Correlation does not imply causation — even Wikipedia says so. Less commonly one hears, “you cannot prove causality with statistics”. Yet, according to the esteemed statistician Frederick Mosteller, “You can *only* prove causality with statistics”. In this book Paul Rosenbaum, one of the field’s leading scholars, discusses how this can be true. He addresses the question of inferring causality in two parts. He first discusses the use of randomized experiments and clinical trials; this is what Mosteller had in mind. The second part takes up observational studies that are not randomized.

Randomized trials are a kind of gold standard. They permit inference about causal effects of treatments on populations of individuals without being able to say anything about the effect on any single individual. But they are not always practical and sometimes would be unethical. Observational studies deal with situations where treatments of interest are not subject to experimental control or where they may be harmful. The absence of control is their most critical limitation.

Rosenbaum’s goal is “to present the concepts of causal inference clearly, with reasonable precision, but with a minimum of technical material”. He approaches more difficult concepts by starting with the simplest non-trivial example, leaving out inessential details and avoiding generalizations. Background material (some probability and a little statistics) and notation are introduced as needed. Sections marked with an asterisk explain in English ideas that would need additional technical detail to explain fully. For readers comfortable with conditional probability he provides endnotes that amplify or clarify arguments in the text.

John Stuart Mill had suggested that causal inference needed to compare identical people under alternative treatments. Ronald Fisher, who introduced randomized trials, knew this was not possible and suggested instead to make the treatment that a person receives unrelated to anything (such a gender, race, health or state of mind) that makes people different. His approach: flip a fair coin to assign individuals to treatment groups. Rosenbaum explores randomized trials in the first four chapters. He uses two examples to introduce the basic techniques. The first compares two emergency treatments for septic shock; one is more aggressive but also more invasive and the other less aggressive. The second describes the successful randomized trials that convinced people that the Salk polio vaccine worked.

Good observational studies are harder to get right and much subtler. Rosenbaum says that the best ones can only be “reasonably compelling”. He develops several approaches intended to address the host of complications in the following nine chapters. Natural experiments provide one attempt to identify circumstances in the world that would resemble well-designed random experiments. Rosenbaum uses the example of the Dutch famine during World War II to explore the effects of very poor prenatal nutrition on cognitive development. Another approach is the use of “elaborate theories” that identify as many different consequences of a treatment as possible and plan observational studies to discover which of these consequences is likely to hold. Here one of Rosenbaum’s examples is the effect of a father’s occupation (manufacturing batteries using lead) on the health of his children.

The author’s voice is an important element in the book’s success. Rosenbaum is consistently clear and direct, and seems at times to be speaking directly to the reader. His excellent set of examples (twenty-five of them altogether) bring the more theoretical discussions to life.

This is not a textbook in the usual sense — there are no exercises. It would be valuable supplementary reading for beginning to intermediate statistics courses. Anyone interested in understanding and evaluating the multitude of studies that we see almost daily in news reports would also find it valuable.

Although the book is intended to be self-contained, it would be tough reading for readers who are less than comfortable with statistical reasoning. The concepts of hypothesis testing and associated test statistics are used freely and are likely to overwhelm less experienced readers.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.