Randomized Experiments

  • Paul R. Rosenbaum
Part of the Springer Series in Statistics book series (SSS)


Observational studies and controlled experiments have the same goal, inference about treatment effects, but random assignment of treatments is present only in experiments. This chapter reviews the role of randomization in experiments, and so prepares for discussion of observational studies in later chapters. A theory of observational studies must have a clear view of the role of randomization, so it can have an equally clear view of the consequences of its absence. Sections 2.1 and 2.2 give two examples: a large controlled clinical trial, and then a small but famous example due to Sir Ronald Fisher, who is usually credited with the invention of randomization, which he called the “reasoned basis for inference” in experiments. Later sections discuss the meaning of this phrase, that is, the link between randomization and statistical methods. Most of the material in this chapter is quite old.


Partial Order Treatment Assignment Multivariate Response Isotonic Function Adjusted Response 
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  1. Ahlswede, R. and Daykin, D. (1978) An inequality for the weights of two families of sets, their unions, and intersections. Z. Wahrsch. Verus Gebiete, 43, 183–185.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Anderson, I. (1987) Combinatorics of Finite Sets. New York: Oxford University Press.zbMATHGoogle Scholar
  3. Birch, M. W. (1964) The detection of partial association, I: The 2 × 2 case. Journal of the Royal Statistical Society, Series B,26, 313–324.MathSciNetzbMATHGoogle Scholar
  4. Birch, M. W. (1965) The detection of partial association, II: The general case. Journal of the Royal Statistical Society, Series B,27, 111–124.MathSciNetzbMATHGoogle Scholar
  5. Bollobas, B. (1986) Combinatorics. New York: Cambridge University Press.zbMATHGoogle Scholar
  6. Campbell, D. and Stanley, J. (1963) Experimental and Quasi-Experimental Designs for Research. Chicago: Rand McNally.Google Scholar
  7. Cochran, W. G. (1963) Sampling Techniques. New York: Wiley.Google Scholar
  8. Cox, D. R. (1958a) Planning of Experiments. New York: Wiley.zbMATHGoogle Scholar
  9. Cox, D. R. (1958b) The interpretation of the effects of non-additivity in the Latin square. Biometrika, 45, 69–73.zbMATHGoogle Scholar
  10. Cox, D. R. (1966) A simple example of a comparison involving quantal data. Biometrika, 53, 215–220.MathSciNetCrossRefGoogle Scholar
  11. Cox, D. R. (1970) The Analysis of Binary Data. London: Methuen.zbMATHGoogle Scholar
  12. Cox, D. R. and Hinkley, D.V. (1974) Theoretical Statistics. London: Chapman & Hall.zbMATHGoogle Scholar
  13. Cox, D. R. and Reid, N. (2000) The Theory of the Design of Experiments. New York: CRC Press.zbMATHGoogle Scholar
  14. Eaton, M. (1967) Some optimum properties of ranking procedures. Annals of Mathematical Statistics, 38, 124–137.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Eaton, M. (1982) A review of selected topics in probability inequalities. Annals of Statistics, 10, 11–43.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Eaton, M. (1987) Lectures on Topics in Probability Inequalities. Amsterdam: Centrum. voor Wiskunde en Informatica.zbMATHGoogle Scholar
  17. Efron, B. (1971) Forcing a sequential experiment to be balanced. Biometrika, 58, 403–417.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Fisher, R. A. (1935, 1949) The Design of Experiments. Edinburgh: Oliver & Boyd.Google Scholar
  19. Fortuin, C., Kasteleyn, P., and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Communications in Mathematical Physics, 22, 89–103.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Freidlin, B. and Gastwirth, J. L. (2000) Should the median test be retired from general use? American Statistician, 54, 161–164.Google Scholar
