Observational Studies pp 19-70 | Cite as

# Randomized Experiments

## Abstract

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.

## Keywords

Partial Order Treatment Assignment Multivariate Response Isotonic Function Adjusted Response## Preview

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## References

- 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.MathSciNetMATHCrossRefGoogle Scholar - Anderson, I. (1987)
*Combinatorics of Finite Sets*. New York: Oxford University Press.MATHGoogle Scholar - 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.MathSciNetMATHGoogle Scholar - Birch, M. W. (1965) The detection of partial association, II: The general case.
*Journal of the Royal Statistical Society*, Series B,**27**, 111–124.MathSciNetMATHGoogle Scholar - Bollobas, B. (1986)
*Combinatorics*. New York: Cambridge University Press.MATHGoogle Scholar - Campbell, D. and Stanley, J. (1963)
*Experimental and Quasi-Experimental Designs for Research*. Chicago: Rand McNally.Google Scholar - Cochran, W. G. (1963)
*Sampling Techniques*. New York: Wiley.Google Scholar - Cox, D. R. (1958a)
*Planning of Experiments*. New York: Wiley.MATHGoogle Scholar - Cox, D. R. (1958b) The interpretation of the effects of non-additivity in the Latin square.
*Biometrika*,**45**, 69–73.MATHGoogle Scholar - Cox, D. R. (1966) A simple example of a comparison involving quantal data.
*Biometrika*,**53**, 215–220.MathSciNetCrossRefGoogle Scholar - Cox, D. R. (1970)
*The Analysis of Binary Data*. London: Methuen.MATHGoogle Scholar - Cox, D. R. and Hinkley, D.V. (1974)
*Theoretical Statistics*. London: Chapman & Hall.MATHGoogle Scholar - Cox, D. R. and Reid, N. (2000)
*The Theory of the Design of Experiments*. New York: CRC Press.MATHGoogle Scholar - Eaton, M. (1967) Some optimum properties of ranking procedures.
*Annals of Mathematical Statistics*,**38**, 124–137.MathSciNetMATHCrossRefGoogle Scholar - Eaton, M. (1982) A review of selected topics in probability inequalities.
*Annals of Statistics*,**10**, 11–43.MathSciNetMATHCrossRefGoogle Scholar - Eaton, M. (1987)
*Lectures on Topics in Probability Inequalities*. Amsterdam: Centrum. voor Wiskunde en Informatica.MATHGoogle Scholar - Efron, B. (1971) Forcing a sequential experiment to be balanced.
*Biometrika*,**58**, 403–417.MathSciNetMATHCrossRefGoogle Scholar - Fisher, R. A. (1935, 1949)
*The Design of Experiments*. Edinburgh: Oliver & Boyd.Google Scholar - Fortuin, C., Kasteleyn, P., and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets.
*Communications in Mathematical Physics*,**22**, 89–103.MathSciNetMATHCrossRefGoogle Scholar - Freidlin, B. and Gastwirth, J. L. (2000) Should the median test be retired from general use?
*American Statistician*,**54**, 161–164.Google Scholar - Friedman, L. M., DeMets, D. L., and Furberg, C. D. (1998)
*Fundamentals of Clinical Trials*. New York: Springer-Verlag.Google Scholar - Gastwirth, J. L. (1966) On robust procedures.
*Journal of the American Statistical Association*,**61**, 929–948.MathSciNetMATHCrossRefGoogle Scholar - Gehan, E. (1965) A generalized Wilcoxon test for comparing arbitrarily singly censored samples.
*Biometrika*,**52**, 203–223.MathSciNetMATHGoogle Scholar - 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 - 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 - Hettmansperger, T. (1984)
*Statistical Inference Based on Ranks*. New York: Wiley.MATHGoogle Scholar - 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.MathSciNetMATHCrossRefGoogle Scholar - Hodges, J. and Lehmann, E. (1963) Estimates of location based on rank tests.
*Annals of Mathematical Statistics*,**34**, 598–611.MathSciNetMATHCrossRefGoogle Scholar - Holland, P. (1986) Statistics and causal inference (with discussion) .
*Journal of the American Statistical Association*,**81**, 945–970.MathSciNetMATHCrossRefGoogle Scholar - Hollander, M., Proschan, F., and Sethuraman, J. (1977) Functions decreasing in transposition and their applications in ranking problems.
*Annals of Statistics*,**5**, 722–733.MathSciNetMATHCrossRefGoogle Scholar - Hollander, M. and Wolfe, D. (1973)
*Nonparametric Statistical Methods*. New York: Wiley.MATHGoogle Scholar - Holley, R. (1974) Remarks on the FKG inequalities.
*Communications in Mathematical Physics*,**36**, 227–231.MathSciNetCrossRefGoogle Scholar - 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 - Kempthorne, O. (1952)
