of dice and men 69
5 Sex and the single patient 91
6 A hale view of pills 108
7 Time's tables 122
8 A dip in the pool 142
9 The things that bug us 162
10 The law is a ass 186
11 The empire of the sum 212
Notes 231