  21. Friedman, L. M., DeMets, D. L., and Furberg, C. D. (1998) Fundamentals of Clinical Trials. New York: Springer-Verlag.Google Scholar
  22. Gastwirth, J. L. (1966) On robust procedures. Journal of the American Statistical Association, 61, 929–948.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Gehan, E. (1965) A generalized Wilcoxon test for comparing arbitrarily singly censored samples. Biometrika, 52, 203–223.MathSciNetzbMATHGoogle Scholar
  24. Gibbons, J. D. (1982) Brown-Mood median test. In: Encyclopedia of Statistical Sciences, Volume 1, S. Kotz and N. Johnson, eds., New York: Wiley, pp. 322–324.Google Scholar
  25. Hamilton, M. (1979) Choosing a parameter for 2 × 2 table or 2 × 2 × 2 table analysis. American Journal of Epidemiology, 109, 362–375.Google Scholar
  26. Hettmansperger, T. (1984) Statistical Inference Based on Ranks. New York: Wiley.zbMATHGoogle Scholar
  27. Hodges, J. and Lehmann, E. (1962) Rank methods for combination of independent experiments in the analysis of variance. Annals of Mathematical Statistics, 33, 482–497.MathSciNetzbMATHCrossRefGoogle Scholar
  28. Hodges, J. and Lehmann, E. (1963) Estimates of location based on rank tests. Annals of Mathematical Statistics, 34, 598–611.MathSciNetzbMATHCrossRefGoogle Scholar
  29. Holland, P. (1986) Statistics and causal inference (with discussion) . Journal of the American Statistical Association, 81, 945–970.MathSciNetzbMATHCrossRefGoogle Scholar
  30. Hollander, M., Proschan, F., and Sethuraman, J. (1977) Functions decreasing in transposition and their applications in ranking problems. Annals of Statistics, 5, 722–733.MathSciNetzbMATHCrossRefGoogle Scholar
  31. Hollander, M. and Wolfe, D. (1973) Nonparametric Statistical Methods. New York: Wiley.zbMATHGoogle Scholar
  32. Holley, R. (1974) Remarks on the FKG inequalities. Communications in Mathematical Physics, 36, 227–231.MathSciNetCrossRefGoogle Scholar
  33. Jureckova, J. (1984) M-, L- and R-estimators. In: Handbook of Statistics, Volume IV, P. R. Krishnaiah and P. K. Sen, eds., New York: Elsevier, pp. 463–485.Google Scholar
  34. Kempthorne, O. (1952) The Design and Analysis of Experiments. New York: Wiley.zbMATHGoogle Scholar
  35. Krieger, A. M. and Rosenbaum, P. R. (1994) A stochastic comparison for arangement increasing functions. Combinatorics, Probability and Computing, 3, 345–348.MathSciNetzbMATHCrossRefGoogle Scholar
  36. Lehmann, E. L. (1959) Testing Statistical Hypotheses. New York: Wiley.zbMATHGoogle Scholar
  37. Lehmann, E. L. (1975) Nonparametrics: Statistical Methods Based on Ranks. San Francisco: Holden-Day.zbMATHGoogle Scholar
  38. MacLane, S. and Birkoff, G. (1988) Algebra. New York: Chelsea.zbMATHGoogle Scholar
  39. Mann, H. and Whitney, D. (1947) On a test of whether one of two random variables is stochastically larger than the other. Annals of Mathematical Statistics, 18, 50–60.MathSciNetzbMATHCrossRefGoogle Scholar
  40. Mantel, N. (1963) Chi-square tests with one degree of freedom: Extensions of the Mantel-Haenszel procedure. Journal of the American Statistical Association, 58, 690–700.MathSciNetzbMATHGoogle Scholar
  41. Mantel, N. (1967) Ranking procedures for arbitrarily restricted observations. Biometrics, 23, 65–78.CrossRefGoogle Scholar
  42. Mantel, N. and Haenszel, W. (1959) Statistical aspects of retrospective studies of disease. Journal of the National Cancer Institute, 22, 719–748.Google Scholar
  43. Maritz, J. (1981) Distribution-Free Statistical Methods. London: Chapman & Hall.zbMATHGoogle Scholar
  44. Marshall, A. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications. New York: Academic.zbMATHGoogle Scholar
  45. McNemar, Q. (1947) Note on the sampling error of the differences between correlated proportions or percentage. Psychometrika, 12, 153–157.CrossRefGoogle Scholar
  46. Murphy, M., Hultgren, H., Detre, K., Thomsen, J., and Takaro, T. (1977) Treatment of chronic stable angina: A preliminary report of survival data of the randomized Veterans Administration Cooperative study. New England Journal of Medicine, 297, 621–627.CrossRefGoogle Scholar
