Some Logic and History of Hypothesis Testing

• I. J. Good
Chapter
Part of the The University of Western Ontario Series in Philosophy of Science book series (WONS, volume 16)

Abstract

The foundations of statistics are controversial, as foundations usually are. The main controversy is between so-called Bayesian methods, or rather neo-Bayesian, on the one hand and the non-Bayesian, or ‘orthodox’, or sampling-theory methods on the other.1 The most essential distinction between these two methods is that the use of Bayesian methods is based on the assumption that you should try to make your subjective or personal probabilities more objective, whereas anti-Bayesians act as if they wished to sweep their subjective probabilities under the carpet. (See, for example, Good (1976).) Most anti-Bayesians will agree, if asked, that they use judgment when they apply statistical methods, and that these judgments must make use of intensities of conviction,2 but that they would prefer not to introduce numerical intensities of conviction into their formal and documented reports. They regard it as politically desirable to give their reports an air of objectivity and they therefore usually suppress some of the background judgments in each of their applications of statistical methods, where these judgments would be regarded as of potential importance by the Bayesian. Nevertheless, the anti-Bayesian will often be saved by his own common sense, if he has any. To clarify what I have just asserted, I shall give some examples in the present article.

Keywords

Null Hypothesis Subjective Probability Multinomial Distribution Iterate Logarithm Logical Disjunction
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. Barnard, G. A., ‘On the Bayesian-antibayesian controversy’, International Meeting on Bayesian Statistics, Valencia, Spain, 1979. To be published in Trabajos de Estadística y de Investigacion Operativa.Google Scholar
2. Bartlett, M. S., ‘The statistical significance of odd bits of information’, Biometrika 39 (1952), 228–237.Google Scholar
3. Bernoulli, D., ‘Recherches physiques et astronomique …’. Recueil des pièces qui ont remporté le prix de l’Academie Royale des Science 3 (1734), 93–122.Google Scholar
4. Bochner, S., Private oral communication, 1955.Google Scholar
5. Borel, E., Le Hasard, Hermann, Paris, 1920.Google Scholar
6. Braithwaite, R. B., A lecture at the 1951 weekend conference of the Royal Statistical Society in Cambridge, England, 1951.Google Scholar
7. Brier, G. W., ‘Verification of forecasts expressed in terms of probability’, Monthly Weather Rev. 78 (1950), 1–3.
8. Crook, J. F. and Good, I. J., ‘On the application of symmetric Dirichlet distributions and their mixtures to contingency tables, Part II’, Annals of Statistics (in press) (1979).Google Scholar
9. Efron, B., ‘Does an observed sequence of numbers follow a simple rule?’, J. Amer. Statist. Assoc. 66 (1971), 552–568 (with discussion).
10. Feller, W., An Introduction to Probability Theory and its Applications, Vol. 1, Wiley, New York, 1950.Google Scholar
11. de Finetti, B., Theory of Probability, Vol. 1, Wiley, New York, 1975.Google Scholar
12. Frazier, K., ‘Schmidt’s airing at the APS’, The Skeptical Inquirer: The Zetetic 3 (1979), No. 4, 2–4.Google Scholar
13. Frieman, J. A., Chalmers, T. C., Smith, Harry, Jr., and Kuebler, R. R., ‘The importance of beta, the type II error and sample size in the design and interpretation of the randomized control trial’, New England J. Medicine (1978), 690–694.Google Scholar
14. Good, I. J., Probability and the Weighing of Evidence, Charles Griffin, London; Hafners, New york, p. 119, 1950Google Scholar
15. Good, I. J., ‘Rational decisions’, J. Roy. Statist. Soc. B14 (1952), 107–114.Google Scholar
16. Good, I. J., ‘The appropriate mathematical tools for describing and measuring uncertainty’, Chapter 3 of Uncertainty and Business Decisions, Liverpool, second edition 1957, 20–36, 1954.Google Scholar
17. Good, I. J., Contribution to the discussion of a paper by G. S. Brown, in Information Theory (ed. C. Cherry), Butterworths, London, p. 13, 1956a.Google Scholar
18. Good, I. J., ‘Which comes first, probability or statistics?’, J. Inst. Actuaries 82 (1956b), 249–255.Google Scholar
19. Good, I. J., ‘The surprise index for the multivariate normal distribution’, Annals Math. Statist. 27 (1956c), 1130–1135.
20. Good, I. J., ‘Saddle-point methods for the multinomial distribution’, Annals Math. Statist. 28 (1954), 861–881.
21. Good, I. J., ‘Significance tests in parallel and in series’, J. Amer. Stat. Assn. 53 (1958), 799–813.
22. Good, I. J., ‘Subjective probability as the measure of a non-measurable set’, Logic, Methodology, and Philosophy of Science: Proc. of the 1960 International Congress, Stanford University Press, pp. 319–329, 1962a.Google Scholar
23. Good, I. J., Contribution to the discussion in The Foundations of Statistics, opened by L. J. Savage, Methuen, London; Wiley, New York, 1962b.Google Scholar
24. Good, I. J., The Estimation of Probabilities: An Essay on Modern Bayesian Methods, M.I.T. Press, 1965.Google Scholar
