Electronic Journal of Statistics

Quasi-Latin designs

C. J. Brien, R. A. Bailey, T. T. Tran, and J. Boland

Full-text: Open access


This paper gives a general method for constructing quasi-Latin square, quasi-Latin rectangle and extended quasi-Latin rectangle designs for symmetric factorial experiments. Two further methods are given for parameter values satisfying certain conditions. The construction of designs for a range of numbers of rows and columns is discussed so that the different construction techniques are covered. For some row and column combinations, different designs are compared. The construction of designs with rows and columns that are nested or contiguous is also discussed.

Article information

Electron. J. Statist. Volume 6 (2012), 1900-1925.

First available in Project Euclid: 12 October 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Design of experiments factorial designs glasshouse experiments latinized designs quasi-Latin designs row-column designs


Brien, C. J.; Bailey, R. A.; Tran, T. T.; Boland, J. Quasi-Latin designs. Electron. J. Statist. 6 (2012), 1900--1925. doi:10.1214/12-EJS732. https://projecteuclid.org/euclid.ejs/1350046857

Export citation


  • [1] Bailey, R. A. (1985). Balance, orthogonality and efficiency factors in factorial design., Journal of the Royal Statistical Society, Series B (Methodological) 47 453–458.
  • [2] Bailey, R. A. (2008)., Design of Comparative Experiments. Cambridge University Press, Cambridge.
  • [3] Brien, C. J. and Bailey, R. A. (2009). Decomposition tables for experiments. I. A chain of randomizations., The Annals of Statistics 37 4184–4213.
  • [4] Brien, C. J., Harch, B. D., Correll, R. L. and Bailey, R. A. (2011). Multiphase experiments with at least one later laboratory phase. I. Orthogonal designs., Journal of Agricultural, Biological and Environmental Statistics 16 422–450.
  • [5] Cochran, W. G. and Cox, G. M. (1957)., Experimental Designs, 2nd ed. John Wiley & Sons, New York.
  • [6] Cox, D. R. and Reid, N. (2000)., The Theory of the Design of Experiments. Chapman and Hall/CRC, Boca Raton.
  • [7] Edmondson, R. N. (1989). Glasshouse design for repeatedly harvested crops., Biometrics 45 301–307.
  • [8] Erdős, P. and Kaplansky, I. (1946). The asymptotic number of Latin rectangles., American Journal of Mathematics 68 230–236.
  • [9] Harshbarger, B. and Davis, L. (1952). Latinized rectangular lattices., Biometrics 8 73–84.
  • [10] Healy, M. J. R. (1951). Latin rectangle designs for $2^n$ factorial experiments on 32 plots., The Journal of Agricultural Science 41 315–316.
  • [11] Houtman, A. M. and Speed, T. P. (1983). Balance in designed experiments with orthogonal block structure., Annals of Statistics 11 1069–1085.
  • [12] 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.
  • [13] Littell, R., Milliken, G., Stroup, W., Wolfinger, R. and Schabenberger, O. (2006)., SAS for Mixed Models, 2nd ed. SAS Press, Cary.
  • [14] Rao, C. R. (1946). Confounded factorial designs in quasi-Latin squares., Sankhyā 7 295–304.
  • [15] Shah, S. M. (1978). On $\alpha$-resolvability and affine $\alpha$-resolvability of incomplete block designs., The Annals of Statistics 6 468–471.
  • [16] Tran, T. T. (2009). Design and Analysis of Experiments for Assessing Indigenous Plant Species. PhD thesis, School of Mathematics and Statistics, The University of South Australia, Adelaide, South, Australia.
  • [17] Williams, E. R. and John, J. A. (1996). Row-column factorial designs for use in agricultural field trials., Journal of the Royal Statistical Society, Series C (Applied Statistics) 45 39–46.
  • [18] Williams, E. R., Matheson, A. C. and Harwood, C. E. (2002)., Experimental Design and Analysis for Tree Improvement, 2nd ed. CSIRO, Melbourne, Australia.
  • [19] Yates, F. (1937). The Design and Analysis of Factorial Experiments., Imperial Bureau of Soil Science Technical Communication 35.
  • [20] Youden, W. J. (1940). Experimental designs to increase accuracy of greenhouse studies., Contributions from Boyce Thompson Institute 11 219–228.