A century of t tests
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The document discusses the historical development of the t-test, highlighting key figures such as William Sealy Gosset (Student), Sir Ronald Fisher, and Arthur Cushny, among others. It examines the origins of the data used by Student in his t-test and the context of its application in medical research. The text also acknowledges contributions from various scholars and emphasizes the significance of proper study design in statistical testing.
A century of t tests
- 1. (c) Stephen Senn 2008 A century of t-tests 1 A Century of t-tests Stephen Senn
- 2. (c) Stephen Senn 2008 A century of t-tests 2 A Letter From America Isidor Greenwald to RA Fisher, 31 December 1934
- 3. (c) Stephen Senn 2008 A century of t-tests 3 Isidor Greenwald • Born 1887 or 1888 • BA Michigan 1909 • PhD Colombia 1911 • Montefiore Hospital, the Bronx, 1911-1914 • Roosevelt Hospital 1914-1928 • New York University Medical School 1932-1952 • “the first person to identify that hyperphosphataemia occurs in people who suffer from kidney disease” (Charles George, personal communication) • Expert on the history of goitre • Helped Severo Ochoa when a refugee in the US • Died 7 February 1976
- 4. (c) Stephen Senn 2008 A century of t-tests 4 Student in 1908
- 5. (c) Stephen Senn 2008 A century of t-tests 5
- 6. (c) Stephen Senn 2008 A century of t-tests 6 See The James Lind Library and Cushny AR, Peebles AR (1905). The action of optical isomers. II. Hyoscines. J Physiology 32:501-510.
- 7. (c) Stephen Senn 2008 A century of t-tests 7 8 1 4 3 6 1 0 87 2 65 5 0 7 432 Hrs sleep Hrssleep Reference line C & P: Isomers of Hyoscine S & F: L-hyoscyamine HBr v L-hyo
- 8. (c) Stephen Senn 2008 A century of t-tests 8 Fisher • Had used the data in Statistical Methods for Research Workers • Had repeated Student’s mistake • Was unaware that the data were in the form of differences to control
- 9. (c) Stephen Senn 2008 A century of t-tests 9 William Sealy Gosset 1876-1937 • Born Canterbury 1876 • Educated Winchester and Oxford • First in mathematical moderations 1897 and first in degree in Chemistry 1899 • Starts with Guinness in 1899 in Dublin • Autumn 1906-spring 1907 with K Pearson at UCL • 1908 publishes ‘The probable error of a mean’
- 10. (c) Stephen Senn 2008 A century of t-tests 10 Student and Fisher . . . of course it doesn’t really matter two straws . . . I have no recollection at all of having selected two columns out of a four column table and if I had not such a genius for making slips I should be inclined to think that I had taken the figures from a notice of the paper . . . I remember I had a good deal of difficulty in getting any figures to illustrate with but I haven’t the faintest recollection of how I managed to run across Cushny and Peebles. (Reference 4, p. 54; see also Reference 5, p. 141)
- 11. (c) Stephen Senn 2008 A century of t-tests 11 Arthur Cushny • Born Fochabers 1866 • Studied medicine at Aberdeen • Worked in Bern (Kronecker) & Strasburg (Schmiedeberg) • Chairs – Michigan 1893-1905 – UCL 1905-1918 – Edinburgh 1918-1926 • Died Peffermill House 1926 Wellcome library London L0012929
- 12. (c) Stephen Senn 2008 A century of t-tests 12 http://www.aboutscotland.co.uk/ecosse/peffermill/index.html Tariff (Weekly Rental): Peffermill House, £ per week: 425 485 545 595 Special rates apply for Hogmanay, Christmas, Festival & Tatoo. Peffermill House
- 13. (c) Stephen Senn 2008 A century of t-tests 13 Cushny’s Scientific Importance • Pioneer investigator of optical isomerism in man • Important work on urinary secretion • Above all he was a scientific sceptic – he reduced the numbers of substances in regular therapeutic use and his students were very grateful to him for that – Much less to remember
- 14. (c) Stephen Senn 2008 A century of t-tests 14
- 15. (c) Stephen Senn 2008 A century of t-tests 15 Alvin Roy Peebles 1884-1917 • Born in Battle Creek Michigan 1884 • Graduated MD Ann Arbor 1906 • Johns Hopkins and back to Ann Arbor before starting at Boulder Colorado • Married 1910 and had one daughter • Died of TB in 1917 aged 33
- 16. (c) Stephen Senn 2008 A century of t-tests 16 Published: November 10, 1912 Copyright © The New York Times
- 17. (c) Stephen Senn 2008 A century of t-tests 17 “…a number of trials of their usefulness for this purpose were made in the Michigan Asylum for Insane at Kalamazoo. The harmlessness of small doses of both alkaloids were first ascertained on ourselves, and then a number of tablets each containing 0.6mg of L-hyoscine or R-hyoscine hydrobromate were used as hypnotics in the wards of Drs Richards and Light under the general supervision of Dr. W. M. Edwards. We are much indebted to these physicians for the results recorded by them. Instead of hyoscine, a certain number of tablets contained 0.6 mg of hyoscyamine hydrobromate, as its usefulness as a hypnotic has not yet been determined. In all, ten patients were treated with the tablets.” (pp. 508-509). Cushny and Peebles (1905)
