Skip to main content
SpringerLink
Log in

Experiments in Rectangular Areas: Design and Randomization

  • Published:
Journal of Agricultural, Biological, and Environmental Statistics Aims and scope Submit manuscript

Abstract

Agricultural experiments are often laid out in a rectangle in 3–5 replicates. Is it better to use a standard randomized complete-block design in rows, or a complete-block design in rows but with restricted randomization, or an efficient row-column design? These approaches differ in the variance of the estimator of a difference between two treatments, and in the bias of the estimator of that variance, as well as in the mechanics of constructing the design and analyzing the data. I conclude that when intra-column correlations are high then the row-column design is best but that when they are moderate the best procedure is to use an improved version of restricted randomization, which gives an unbiased estimator of the average variance in the single experiment performed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bailey, R. A. (1981), “A Unified Approach to Design of Experiments,” Journal of the Royal Statistical Society, Series A, 144, 214–223.

    Article  MATH  Google Scholar 

  • — (1983), “Restricted Randomization,” Biometrika, 70, 183–198.

    Article  MathSciNet  MATH  Google Scholar 

  • — (1985a), “Restricted Randomization Versus Blocking,” International Statistical Review, 53, 171–182.

    Article  MathSciNet  Google Scholar 

  • Bailey, R. A. (1985b), “Restricted Randomization for Neighbour-Balanced Designs,” Statistics & Decisions, Supplement 2, 237–248.

    Google Scholar 

  • — (1987), “One-Way Blocks in Two-Way Layouts,” Biometrika, 74, 27–32.

    Article  MathSciNet  MATH  Google Scholar 

  • — (1991), “Strata for Randomized Experiments (with discussion),” Journal of the Royal Statistical Society, Series B, 53, 27–78.

    MATH  Google Scholar 

  • Bailey, R. A., and Nelson, P. R. (2003), “Hadamard Randomization: A Valid Restriction of Random Permuted Blocks,” Biometrical Journal, 45, 554–560.

    Article  MathSciNet  Google Scholar 

  • Bailey, R. A., and Rowley, C. A. (1987), “Valid Randomization,” Proceedings of the Royal Society of London, Series A, 410, 105–124.

    Article  MATH  Google Scholar 

  • Bowman, D. T. (2000), “TFPlan: Software for Restricted Randomization in Field Plot Design,” Agronomy Journal, 92, 1276–1278.

    Article  Google Scholar 

  • Caliński, T., and Kageyama, S. (2000), Block Designs: A Randomization Approach. Volume I: Analysis, New York: Springer.

    Google Scholar 

  • Cheng, C.-S. (1978), “Optimality of Certain Asymmetrical Experimental Designs,” Annals of Statistics, 6, 1239–1261.

    Article  MathSciNet  MATH  Google Scholar 

  • Cheng, C.-S., and Bailey, R. A. (1991), “Optimality of Some Two-Associate-Class Partially Balanced Incomplete-Block Designs,” Annals of Statistics, 19, 1667–1671.

    Article  MathSciNet  MATH  Google Scholar 

  • Cox, D. R. (1958), Planning of Experiments, New York: John Wiley & Sons.

    MATH  Google Scholar 

  • Federer, W. T. (1955), Experimental Design, New York: Macmillan.

    Google Scholar 

  • Fisher, R. A. (1935), Design of Experiments, Edinburgh: Oliver and Boyd.

    Google Scholar 

  • Gilmour, S. G., and Goos, P. (2009), “Analysis of Data From Nonorthogonal Multi-Stratum Designs,” Journal of the Royal Statistical Society, Series C, 58, 467–484.

    Article  MathSciNet  Google Scholar 

  • Grundy, P. M., and Healy, M. J. R. (1950), “Restricted Randomization and Quasi-Latin Squares,” Journal of the Royal Statistical Society, Series B, 12, 286–291.

    MATH  Google Scholar 

  • Hall, P. (1935), “On Representatives of Subsets,” Journal of the London Mathematical Society, 10, 26–30.

    Article  Google Scholar 

  • John, J. A., and Mitchell, T. J. (1977), “Optimal Incomplete Block Designs,” Journal of the Royal Statistical Society, Series B, 39, 39–43.

    MathSciNet  MATH  Google Scholar 

  • Kempthorne, O. (1961), “Chapter VIII: Constrained Randomization,” in Analysis of Variance Procedures, ARL149, eds. O. Kempthorne, G. Zyskind, S. Addelman, T. N. Throckmorton, and R. F. White, Ohio: Aeronautical Research Lab Office of Aerospace Research, U.S. Air Force, Wright-Patterson Air Force Base, pp. 190–202.

    Google Scholar 

  • Kshirsagar, A. M. (1958), “A Note on Incomplete Block Designs,” Annals of Mathematical Statistics, 29, 907–910.

    Article  MathSciNet  MATH  Google Scholar 

  • Monod, H., Azaïs, J.-M., and Bailey, R. A. (1996), “Valid Randomisation for the First Difference Analysis,” Australian Journal of Statistics, 33, 91–106.

    Article  Google Scholar 

  • Monod, H., and Bailey, R. A. (1993), “Valid Restricted Randomization for Unbalanced Designs,” Journal of the Royal Statistical Society, Series B, 55, 237–251.

    MathSciNet  MATH  Google Scholar 

  • Nelder, J. A. (1965), “The Analysis of Randomized Experiments with Orthogonal Block Structure. I. Block Structure and the Null Analysis of Variance,” Proceedings of the Royal Society of London, Series A, 283, 147–162.

    Article  MathSciNet  MATH  Google Scholar 

  • — (1968), “The Combination of Information in Generally Balanced Designs,” Journal of the Royal Statistical Society, Series B, 30, 303–311.

    MathSciNet  MATH  Google Scholar 

  • Patterson, H. D., and Thompson, R. (1971), “Recovery of Inter-Block Information When Block Sizes Are Unequal,” Biometrika, 58, 545–554.

    Article  MathSciNet  MATH  Google Scholar 

  • Shah, K., and Sinha, B. K. (1989), Theory of Optimal Designs, New York: Springer.

    Book  MATH  Google Scholar 

  • Speed, T. P. (1987), “What Is an Analysis of Variance?,” Annals of Statistics, 15, 885–910.

    Article  MathSciNet  MATH  Google Scholar 

  • Whitaker, D. E. R., Williams, E. R., and John, J. A. (2006), “CycDesigN: A Package for the Computer Generation Of Experimental Designs,” http://cycdesign.co.nz.

  • White, L. V., and Welch, W. J. (1981), “A Method for Constructing Valid Restricted Randomization Schemes Using the Theory of D-Optimal Design of Experiments,” Journal of the Royal Statistical Society, Series B, 43, 167–172.

    MathSciNet  MATH  Google Scholar 

  • Yates, F. (1939), “The Comparative Advantages of Systematic and Randomized Arrangements in the Design of Agricultural and Biological Experiments,” Biometrika, 30, 440–466.

    Google Scholar 

  • — (1948), “Contribution to the Discussion of ‘The Validity of Comparative Experiments’ by F. J. Anscombe,” Journal of the Royal Statistical Society, Series A, 111, 204–205.

    Google Scholar 

  • Youden, W. J. (1972), “Randomization and Experimentation,” Technometrics, 14, 13–22.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, UK

    R. A. Bailey

Authors
  1. R. A. Bailey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. A. Bailey.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bailey, R.A. Experiments in Rectangular Areas: Design and Randomization. JABES 17, 176–191 (2012). https://doi.org/10.1007/s13253-011-0082-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13253-011-0082-6

Key Words

Search

Navigation