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Statistical Science 1999, Vol. 14, No.4, 418- 426 "Student" and Small-Sample Theory E. L. Lehmann Abstract. The paper discusses the contributions Student (W. S. Gosset) made to the three stages in which small-sample methodology was established in the period 1908-1933: (i) the distributions of the test­ statistics under the assumption of normality, (ii) the robustness of these distributions against nonnormality, (iii) the optimal choice of test statis­ tics. The conclusions are based on a careful reading of the correspon­ dence of Gosset with Fisher and E. S. Pearson. Key words and phrases: History of statistics, "exact" distribution theory, assumption of normality, robustness, hypothesis testing, Neyman-Pear­ son theory. 1. INTRODUCTION determined the distributions of the statistics used to test means, variances, correlation and regression In an interview published in Statistical Science coefficients under the assumption of normality. At [Laird (1989)], F. N. David talks about statistics in the second stage (Pearson), the robustness of these the 1920s and 30s as developed by Gosset (Student), distributions under nonnormality was investigated. Fisher, Egon Pearson and Neyman. [In the remain­ Finally, at the last stage Neyman and Pearson laid der of this paper we shall usually refer to E(gon) S. the foundation for a rational choice of test statistics Pearson simply as Pearson and to his father as not only in the normal case but quite generally. In Karl Pearson or occasionally asK. P.] She describes the following sections we shall consider the contri­ herself as a contemporary observer who saw "all butions Gosset made to each of these stages. the protagonists from a worm's eye point of view." An author writing about this period is fortunate (For anyone like myself who knew the feisty David to have available a wealth of material that fre­ later in her life it is hard to imagine her ever quently makes it possible to trace mutual influ­ playing the role of a "worm.") Her surprising as­ ences and the development of ideas in considerable sessment: detail. The principal sources I have used are ac­ I think he [Gosset] was really the big influence knowledged at the end of the paper. in statistics . He asked the questions and Pearson or Fisher put them into statistical lan­ 2. THE NEW METHODOLOGY guage and then Neyman came to work with the mathematics. But I think most of it stems from 2.1 Gosset's 1908 Papers Gosset. (1.1) The event that with little fanfare and no particu­ (Note: Numbers have been added to quotations for lar enthusiasm on the part of the Editor ushered in easier cross-reference.) a new era in statistics was the Biometrika publica­ Here she is of course not talking about all of tion in 1908 of a paper "The probable error of the statistics in this period, but of the development of mean" by "Student," the pseudonym of William the new small-sample approach. Nevertheless, her Sealy Gosset. The reason for the pseudonym was a claim is surprising because Gosset is mainly known policy by Gosset's employer, the brewery Arthur for only one, although a pathbreaking, contribution: Guinness Sons and Co., against work done for the Student's t-test. The aim of this paper is to consider firm being made public. Allowing Gosset to publish to what extent David's conclusion is justified. under a pseudonym was a concession that resulted The basis for the new methodology was estab­ in the birth of the statistician "Student," the cre­ lished in three stages. Stage 1 (Student-Fisher) ator of Student's t-test. Today the pathbreaking nature of this paper is generally recognized and has been widely com­ E. L. Lehmann is· Professor Emeritus, Department mented upon, among others by Pearson (1939), of Statistics, University of California, Berkeley, Cal­ Fisher (1939), Welch (1958), Mosteller and Tukey ifornia 94720. (1977, Section B), Box (1981), Tankard (1984), J. Rojo (ed.), Selected Works of E. L. Lehmann, Selected Works in Probability 1001 and Statistics, DOI 10.1007/978-1-4614-1412-4_83, © Springer Science+Business Media, LLC 2012 "STUDENT" AND SMALL-SAMPLE THEORY Pearson (1990), Lehmann (1993) and Hald (1998, of errors does not exactly hold except the number of Section 27 .5). errors (i.e., components of error) is indefinitely The core of the paper consists of the derivation of great. With a limited number of errors, the law will the distribution of be imperfectly followed; and the deductions, made (2.1) z =(.X- JL)/S on the supposition that the law is strictly followed, will be or may be inaccurate or inconsistent." where X1 , ... , x. are i.i.d. with normal distribution Despite these reservations, Student's assumption N( f.L, o- 