- 18. (c) Stephen Senn 2008 A century of t-tests 18
- 19. (c) Stephen Senn 2008 A century of t-tests 19 S Rudolph Light • Born 1877 Lebanon , Pennsylvania one of eight children • Studied medicine Michigan 1899-1904 (taught by Cushny) • Appointment at Kalamazoo 1904 • Upjohn company 1907 • Marries into Upjohn family 1908 • Friend of Wright Brothers • Dies 1961 • Leaves $1.5m to Kalamazoo College and $350,000 to Vanderbilt University
- 20. (c) Stephen Senn 2008 A century of t-tests 20 Richards? • Probably George Gill Richards 1883-1950 • Born Mendon, Utah • Worked in Vienna and Utah
- 21. (c) Stephen Senn 2008 A century of t-tests 21 “As a general rule a tablet was given on each alternate evening and the duration of sleep and other features were noted and compared with those of the intervening control night on which no hypnotic was given. Hyocyamine was thus used on three occasions, and then racemic hyoscine, and then laevohyoscine. Then a tablet was given each evening for a week or more, the different alkaloids following each other in succession. The results may be given shortly in tabular form, details being reserved for publication elsewhere. (Reference 3, p. 509)” If we use the symbol - to stand for ‘control night’, H for hyoscyamine, R for R- hyoscine and L for L-hyoscine, a typical sequence seems to have been of the form H – H – H – R – R – R – L – L – L - H R L H R L H R L
- 22. (c) Stephen Senn 2008 A century of t-tests 22 A centenary example The Cushny and Peebles (1905) data used by Student (1908) Hours of sleep for 11 subjects at the insane asylum at Kalamazoo 21 4 6 1 8 2 10 3 4 5 6 7 8 5 93 7 11 Patient H0.6 mg L-hyoscyamine HBr v Patient L0.6 mg L-hyoscine HBr v Patient R0.6 mg R-hyoscine HBr v Patient Control v Patient
- 23. (c) Stephen Senn 2008 A century of t-tests 23 8 1 4 3 6 1 0 87 2 65 5 0 7 432 Hrs sleep Hrssleep Reference line C & P: Isomers of Hyoscine S & F: L-hyoscyamine HBr v L-hyo
- 24. (c) Stephen Senn 2008 A century of t-tests 24 Fisher (again) • Corrected the mistake in the sixth edition • Kept the numbers but removed the labels • Made one interesting and revealing change to the text Window at Caius Cambridge
- 25. (c) Stephen Senn 2008 A century of t-tests 25 What a difference a word makes ‘Let us suppose that the above figures. . . had been obtained using different patients for the two drugs; the experiment would have been less well controlled, and we should expect to obtain less certain results from the same number of observations, for it is a priori probable, and the above figures suggest, that personal variations in response to the drugs will be to some extent correlated’ (Reference 2, p. 110).
- 26. (c) Stephen Senn 2008 A century of t-tests 26 What a difference a word makes ‘Let us suppose that the above figures. . . had been obtained using different patients for the two drugs; the experiment would have been less well controlled, and we should expect to obtain less certain results from the same number of observations, for it is a priori probable, and the above figures suggest, that personal variations in response to the drugs will be to some extent similar’ (Reference 24, p. 130).
- 27. (c) Stephen Senn 2008 A century of t-tests 27 What is Fisher Thinking? 0 0 0 2 , 0, for control var , cov , 0, cov 0, , , Then two things follow 1. 0.5 2.var var var ij ij ij i j ij ij ij i j ij i ij i ij ij D D ij ij ij ij Y S j D Y Y and S j j j j j D D D D Suppose we have an additive hierarchical model Hence the data are correlated anyway by virtue of being expressed as differences to a common control and this correlation is not the origin of the reduced variance when active treatments are compared. There must be a further explanation in terms of patient by treatment interaction. Patients are showing similar responses.
- 28. (c) Stephen Senn 2008 A century of t-tests 28 1 0 4 8 876 2 50 6 432 Hrs sleepfor 0.6mg R-hyoscineHBr Raw values Difference to control Reference line
- 29. (c) Stephen Senn 2008 A century of t-tests 29 Correlations Control H 0.5860 L 0.3238 0.8113 R 0.2458 0.7643 0.9520 Control H L R Number of observations: 10 Correlations D_H D_L 0.7952 D_R 0.7902 0.9674 D_H D_L D_R Number of observations: 10
- 30. (c) Stephen Senn 2008 A century of t-tests 30 Senn and Richardson(1994) based on Preece (1982)
- 31. (c) Stephen Senn 2008 A century of t-tests 31 Two Mysteries • Where did Student get the data? – Directly from Cushny who had moved to UCL from Michigan shortly before Student arrived there • Where are the original data? – C&P state “The results may be given shortly in tabular form, details being reserved for publication elsewhere.” • Was is down to Peebles to publish them and did he never do so?