*The Design and Analysis of Experiments*. New York: Wiley.MATHGoogle Scholar - Krieger, A. M. and Rosenbaum, P. R. (1994) A stochastic comparison for arangement increasing functions.
*Combinatorics, Probability and Computing*,**3**, 345–348.MathSciNetMATHCrossRefGoogle Scholar - Lehmann, E. L. (1959)
*Testing Statistical Hypotheses*. New York: Wiley.MATHGoogle Scholar - Lehmann, E. L. (1975)
*Nonparametrics: Statistical Methods Based on Ranks*. San Francisco: Holden-Day.MATHGoogle Scholar - MacLane, S. and Birkoff, G. (1988)
*Algebra*. New York: Chelsea.MATHGoogle Scholar - 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.MathSciNetMATHCrossRefGoogle Scholar - 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.MathSciNetMATHGoogle Scholar - Mantel, N. (1967) Ranking procedures for arbitrarily restricted observations.
*Biometrics*,**23**, 65–78.CrossRefGoogle Scholar - Mantel, N. and Haenszel, W. (1959) Statistical aspects of retrospective studies of disease.
*Journal of the National Cancer Institute*,**22**, 719–748.Google Scholar - Maritz, J. (1981)
*Distribution-Free Statistical Methods*. London: Chapman & Hall.MATHGoogle Scholar - Marshall, A. and Olkin, I. (1979)
*Inequalities: Theory of Majorization and Its Applications*. New York: Academic.MATHGoogle Scholar - McNemar, Q. (1947) Note on the sampling error of the differences between correlated proportions or percentage.
*Psychometrika*,**12**, 153–157.CrossRefGoogle Scholar - 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 - 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 - Reprinted in
*Statistical Science*1990,**5**, 463–480, with discussion by T. Speed and D. Rubin.MathSciNetMATHGoogle Scholar - Neyman, J. (1935) Statistical problems in agricultural experimentation.
*Supplement to the Journal of the Royal Statistical Society*,**2**, 107–180.MATHCrossRefGoogle Scholar - Pagano, M. and Tritchler, D. (1983) Obtaining permutation distributions in polynomial time.
*Journal of the American Statistical Association*,**78**, 435–440.MathSciNetMATHCrossRefGoogle Scholar - Robinson, J. (1973) The large sample power of permutation tests for randomization models.
*Annals of Statistics*,**1**, 291–296.MathSciNetMATHCrossRefGoogle Scholar - Rosenbaum, P. R. (1988) Sensitivity analysis for matching with multiple controls.
*Biometrika*,**75**, 577–581.MathSciNetMATHCrossRefGoogle Scholar - 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.MathSciNetMATHCrossRefGoogle Scholar - Rosenbaum, P. R. (1991) Some poset statistics.
*Annals of Statistics*,**19**, 1091–1097.MathSciNetMATHCrossRefGoogle Scholar - Rosenbaum, P. R. (1994) Coherence in observational studies.
*Biometrics*,**50**, 368–374.MATHCrossRefGoogle Scholar - Rosenbaum, P. R. (1995) Quantiles in nonrandom samples and observational studies.
*Journal of the American Statistical Association*,**90**, 1424–1431.MathSciNetMATHCrossRefGoogle Scholar - 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 - Rubin, D. B. (1974) Estimating the causal effects of treatments in randomized and nonrandomized studies.
*Journal of Educational Psychology*,**66**, 688–701.CrossRefGoogle Scholar - Rubin, D. B. (1977) Assignment to treatment group on the basis of a covariate.
*Journal of Educational Statistics*,**2**, 1–26.CrossRefGoogle Scholar - Rubin, D. B. (1986) Which ifs have causal answers?
*Journal of the American Statistical Association*,**81**, 961–962.Google Scholar - Savage, I. R. (1957) Contributions to the theory of rank order statistics: The trend case.
*Annals of Mathematical Statistics*,**28**, 968–977.MathSciNetMATHCrossRefGoogle Scholar - 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.MathSciNetMATHCrossRefGoogle Scholar - 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 - Welch, B. L. (1937) On the z-test in randomized blocks and Latin squares.
*Biometrika*,**29**, 21–52.MATHGoogle Scholar - Wilk, M. B. (1955) The randomization analysis of a generalized randomized block design.
*Biometrika*,**42**, 70–79.MathSciNetMATHGoogle Scholar - Wittgenstein, L. (1958)
*Philosophical Investigations*(Third Edition) . Englewood Cliffs, NJ: Prentice-Hall.Google Scholar - Zelen, M. (1974) The randomization and stratification of patients to clinical trials.
*Journal of Chronic Diseases*,**27**, 365–375.MathSciNetCrossRefGoogle Scholar