2) and where S 2 = l(X; - X)2 jn. (The defi­ of normality is thus grounded in a well-established nitions of z and S 2 do not agree with current usage tradition. but are those used by Student.) This derivation was Student illustrated the use of his distribution by a difficult task for Gosset who was a chemist, not a three examples, including one of a paired compari­ mathematician, and although he obtained the cor­ son experiment which he reduces to the one-sample rect answer, he was not able to give a rigorous situation by taking differences. Finally, he gave a proof. table of the z-distribution for sample sizes 4 to 10. However, this distributional result was not the He later extended it to sample sizes 2 to 30 [Stu­ reason for the enormous influence of the paper. The dent, (1917)]. The change from z to t = z~ principal contribution was that it brought a new which is now called Student's t , is due to Fisher point of view. It stated the need for methods deal­ (1925a, b) and is discussed in Eisenhart (1979). ing with small samples, for which the normal ap­ Student provided tables for t in 1925. proximations of the theory of errors were not ade­ The paper on z was followed by another paper quate, and it brought the crucial insight that exact (1908b) in which Student tackled the small-sample results can be obtained by making an additional distribution of the sample correlation coefficient in assumption: that the form of the distribution of the the normal case when the population correlation observations is known. Concerning this assump­ coefficient is 0. Because a mathematical derivation tion, Student wrote in the Introduction to his paper was beyond his powers, he decided to "fit a Pearson [Student (1908a)]: curve" and, using some elementary properties of . the sample is not sufficiently large to deter­ correlation coefficients, he "guessed" (his own word) mine what is the law of distribution of individ­ the correct form. uals. It is usual, however, to assume a normal For the case that the population correlation coef­ distribution . .. : since some law of distribution ficient is different from 0, he came to the conclusion must be assumed it is better to work with a that it "probably cannot be represented by any of curve whose area and ordinates are tabled, and Professor Pearson's types of frequency curves" and whose properties are well known. This assump­ admits that he "cannot suggest an equation which tion is accordingly made in the present paper, will accord with the facts." so that its conclusions are not strictly applica­ Gosset wrote no further papers on small-sample ble to populations known not to be normally distributions (except for providing tables of z and distributed; yet it appears probable that the t). An obvious explanation is that he had a full-time deviation from normality must be very extreme job as brewer. However, he himself denies that this to lead to serious error. (2.2) was the reason, explaining to Fisher (July 14, 1924): "By the way it is not time but ability which has What did Student mean when he wrote that "it is prevented me following up my work by more on usual to assume a normal distribution"? He learned your lines." Gosset (1970) His 1908 papers did not statistics by reading two books: Airy's Theory of receive much attention and his ideas might have Errors of Observations (1879) and Merriman's continued to go unnoticed had they not acquired a Method of Least Squares (1884). Both emphasize new champion of exceptional brilliance and enor­ that errors are typically sums of a large number of mous energy. independent small components and hence are ap­ 2.2 Fisher's Proof proximately normally distributed. In fact, the nor­ mal distribution is called the Law of Probability of In 1912 R. A. Fisher, then 22 years old and a Errors. Merriman says about it (page 33), "What­ Cambridge undergraduate, was put into contact ever may be thought of the theoretical deductions of with Gosset through Fisher's teacher, the as­ the law of probability of error, there can be no tronomer F. J. M. Stratton. As a result, Gosset doubt but that its practical demonstration by expe­ received from Fisher a proof of the z-distribution rience is entirely satisfactory." Airy (page 24) is and asked Karl Pearson to look at it, admitting that slightly more cautious because he warns, "It must he could not follow the argument (which was based always be borne in mind that the law of frequency on n-dimensional geometry) and suggesting, "It 1002 E. L. LEHMANN seemed to me that if it's alright perhaps you might cult and also that he had no plans to work on them like to put the proof in a note [in Biometrika of himself.
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