Analysis of a three-way interaction including multi-attributes
Mario Varela A , Jose Crossa B E , Jagdish Rane C , Arun Kumar Joshi B and Richard Trethowan DA Departamento de Matemática del Instituto Nacional de Ciencias Agrícolas, La Habana, Carretera Tapaste, Km 3½, San José de Las Lajas, Apdo Postal 32700, Habana, Cuba.
B Biometrics and Statistics Unit of the Crop Informatics Laboratory, International Maize and Wheat Improvement Center (CIMMYT), Apdo, Postal 6-641, Mexico DF, Mexico.
C Directorate of Wheat Research, ICAR, Karnal 132 001, India.
D Plant Breeding Institute, The University of Sydney, PMB11, Camden, NSW 2570, Australia.
E Corresponding author. Email: j.crossa@cgiar.org
Australian Journal of Agricultural Research 57(11) 1185-1193 https://doi.org/10.1071/AR06081
Submitted: 13 March 2006 Accepted: 10 July 2006 Published: 27 October 2006
Abstract
The additive main effect and multiplicative interaction (AMMI) has been widely used for studying and interpreting genotype × environment interaction (GEI) in agricultural experiments using multi-environment trials (METs). When METs are performed across several years the interaction is referred to as a 3-mode (3-way) data array, in which the modes are genotypes, environments, and years. The 3-way array can be applied to other conditions or factors artificially created by the researcher, such as different sowing dates or plant densities, etc. Three-way interaction data can be studied using the AMMI analysis. The objective of this study is to apply the 3-mode AMMI to 2 datasets. Dataset 1 comprises genotype (25) × location (4) × sowing time (4) interaction; 8 traits were measured. The structure of dataset 2 is genotype (20) × irrigation regimes (4) × year (3) on grain yield. Results of the 3-way AMMI on dataset 1 show that several important 3-way interactions were not detected when condensing location (4) × sowing time (4) into environments (16). An alternative 3-way array, genotype × attribute × locations for the early sowing date in Year 1, is considered. Results of the 3-way AMMI on dataset 2 show that different patterns of response of genotypes can be found at different irrigation methods and years.
Additional keywords: Three-mode interaction; principal component analyses.
Basford KE,
Kroonenberg PM,
DeLacy IH, Lawrence PK
(1990) Multi-attribute evaluation of regional cotton variety trials. Theoretical and Applied Genetics 79, 225–324.
| Crossref | GoogleScholarGoogle Scholar |
Crossa J,
Basford K,
Taba S,
Delacy I, Silva E
(1995) Three-mode analyses of maize using morphological and agronomic attributes measured in multilocation trials. Crop Science 35, 1483–1491.
Eberhart SA, Russell WA
(1966) Stability parameters for comparing varieties. Crop Science 6, 36–40.
van Eeuwijk FA, Kroonenberg PM
(1998) Multiplicative models for interaction in three-way ANOVA, with applications to plant-breeding. Biometrics 54, 1315–1333.
| Crossref | GoogleScholarGoogle Scholar |
Finlay KW, Wilkinson GN
(1963) The analysis of adaptation in a plant breeding programme. Australian Journal of Agricultural Research 14, 742–754.
| Crossref | GoogleScholarGoogle Scholar |
Gabriel KR
(1971) The biplot graphic display of matrices with applications to principal components analysis. Biometrika 58, 453–467.
| Crossref | GoogleScholarGoogle Scholar |
Gabriel KR
(1978) Least squares approximation of matrices by additive and multiplicative models. Journal of the Royal Statistical Society: Series B 40, 186–196.
Gauch HG
(1988) Model selection and validation for yield trials with interaction. Biometrics 44, 705–715.
| Crossref | GoogleScholarGoogle Scholar |
Gollob HF
(1968) A statistical model which combines features of factor analytic and analyses of variance techniques. Psychometrika 33, 73–115.
| Crossref | GoogleScholarGoogle Scholar | PubMed |
Kroonenberg PM, Basford KE
(1989) An investigation of multi-attribute genotype response across environments using three-mode principal component analysis. Euphytica 44, 109–123.
| Crossref | GoogleScholarGoogle Scholar |
Kroonenberg PM, De Leeuw J
(1980) Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika 45, 69–97.
| Crossref | GoogleScholarGoogle Scholar |
Mandel J
(1969) The partitioning of interaction in analysis of variance. Journal of Research of the National Bureau of Standards,Series B 73, 309–328.
Mandel J
(1971) A new analysis of variance model for non-additive data. Technometrics 13, 1–18.
| Crossref | GoogleScholarGoogle Scholar |
Timmerman ME, Kiers HAL
(2000) Three-mode principal components analysis. Choosing the numbers of components and sensitivity to local optima. British Journal of Mathematical and Statistical Psychology 53, 1–16.
| Crossref | GoogleScholarGoogle Scholar | PubMed |
Tucker LR
(1966) Some mathematical notes on three-mode factor analysis. Psychometrika 31, 279–311.
| Crossref | GoogleScholarGoogle Scholar | PubMed |
Varela M, Torres V
(2005) Aplicación del Análisis de Componentes Principales de Tres Modos en la caracterización multivariada de somaclones de King grass. Revista Cubana de Ciencia Agrícola 39, 12–19.
Williams EJ
(1952) The interpretation of interactions in factorial experiments. Biometrika 39, 65–81.
| Crossref | GoogleScholarGoogle Scholar |
Yates F, Cochran WG
(1938) The analysis of group of experiments. Journal of Agricultural Science, Cambridge 28, 556–580.
Zobel RW,
Wright MJ, Gauch HG
(1988) Statistical analysis of a yield trial. Agronomy Journal 80, 388–393.