- 32. (c) Stephen Senn 2008 A century of t-tests 32 Student’s T-Test? • The modern form we use of ratio of mean to standard error is due to Fisher • Student was anticipated by Jakob Luroth (1844-1910) who derived (essentially) the same result in 1876
- 33. (c) Stephen Senn 2008 A century of t-tests 33 Student’s T-Test! • Student derived it himself • Because he needed it • And he used it • Although Luroth anticipated him and Fisher improved (and proved) his test (in 1925) Student was the true original implementer of the t-test
- 34. (c) Stephen Senn 2008 A century of t-tests 34 The Trial at Kalamazoo • An innovative investigation – Interesting scientific hypothesis – Clever use of control – Intelligent analysis • Without benefit of statistics! – Sensible conclusion – Ethical ground prepared • Only randomisation is missing
- 35. (c) Stephen Senn 2008 A century of t-tests 35 Arthur Cushny (reprise) Long after doses and actions have faded from the memory will there remain in the mind the vision of a true scientist . . . of one who founded the modern science of Pharmacology, and filled his students with his own enthusiasm . . . We, too, have our giants, and in the days to come we will look back upon Cushny as one of the greatest of them all (Reference 4 1, p. 204) From the Edinburgh University student magazine The name of this magazine?
- 36. (c) Stephen Senn 2008 A century of t-tests 36 The Student
- 37. (c) Stephen Senn 2008 A century of t-tests 37 ACKNOWLEDGEMENTS It would have been impossible for us to write this article without the assistance of many persons who carried out research on our behalf or otherwise provided us with information or comments. We thank the following for their help: Ms Jean Archibald, Edinburgh University Library; Professor Peter Armitage, Oxford; Professor George Barnard, Essex; Mr. Le Roy Barnett, State of Michigan Archives; Ms Carol Bowman, University College Records Office; Mr. Alan Clark, The Royal Society; Mr. Dilip Chaterji, University College Library; Dr. James Coleman, Kalamazoo Regional Psychiatric Hospital; Professor D. Colquhoun, University College; Professor Richard Cornell, University of Michigan; the Daily Camera Obituary Department, Boulder, Colorado; Ms. Sheila Edwards, The Royal Society; Ms. Amanda Engineer and Mr. F. Friend, University College Library; Dr. Fred W. Hunter, American Medical Association; Ms. Karen Jania, Bentley Historical Library, University of Michigan; Dr. Walter Kremers, CIBA, Basle; Ms. Catherine Larson, Kalamazoo Public Library; Dr. Richard Light, Kalamazoo; Major Charles O’Leary, The Worshipful Society of Apothecaries of London; Professor J. Parascandola, National Library of Medicine, Bethesda; Dr. Nicholas Part, University of Dundee; Professor Robin Plackett, Newcastle; Professor D. Preece, University of Kent; Ms. B. Rothwell, Register and Annual Retention Fee Division of the General Medical Council; Dr. Stefan Schwabe, CIBA, Basle; Mrs. M. Shaw, The Royal Statistical Society; Miss J. Sheppard, The Wellcome Institute for the History of Medicine; Dr. Tilli Tansey, The Physiological Society; and Ms. Bette Thompson, Ann Arbor Public Library. Documents held at the Contemporary Medical Archives Centre, Wellcome Institute for the History of Medicine, were consulted by courtesy of the Wellcome Trustee. We are also extremely grateful to Miss Marian B. Gosset, Oxford for permission to quote from her father’s correspondence to Dr. H. Wehrli, CIBA, Basle for preparing Figure 1 and for helpful comments and to an anonymous referee for a very careful reading of and detailed comments on an earlier draft. And my co-author (in 1994) Bill Richardson
- 38. (c) Stephen Senn 2008 A century of t-tests 38 1 1 1 2 2 3 3 4 4 5 5 6 7
- 39. (c) Stephen Senn 2008 A century of t-tests 39 1 1 1 2 2 3 3 4 4 5 5 6 7
- 40. (c) Stephen Senn 2008 A century of t-tests 40 One-sample t-test Variate: Difference. Summary Standard Standard error Sample Size Mean Variance deviation of mean Difference 13 45.38 1648 40.59 11.26 95% confidence interval for mean: (20.85, 69.91) Test of null hypothesis that mean of Difference is equal to 0 Test statistic t = 4.03 on 12 d.f. Probability = 0.002
- 41. (c) Stephen Senn 2008 A century of t-tests 41 Data from a cross-over trial in asthma Peak expiratory flow in litres/minute Patient Formoterol Salbutamol Difference 1 310 270 40 4 310 260 50 6 370 300 70 7 410 390 20 10 250 210 40 11 380 350 30 14 330 365 -35 2 385 370 15 3 400 310 90 5 410 380 30 9 320 290 30 12 340 260 80 13 220 90 130