25. Good, I. J., ‘A Bayesian significance test for multinomial distributions’, J. Roy. Statist. Soc. B29 (1967), 399–431. (With discussion.)Google Scholar
26. Good, I. J., ‘Corroboration, explanation, evolving probability, simplicity, and a sharpened razor’, Brit. J. Philos. Sci. 19 (1968), 123–143.
27. Good, I. J., ‘A subjective evaluation of Bode’s Law and an “objective” test for approximate numerical rationality’, J. Amer. Statist. Assoc. 64 (1969), 23–66.
28. Good, I. J., ‘Twenty-seven principles of rationality’, 1971, Appendix to ‘The probabilitistic explication of information, evidence, surprise, causality, explanation, and utility’, in V. P. Godambe and D. A. Sprott (eds.), Foundations of Statistical Inference (Proceedings of an international symposium at Waterloo, April 1979 ). Holt, Reinhart and Winston of Canada, Toronto, 1979, pp. 124–127.Google Scholar
29. Good, L J., ‘Information, rewards, and quasi-utilities’, in J. J. Leach, R. Butts, and G. Pearce (eds.), Science, Decision and Value, D. Reidel, Dordrecht, 1973, 115–127.Google Scholar
30. Good, I. J., ‘The Bayesian influence, or how to sweep subjectivism under the carpet’, in Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science (Proc. of a Conference in May 1973 at the Univ. of W. Ontario; Eds. C. A. Hooker and W. Harper), Vol. 2, D. Reidel, Dordrecht, Holland, 1976, pp. 125–174.Google Scholar
31. Good, I. J., ‘Explicativity: a mathematical theory of explanation with statistical applications’, Proc. Roy. Soc. (London) A354 (1977), 303–330. Reprinted in Zellner (1980).Google Scholar
32. Good, I. J., ‘Ethical treatments’, J. Statist. Comput. Simul. 7 (1978) 292–295.
33. Good, I. J., ‘Some history of the hierarchical Bayesian methodology’, invited paper for the International Meeting on Bayesian Statistics, May 28–June 2, 1979, Valencia, Spain. Trabajos de Estadística y de Investigación Operativa (in press).Google Scholar
34. Good, I. J., ‘The philosophy of exploratory datum analysis’. In American Statistical Association 1980 Proceedings of the Business and Economic Statistics Section 1980.Google Scholar
35. Good, I. J., and Crook, J. F., ‘The Bayes/non-Bayes compromise and the multinomial distribution’, J. Amer. Statist Assoc. 69 (1974), 711–720.
36. Good, I. J., and Crook, J. F., ‘The Rank II powers of tests for multinomials and contingency tables’, In preparation, 1979.Google Scholar
37. Hendrickson, A. and Buehler, R. J., ‘Elicitation of subjective probabilities by sequential choices’, J. Amer. Statist. Assoc. 67 (1972), 880–883.
38. Jeffreys, H., Theory of Probability, Clarendon Press, Oxford, 1939. (The third edn. appeared in 1961.)Google Scholar
39. Kalbfleisch, J. G. and Sprott, D. A., ‘On tests of significance’, in C. A. Hooker and W. Harper (eds.), Foundations of Probability Theory, Statistical Theory, Statistical Inference, and Statistical Theories of Science, Vol 2, D. Reidel, Dordrecht, Holland, 259–272, 1976.Google Scholar
40. Kempthorne, O and Folks, L., Probability, Statistics, and Data Analysis, Iowa State Univ. Press, 1971.Google Scholar
41. Kendall, M. G. and Stuart. A., The Advanced Theory of Statistics, Vol. 2, Charles Griffin, London, 1960.Google Scholar
42. Lehmann, E. L., Testing Statistical Hypotheses, Wiley, New York, 1959.Google Scholar
43. Lindley, D. V., ‘A statistical paradox’, Biometrika 44 (1957), 187–192.Google Scholar
44. Minsky, M. and Selfridge, O. G., ‘Learning in random nets’, in Colin Cherry (ed.), Information Theory, Butterworths, London, 1961, pp. 335–347.Google Scholar
45. Neyman, J., ‘Frequentist probability and frequentist statistics’, Syntheses 36 (1977), 97–131.
46. Neyman, J. and Pearson, E. S., ‘On the problem of the most efficient tests of statistical hypotheses’, Philosophical Transactions of the Royal Society of London, Series A 231 (1933), 289–337.
47. Patil, G. P., ‘On the evaluation of the negative binomial distribution with examples’, Technometrics 2 (1960), 501–505.
48. Peirce, C. S., ‘The probability of induction’, Popular Science Monthly (1878), reprinted in James R. Newman (ed.), The World of Mathematics, 2, Simon and Schuster, New York, 1956, pp. 1341–1354.Google Scholar
49. Rényi, A., ‘On measures of entropy and information’, Proc. Fourth Berkeley Sympos. Math. Statist. and Prob., Vol. 1, Univ. Press, Berkeley, Calif., pp. 547–561, 1961.Google Scholar
50. Savage L. J., ‘Elicitation of personal probabilities and expectations’, J. Amer. Statist. Assoc. 66 (1971), 783–801.
51. Todhunter, I., A History of the Mathematical theory of Probability, 1865. Reprint Chelsea Publishing Co., New York, 1949 and 1965.Google Scholar
52. Tullock, G., Private oral communication (1979).Google Scholar
53. Turing, A. M., Private communication (1941).Google Scholar
54. Weaver, W., ‘Probability, rarity, interest and surprise’, Scientific Monthly 67 (1948), 390–392.Google Scholar
55. Zellner, A. (ed.), Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys. North Holland, Amsterdam (1980).Google Scholar