  47. Neyman, J. (1923) On the application of probability theory to agricultural experiments. Essay on principles. Section 9. (In Polish) Roczniki Nauk Roiniczych, Tom X, pp. 1–51Google Scholar
  48. Reprinted in Statistical Science 1990, 5, 463–480, with discussion by T. Speed and D. Rubin.MathSciNetzbMATHGoogle Scholar
  49. Neyman, J. (1935) Statistical problems in agricultural experimentation. Supplement to the Journal of the Royal Statistical Society, 2, 107–180.zbMATHCrossRefGoogle Scholar
  50. Pagano, M. and Tritchler, D. (1983) Obtaining permutation distributions in polynomial time. Journal of the American Statistical Association, 78, 435–440.MathSciNetzbMATHCrossRefGoogle Scholar
  51. Robinson, J. (1973) The large sample power of permutation tests for randomization models. Annals of Statistics, 1, 291–296.MathSciNetzbMATHCrossRefGoogle Scholar
  52. Rosenbaum, P. R. (1988) Sensitivity analysis for matching with multiple controls. Biometrika, 75, 577–581.MathSciNetzbMATHCrossRefGoogle Scholar
  53. Rosenbaum, P. R. (1989) On permutation tests for hidden biases in observational studies: An application of Holley’s inequality to the Savage lattice. Annals of Statistics, 17, 643–653.MathSciNetzbMATHCrossRefGoogle Scholar
  54. Rosenbaum, P. R. (1991) Some poset statistics. Annals of Statistics, 19, 1091–1097.MathSciNetzbMATHCrossRefGoogle Scholar
  55. Rosenbaum, P. R. (1994) Coherence in observational studies. Biometrics, 50, 368–374.zbMATHCrossRefGoogle Scholar
  56. Rosenbaum, P. R. (1995) Quantiles in nonrandom samples and observational studies. Journal of the American Statistical Association, 90, 1424–1431.MathSciNetzbMATHCrossRefGoogle Scholar
  57. Rosenbaum, P. R. (1999) Holley’s inequality. Encyclopedia of Statistical Sciences, Update Volume 3, S. Kotz, C. B. Read, D. L. Banks, eds., New York: Wiley, pp. 328–331.Google Scholar
  58. Rubin, D. B. (1974) Estimating the causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology, 66, 688–701.CrossRefGoogle Scholar
  59. Rubin, D. B. (1977) Assignment to treatment group on the basis of a covariate. Journal of Educational Statistics, 2, 1–26.CrossRefGoogle Scholar
  60. Rubin, D. B. (1986) Which ifs have causal answers? Journal of the American Statistical Association, 81, 961–962.Google Scholar
  61. Savage, I. R. (1957) Contributions to the theory of rank order statistics: The trend case. Annals of Mathematical Statistics, 28, 968–977.MathSciNetzbMATHCrossRefGoogle Scholar
  62. Savage, I. R. (1964) Contributions to the theory of rank order statistics: Applications of lattice theory. Review of the International Statistical Institute, 32, 52–63.MathSciNetzbMATHCrossRefGoogle Scholar
  63. Tukey, J. W. (1985) Improving crucial randomized experiments-especially in weather modification-by double randomization and rank combination. In: Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, L. Le Cam and R. Olshen, eds., Volume 1, Belmont, CA: Wadsworth, pp. 79–108.Google Scholar
  64. Welch, B. L. (1937) On the z-test in randomized blocks and Latin squares. Biometrika, 29, 21–52.zbMATHGoogle Scholar
  65. Wilcoxon, F. (1945) Individual comparisons by ranking methods. Biometrics, 1, 8083.Google Scholar
  66. Wilk, M. B. (1955) The randomization analysis of a generalized randomized block design. Biometrika, 42, 70–79.MathSciNetzbMATHGoogle Scholar
  67. Wittgenstein, L. (1958) Philosophical Investigations (Third Edition) . Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  68. Zelen, M. (1974) The randomization and stratification of patients to clinical trials. Journal of Chronic Diseases, 27, 365–375.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Paul R. Rosenbaum
    • 1
  1. 1.Department of Statistics, The Wharton SchoolUniversity of PennsylvaniaPhiladelphiaUSA

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