The Annals of Statistics

Decomposition tables for experiments. II. Two–one randomizations

C. J. Brien and R. A. Bailey

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We investigate structure for pairs of randomizations that do not follow each other in a chain. These are unrandomized-inclusive, independent, coincident or double randomizations. This involves taking several structures that satisfy particular relations and combining them to form the appropriate orthogonal decomposition of the data space for the experiment. We show how to establish the decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated. This leads to recommendations for when the different types of multiple randomization should be used.

Article information

Ann. Statist. Volume 38, Number 5 (2010), 3164-3190.

First available in Project Euclid: 13 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J10: Analysis of variance and covariance
Secondary: 62K99: None of the above, but in this section

Analysis of variance balance decomposition table design of experiments efficiency factor intertier interaction multiphase experiments multitiered experiments orthogonal decomposition pseudofactor structure tier


Brien, C. J.; Bailey, R. A. Decomposition tables for experiments. II. Two–one randomizations. Ann. Statist. 38 (2010), no. 5, 3164--3190. doi:10.1214/09-AOS785.

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  • [1] Bailey, R. A. (1992). Efficient semi-Latin squares. Statist. Sinica 2 413–437.
  • [2] Bailey, R. A. (1996). Orthogonal partitions in designed experiments. Des. Codes Cryptogr. 8 45–77.
  • [3] Bailey, R. A. (2004). Association Schemes: Designed Experiments, Algebra and Combinatorics. Cambridge Studies in Advanced Mathematics 84. Cambridge Univ. Press, Cambridge.
  • [4] Bailey, R. A. and Speed, T. P. (1986). Rectangular lattice designs: Efficiency factors and analysis. Ann. Statist. 14 874–895.
  • [5] Bailey, T. B. (2006). Contribution to the discussion of “Multiple randomizations” by C. J. Brien and R. A. Bailey. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 604.
  • [6] Bose, R. C. and Mesner, D. M. (1959). On linear associative algebras corresponding to association schemes of partially balanced designs. Ann. Math. Statist. 30 21–38.
  • [7] Bose, R. C. and Shimamoto, T. (1952). Classification and analysis of partially balanced incomplete block designs with two associate classes. J. Amer. Statist. Assoc. 47 151–284.
  • [8] Brien, C. J. (1983). Analysis of variance tables based on experimental structure. Biometrics 39 53–59.
  • [9] Brien, C. J. and Bailey, R. A. (2006). Multiple randomizations (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 68 571–609.
  • [10] Brien, C. J. and Bailey, R. A. (2009). Decomposition tables for experiments I. A chain of randomizations. Ann. Statist. 37 4184–4213.
  • [11] Brien, C. J. and Demétrio, C. G. B. (2009). Formulating mixed models for experiments, including longitudinal experiments. J. Agric. Biol. Environ. Stat. 14 253–280.
  • [12] Brien, C. J. and Payne, R. W. (1999). Tiers, structure formulae and the analysis of complicated experiments. The Statistician 48 41–52.
  • [13] Cochran, W. G. and Cox, G. M. (1957). Experimental Designs, 2nd ed. Wiley, New York.
  • [14] Eccleston, J. and Russell, K. (1975). Connectedness and orthogonality in multi-factor designs. Biometrika 62 341–345.
  • [15] James, A. T. and Wilkinson, G. N. (1971). Factorization of the residual operator and canonical decomposition of nonorthogonal factors in the analysis of variance. Biometrika 58 279–294.
  • [16] Patterson, H. D. and Thompson, R. (1971). Recovery of inter-block information when block sizes are unequal. Biometrika 58 545–554.
  • [17] Tjur, T. (1984). Analysis of variance models in orthogonal designs (with discussion). Internat. Statist. Rev. 52 33–81.
  • [18] Wood, J. T., Williams, E. R. and Speed, T. P. (1988). Non-orthogonal block structure in two-phase designs. Aust. J. Stat. 30A 225–237.
  • [19] Yates, F. (1935). Complex experiments (with discussion). J. R. Statist. Soc. Suppl. 2 181–247.