Abraham Wald invented the test that can stop early

Person: Abraham Wald
Lived: 1902–1950
Field: Statistics
From: Hungarian-American

The story

Abraham Wald arrived in New York in 1938 as a refugee from Nazi-occupied Austria. Born in 1902 in Kolozsvár, now Cluj-Napoca in Romania, he had earned his doctorate in Vienna but, as a Jew, could not obtain a university post there. At Columbia University he retrained in modern statistics under Harold Hotelling, joined the faculty in 1941 and, once the United States entered the Second World War, became a member of the Statistical Research Group, a team of about 18 researchers working on military problems.

One of those problems was quality control. Wartime factories had to inspect enormous quantities of munitions and equipment, and every test used up time and materials. The standard statistical approach fixed the number of observations in advance, gathered all the data and only then analysed it. Wald's idea was simple, yet no one had built it into a statistical theory: examine the evidence as each result comes in, and stop as soon as it justifies a decision.

His answer was the sequential probability ratio test. After each new observation the test updates a running score of how strongly the evidence favours one hypothesis over another; when the score crosses an upper or lower boundary, set by the error rates the user will accept, testing stops. It often reached reliable decisions with far fewer observations than classical methods. Its value to the war effort was recognised at once, and the method was given a restricted classification. Wald solved the general problem of sequential tests of statistical hypotheses, published a long paper on them in 1945 and gathered his results in the book Sequential Analysis in 1947. He had conjectured in 1943 that the test was optimal, and he proved it with his colleague Jacob Wolfowitz in a joint paper in 1948.

Sequential methods quickly made industrial inspection cheaper and faster, and Wald's theorems on the number of observations required and on error probabilities found immediate practical use. After the war the statistician Peter Armitage brought sequential analysis into medical research. Today clinical trials routinely include planned interim analyses, so that a trial can be stopped early when a treatment is clearly working or clearly harmful, sparing patients and cost. The sequential probability ratio test is also used as a stopping rule in computerised testing of examinees.

Wald himself did not live to see most of this. He and his wife died in a plane crash in India in December 1950, while he was on a lecture tour at the invitation of the Indian government.

Why it matters

Sequential analysis lets researchers reach sound decisions as soon as the evidence allows, saving time and money in industry and helping protect patients in clinical trials.

About Abraham Wald

Refugee mathematician who founded sequential analysis and statistical decision theory, and whose wartime study of damaged bombers became the classic lesson in survivorship bias.

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Abraham Wald

1902–1950 · Statistics

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