, Volume 183, Issue 3, pp 401–413 | Cite as

Incomplete split-plot designs based on α-designs: a compromise between traditional split-plot designs and randomised complete block design

  • Kristian Kristensen


The paper shows how the α-design (also known as generalised lattice) may be used for constructing incomplete split-plot designs and describes four different methods (A, B, C and D) of construction. Intra-block efficiency factors and theoretical considerations are used to compare the methods. Based on those considerations method B was considered to be the most appropriate method for trials where tests for interaction between the two factors were important and thus this method was used and most of the paper deals with trials based on this construction method. The incomplete split-plots were superior to traditional split-plots in most cases—and the increase in efficiency of the designs can be quite large—especially for comparisons involving the whole-plot treatment. The efficiency for the comparison of the main effect of the whole-plot treatment was in most cases larger for randomized complete block design than for the incomplete split-plot design, but for other comparisons the proposed designs were in most cases more efficient than a randomized complete block design. The efficiency of the designs was compared to traditional split-plot designs and randomized complete block designs using three types of data. The three types were simulated data with known covariance structure, data from uniformity trials and data from actual trials using incomplete split-plot designs for comparing cereal varieties under different growing conditions. It is concluded that the incomplete split-plot designs may be a good alternative to traditional split-plots and a good compromise between split-plots and randomised complete blocks.


Incomplete split-plot Construction Efficiency Uniformity trials Variety trials 



The author would like to thank Hanne Østergaard and Frederica Bigongiali for accepting their data to be used. Hanne Østergaard is also thanked for valuable comments on a draft for this paper. The work was partly funded by DARCOF II, the project BAR-OF: Characteristics of spring barley varieties for organic farming.


  1. Bigongiali F (2009) Competition between wheat cultivar and weeds in organic agriculture. PhD thesis. Scuola Superiore Sant’Anna, Piza. 148 pp, ISBN 88-901624-6-5Google Scholar
  2. Box GEP, Hunter WG, Hunter JS (1978) Statistics for experimenters. Wiley, New York 653 ppGoogle Scholar
  3. Dorph-Petersen K (1949) Parcelfordeling i markforsøg (Distribution of plots in field trials). Tidsskr Planteavl 52:111–175Google Scholar
  4. Heidmann T (1988) Startkarakterisering af arealer til systemforskning: I Forsøgsarealer, måleprogram og metoder. Tidsskr Planteavl Special issue. S 1958, 89 ppGoogle Scholar
  5. Heidmann T (1989) Startkarakterisering af arealer til systemforskning: IV Resultater fra arealet ved Jyndevad. Tidsskr Planteavl Special issue. S 2021, 163 ppGoogle Scholar
  6. Hering F, Mejza S (2002) An incomplete split-plot design generated by GDPBIBD(2)s. J Stat Plan Inference 106:125–134CrossRefGoogle Scholar
  7. John PWM (1971) Statistical design and analysis of experiments. The Macmillan Company, New York 356 ppGoogle Scholar
  8. John JA, Eccleston JA (1986) Row-column α-designs. Biometrika 73:301–306Google Scholar
  9. John JA, Williams ER (1995) Cyclic and computer generated designs, 2nd edn. Chapman & Hall, London 255 ppGoogle Scholar
  10. Jørgensen LN, Olesen JE (2002) Fungicide treatments affect yield and moisture content of grain and straw in Winter wheat. Crop Prot 21:1023–1032CrossRefGoogle Scholar
  11. Kachlicka D, Mejza S (1990) Incomplete split-plot experiments—whole-plot treatments in a row-column design. Comput Stat Data Anal 9:135–146CrossRefGoogle Scholar
  12. Kenward MG, Roger JH (1997) Small sample inference for fixed effects from restricted maximum likelihood. Biometrics 53:983–997PubMedCrossRefGoogle Scholar
  13. Kristensen K (2003) Incomplete split-plots in variety trials—based on α-designs. Seventh working seminar on statistical methods in variety testing. Biuletyn Oceny Odmian (Cultivar Testing Bulletin) 31: 7–17Google Scholar
  14. Kristensen K, Ericson L (2008) Importance of growth characteristics for yield of barley in different growing systems: will growth characteristics describe yield differently in different growing systems? Euphytica 163:367–380CrossRefGoogle Scholar
  15. Kristensen K, Ersbøll AK (1995) The use of geostatistical methods in variety trials, where some varieties are unreplicated. Biuletyn Oceny Odmian (Cultivar Testing Bulletin) 26–27:113–122Google Scholar
  16. Mejza I, Mejza S (1984) Incomplete split-plot designs. Stat Prob Lett 2:327–332CrossRefGoogle Scholar
  17. Mercer WB, Hall AD (1911) The experimental error of field trials. J Agric Sci Camb 4:107–132CrossRefGoogle Scholar
  18. Østergaard H, Kristensen K, Pinnschmidt HO, Hansen PK, Hovmøller MS (2008) Predicting spring barley yield from variety specific yield potential, disease resistance and straw length, and from environment-specific disease loads and weed pressure. Euphytica 163:391–408CrossRefGoogle Scholar
  19. Ozawa K, Jimbo M, Mejza I, Kuriki S, Mejza S (2002) Optimality and construction of incomplete split-block designs. J Stat Plan Inference 106:135–157CrossRefGoogle Scholar
  20. Ozawa K, Mejza S, Jimbo M, Mejza I, Kuriki S (2004) Incomplete split-plot designs generated by some resolvable balanced designs. Stat Prob Lett 68:9–15CrossRefGoogle Scholar
  21. Patterson HD, Williams ER (1976) A new class of resolvable incomplete block designs. Biometrika 63:83–92CrossRefGoogle Scholar
  22. Patterson HD, Williams ER, Hunter EA (1978) Block designs for variety trials. J Agric Sci Camb 90:395–400CrossRefGoogle Scholar
  23. Rees HD (1969) The analysis of variance on some non-orthogonal designs with split-plot. Biometrika 56:43–54CrossRefGoogle Scholar
  24. Robinson J (1967) Incomplete split-plot designs. Biometrics 23:793–802PubMedCrossRefGoogle Scholar
  25. Robinson J (1970) Blocking in incomplete split-plot designs. Biometrika 57:347–350CrossRefGoogle Scholar
  26. Spilke J, Piepho HP, Meyer U (2004) Approximating the degrees of freedom for contrasts of genotypes laid out as subplots in an alpha-design in a split-plot experiment. Plant Breed 123:193–197CrossRefGoogle Scholar
  27. Whitaker D, Williams ER, John JA (2006) CycDesigN: a package for computer generation of experimental designs. Accessed 2 December 2009
  28. Williams ER, Talbot M (1994) ALPHA+. Experimental designs for variety trials. CSIRO/SASS, Canberra/EdinburghGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Genetics and Biotechnology, Faculty of Agricultural SciencesUniversity of AarhusFoulumDenmark

Personalised recommendations