Abstract
When generating experimental designs for field trials laid out on a rectangular grid of plots, it is useful to allow for blocking in both rows and columns. When the design is nonresolvable, randomized classical row–column designs may occasionally involve clustered placement of several replications of a treatment. In our experience, this feature prevents the more frequent use of these useful designs in practice. Practitioners often prefer a more even distribution of treatment replications. In this paper we illustrate how spatial variance–covariance structures can be used to achieve a more even distribution of treatment replications across the field and how such designs compare with classical row–column designs in terms of efficiency factors. We consider both equally and unequally replicated designs, including partially replicated designs. Supplementary materials accompanying this paper appear online.
Similar content being viewed by others
References
Atkinson, A. C., Donev, A. N., and Tobias, R. D. (2009), “Optimum experimental designs with SAS”, Oxford: Oxford University Press.
Bueno Filho, J. S. D. S., and Gilmour, S. G. (2003), “Planning incomplete block experiments when treatments are genetically related”, Biometrics, 59, 375–381.
Butler, D. G., Smith, A. B., and Cullis, B. R. (2014), “On the design of field experiments with correlated treatment effects”, Journal of Agricultural, Biological and Environmental Statistics, 19, 541–557.
Cullis, B. R., Smith, A. B., and Coombes, N. E. (2006), “On the design of early generation variety trials with correlated data”, Journal of Agricultural, Biological and Environmental Statistics, 11, 381–393.
Eccleston, J. A. and Chan, B. S. P. (1998), “Design algorithms for correlated data”, In Payne, R. and Green P. J., editors, COMPSTAT98: Proceedings in Computational Statistics, pp. 41–52. Physica-Verlag, Heidelberg.
Gilmour, A. R., Cullis, B. R., and Verbyla, A. P. (1997), “Accounting for natural and extraneous variation in the analysis of field experiments”, Journal of Agricultural Biological and Environmental Statistics, 2, 269–293.
Herzberg, A. M., and Jarrett, R. G. (2007), “A-optimal block designs with additional singly replicated treatments”, Journal of Applied Statistics, 34, 61–70.
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, 258–294.
John, J. A., and Williams, E. R. (1995), “Cyclic and computer generated designs, Second edition”, London: Chapman and Hall.
John, J. A., and Williams, E. R. (2002), “t-latinized designs”, Australian and New Zealand Journal of Statistics, 40, 111–118.
Martin, R. J. (1986), “On the design of experiments under spatial correlation”, Biometrika, 73, 247–277.
Piepho, H. P. (2015), “Generating efficient designs for comparative experiments using the SAS procedure OPTEX”, Communications in Biometry and Crop Science, 10, 96–114.
Piepho, H. P., Michel, V., and Williams, E. R. (2015), “Beyond Latin squares: A brief tour to row-column designs”, Agronomy Journal, 107, 2263–2270.
Piepho, H. P., Williams, E. R., and Ogutu, J. O. (2013), “A two-stage approach to recovery of inter-block information and shrinkage of block effect estimates”, Communications in Biometry and Crop Science, 8, 10–22.
Smith, A. B., Lim, P., and Cullis, B. R. (2006), “The design and analysis of multi-phase plant breeding experiments,” Journal of Agricultural Science, 144, 393–409.
Williams, E. R. (1986a), “Row and column designs with contiguous replicates,” Australian Journal of Statistics, 28, 154–163.
Williams, E. R. (1986b), “A neighbour model for field experiments”, Biometrika, 73, 279–287.
Williams, E. R., John, J. A., and Whitaker, D. (2006), “Construction of resolvable spatial row-column designs”, Biometrics, 62, 103–108.
Williams, E. R., John, J. A., and Whitaker, D. (2014), “Construction of more flexible and efficient p-rep designs”, Australian and New Zealand Journal of Statistics, 56, 89–96.
Williams, E. R., and Piepho, H. P. (2013), “A comparison of spatial designs for field variety trials”, Australian and New Zealand Journal of Statistics, 55, 253–258.
Williams, E. R., and Piepho, H. P. (2015), “Optimality and contrasts in block designs with unequal treatment replication”, Australian and New Zealand Journal of Statistics , 57, 203–209 (DOI:10.1111/anzs.1211).
Williams, E. R., Piepho, H. P., and Whitaker, D. (2011), “Augmented p-rep designs”, Biometrical Journal, 53, 19–27.
Yates, F. (1939), “The comparative advantages of systematic and randomized arrangements in the design of agricultural and biological experiments”, Biometrika, 30, 440–466.
Acknowledgments
We thank Randy Tobias (SAS Institute Inc.) for useful hints on the usage of the OPTEX procedure. Two referees are thanked for very helpful comments. One referee is thanked in particular for suggesting the addition of blocking effects for groups of columns.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Piepho, HP., Williams, E.R. & Michel, V. Nonresolvable Row–Column Designs with an Even Distribution of Treatment Replications. JABES 21, 227–242 (2016). https://doi.org/10.1007/s13253-015-0241-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13253-015-0241-2