The Annals of Statistics

Decomposition tables for experiments I. A chain of randomizations

C. J. Brien and R. A. Bailey

Full-text: Open access

Abstract

One aspect of evaluating the design for an experiment is the discovery of the relationships between subspaces of the data space. Initially we establish the notation and methods for evaluating an experiment with a single randomization. Starting with two structures, or orthogonal decompositions of the data space, we describe how to combine them to form the overall decomposition for a single-randomization experiment that is “structure balanced.” The relationships between the two structures are characterized using efficiency factors. The decomposition is encapsulated in a decomposition table. Then, for experiments that involve multiple randomizations forming a chain, we take several structures that pairwise are structure balanced and combine them to establish the form of the orthogonal decomposition for the experiment. In particular, it is proven that the properties of the design for such an experiment are derived in a straightforward manner from those of the individual designs. We show how to formulate an extended decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated.

Article information

Source
Ann. Statist. Volume 37, Number 6B (2009), 4184-4213.

Dates
First available in Project Euclid: 23 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1256303541

Digital Object Identifier
doi:10.1214/09-AOS717

Mathematical Reviews number (MathSciNet)
MR2572457

Zentralblatt MATH identifier
1191.62139

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

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

Citation

Brien, C. J.; Bailey, R. A. Decomposition tables for experiments I. A chain of randomizations. Ann. Statist. 37 (2009), no. 6B, 4184--4213. doi:10.1214/09-AOS717. https://projecteuclid.org/euclid.aos/1256303541


Export citation

References

  • [1] Bailey, R. A. (1981). A unified approach to design of experiments. J. Roy. Statist. Soc. Ser. A 144 214–223.
  • [2] Bailey, R. A. (1982). Block structures for designed experiments. In Applications of Combinatorics (R. J. Wilson, ed.) 1–18. Shiva, Nantwich.
  • [3] Bailey, R. A. (1984). Contribution to the discussion of “Analysis of variance models in orthogonal designs” by T. Tjur. Internat. Statist. Rev. 52 65–77.
  • [4] Bailey, R. A. (1989). Designs: mappings between structured sets. In Surveys in Combinatorics, 1989 (J. Siemons, ed.). Lond. Math. Soc. Lect. Note Ser. 141 22–51. Cambridge Univ. Press, Cambridge.
  • [5] Bailey, R. A. (1994). General balance: artificial theory or practical relevance? In Proceedings of the International Conference on Linear Statistical Inference LINSTAT ’93 (T. Caliński and R. Kala, eds.) 171–184. Kluwer, Dordrecht.
  • [6] Bailey, R. A. (1996). Orthogonal partitions in designed experiments. Des. Codes Cryptogr. 8 45–77.
  • [7] Bailey, R. A. (2004). Association Schemes: Designed Experiments, Algebra and Combinatorics. Cambridge Univ. Press, Cambridge.
  • [8] Bailey, R. A. (2004). Principles of designed experiments in J. A. Nelder’s papers. In Methods and Models in Statistics: In Honour of Professor John Nelder FRS (N. M. Adams, M. J. Crowder, D. J. Hand and D. A. Stephens, eds.) 171–194. Imperial College Press, London.
  • [9] Bailey, R. A. (2008). Design of Comparative Experiments. Cambridge Univ. Press, Cambridge.
  • [10] Birkhoff, G. and Maclane, S. (1965). A Survey of Modern Algebra, 3rd ed. Macmillan, New York.
  • [11] Brien, C. J. (1983). Analysis of variance tables based on experimental structure. Biometrics 39 51–59.
  • [12] Brien, C. J. (1989). A model comparison approach to linear models. Util. Math. 36 225–254.
  • [13] Brien, C. J. (1992). Factorial linear model analysis. Ph.D. thesis, Dept. Plant Science, Univ. Adelaide, Adelaide, South Australia. Available at http://thesis.library.adelaide.edu.au/adt-SUA/public/adt-SUA20010530.175833/index.html (accessed July 9, 2008).
  • [14] Brien, C. J. and Bailey, R. A. (2006). Multiple randomizations (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 68 571–609.
  • [15] Brien, C. J. and Payne, R. W. (1999). Tiers, structure formulae and the analysis of complicated experiments. The Statistician 48 41–52.
  • [16] Brien, C. J. and Payne, R. W. (2008). AMTIER Procedure. In GenStat Reference Manual Release 11, Part 3. Procedure Library PL 19 (R. W. Payne and P. W. Lane, eds.) 84–88. VSN International, Hemel Hempstead.
  • [17] Cox, D. R. (1958). Planning of Experiments. Wiley, New York.
  • [18] Eccleston, J. A. and Russell, K. G. (1975). Connectedness and orthogonality in multi-factor designs. Biometrika 62 341–345.
  • [19] Federer, W. T. (1975). The misunderstood split plot. In Applied Statistics (R. P. Gupta, ed.) 9–39. North Holland, Amsterdam.
  • [20] Fisher, R. A. (1935). Contribution to the discussion of “Complex experiments” by F. Yates. J. Roy. Statist. Soc. Suppl. 2 229–231.
  • [21] Fisher, R. A. (1935). Design of Experiments. Oliver and Boyd, Edinburgh.
  • [22] Heiberger, R. M. (1989). Computation for the Analysis of Designed Experiments. Wiley, New York.
  • [23] Hinkelmann, K. and Kempthorne, O. (2008). Design and Analysis of Experiments: Volume I: Introduction to Experimental Design, 2nd ed. Wiley, New York.
  • [24] Houtman, A. M. and Speed, T. P. (1983). Balance in designed experiments with orthogonal block structure. Ann. Statist. 11 1069–1085.
  • [25] Insightful Corporation. (2007). S-PLUS 8.0 for Windows. Insightful Corporation, Seattle, Washington.
  • [26] 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.
  • [27] McIntyre, G. A. (1955). Design and analysis of two phase experiments. Biometrics 11 324–334.
  • [28] Nelder, J. A. (1965). The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 283 147–162.
  • [29] Nelder, J. A. (1965). The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 283 163–178.
  • [30] Nelder, J. A. (1968). The combination of information in generally balanced designs. J. Roy. Statist. Soc. Ser. B 30 303–311.
  • [31] Nelder, J. A. (1977). A reformulation of linear models. J. Roy. Statist. Soc. Ser. A 140 48–76.
  • [32] Payne, R. W., Harding, S. A., Murray, D. A., Soutar, D. M., Baird, D. B., Glaser, A. I., Channing, I. C., Welham, S. J., Gilmour, A. R., Thompson, R. and Webster, R. (2008). The Guide to GenStat Release 11, Part 2 Statistics. VSN International, Hemel Hempstead.
  • [33] Payne, R. W. and Tobias, R. D. (1992). General balance, combination of information and the analysis of covariance. Scand. J. Statist. 19 3–23.
  • [34] Payne, R. W. and Wilkinson, G. N. (1977). A general algorithm for analysis of variance. J. Roy. Statist. Soc. Ser. C 26 251–260.
  • [35] Pearce, S. C. (1983). The Agricultural Field Experiment. Wiley, Chichester.
  • [36] R Core Development Team (2008). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna.
  • [37] Tjur, T. (1984). Analysis of variance models in orthogonal designs. Internat. Statist. Rev. 52 33–81.
  • [38] Wilk, M. B. and Kempthorne, O. (1956). Some aspects of the analysis of factorial experiments in a completely randomized design. Ann. Math. Statist. 27 950–985.
  • [39] Wilkinson, G. N. (1970). A general recursive procedure for analysis of variance. Biometrika 57 19–46.
  • [40] 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.