Theoretical and Applied Genetics

, Volume 84, Issue 1–2, pp 161–172 | Cite as

Using the shifted multiplicative model to search for “separability” in crop cultivar trials

  • P. L. Cornelius
  • M. Seyedsadr
  • J. Crossa


The shifted multiplicative model (SHMM) is used in an exploratory step-down method for identifying subsets of environments in which genotypic effects are “separable” from environmental effects. Subsets of environments are chosen on the basis of a SHMM analysis of the entire data set. SHMM analyses of the subsets may indicate a need for further subdivision and/or suggest that a different subdivision at the previous stage should be tried. The process continues until SHMM analysis indicates that a SHMM with only one multiplicative term and its “point of concurrence” outside (left or right) of the cluster of data points adequately fits the data in all subsets. The method is first illustrated with a simple example using a small data set from the statistical literature. Then results obtained in an international maize (Zea mays L.) yield trial with 20 sites and nine cultivars is presented and discussed.

Key words

Genotype x environment interaction Shifted multiplicative model Separability Concurrent regression model Crossover interaction Qualitative interaction 


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  1. Azzalini A, Cox DR (1984) Two new tests associated with analysis of variance. J R Statist Soc B 46:335–343Google Scholar
  2. Baker RJ (1988) Tests for crossover genotype-environmental interactions. Can J Plant Sci 68:405–410Google Scholar
  3. Baker RJ (1990) Crossover genotype-environmental interaction in spring wheat. In: Kang MS (ed) Genotype-by-environment interaction and plant breeding. Department of Agronomy, Louisiana Agric Exp Stn, LSU Agric Center, Baton Rouge, La.Google Scholar
  4. Boik RJ (1985) A new approximation to the distribution function of the studentized maximum root. Commun in Stat B Simul Comput 14:759–767Google Scholar
  5. Gauch HG Jr (1988) Model selection and validation for yield trials with interaction. Biometrics 44:705–715Google Scholar
  6. Gauch HG Jr, Zobel RW (1988) Predictive and postdictive success of statistical analyses of yield trials. Theor Appl Genet 76:1–40Google Scholar
  7. Gregorius H-R, Namkoong G (1986) Joint analysis of genotypic and environmental effects. Theor Appl Genet 12:413–422Google Scholar
  8. Goodman LA, Haberman ST (1990) The analysis of nonadditivity in two-way analysis of variance. J Am Stat Assoc 85:139–145Google Scholar
  9. Johnson DE (1976) Some new multiple comparison procedures for the two-way AOV model with interaction. Biometrics 32:929–934Google Scholar
  10. Johnson DE, Graybill FA (1972) An analysis of a two-way model with interaction and no replication. J Am Stat Assoc 67:862–868Google Scholar
  11. Mandel J (1961) Nonadditivity in two-way analysis of variance. J Am Stat Assoc 56:878–888Google Scholar
  12. Mandel J (1971) A new analysis of variance model for nonadditive data. Technometrics 13:1–8Google Scholar
  13. Marasinghe MG (1985) Asymptotic tests and Monte Carlo studies associated with the multiplicative interaction model. Commun Stat A Theory Methods 14:2219–2231Google Scholar
  14. Schott JR (1986) A note on the critical values used in stepwise tests for multiplicative components of interaction. Commun Stat A Theory Methods 15:1561–1570Google Scholar
  15. Seyedsadr SM (1987) Statistical and computational procedures for estimation and hypothesis testing with respect to the shifted multiplicative model. Ph.D thesis, University of KentuckyGoogle Scholar
  16. Seyedsadr M, Cornelius PL (1991a) Functions approximating the expectations and standard deviations of sequential sums of squates in the shifted multiplicative model for a two-way table. University of Kentucky. Department of Statistics Technical Report 322Google Scholar
  17. Seyedsadr M, Cornelius PL (1991b) Hypothesis testing for components of the shifted multiplicative model for a nonadditive two-way table. University of Kentucky. Department of Statistics Technical Report 315Google Scholar
  18. Seyedsadr M, Cornelius PL (1991c) Shifted multiplicative models for nonadditive two-way tables. Commun Stat B Simul Comput 2/(3): (in press)Google Scholar
  19. Snee RD (1982) Nonadditivity in a two-way classification: Is it interaction or nonhomogeneous variance? J Am Stat Assoc 77:515–518Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • P. L. Cornelius
    • 1
  • M. Seyedsadr
    • 2
  • J. Crossa
    • 3
  1. 1.Departments of Agronomy and StatisticsUniversity of KentuckyLexingtonUSA
  2. 2.Gershenson Radiation Oncology Center, School of MedicineWayne State UniversityDetroitUSA
  3. 3.International Maize and Wheat Improvement Center (CIMMYT)Mexico, D.F.Mexico

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