Constrained AMMI Model: Application to Polish Winter Wheat Post‐Registration Data

Abstract

Constrained principal component analysis (C‐PCA) describes a two‐dimensional data table and assumes a linear dependence of the principal component scores on known additional parameters (i.e., explanatory matrices). In this study, we used C‐PCA to generalize the additive main effects and multiplicative interaction (AMMI) model and propose the constrained AMMI model. The constrained AMMI model is interpreted and illustrated when (i) only the environmental principal component parameters have an explanatory data matrix, (ii) only the genotype principal component parameters have an explanatory data matrix, and (iii) both types of parameters have explanatory data matrices. The cross‐validation procedure is adapted for model diagnosis. Data for winter wheat (Triticum aestivum L.) genotype × location × management × year grain yield, recorded in Poland from multienvironment trials conducted in the post‐registration variety testing system, were analyzed and used for model comparison.

The Characterization of 10 Spring Camelina Genotypes Grown in Environmental Conditions in North-Eastern Poland

“…The high average yields of genotype 13CS0787-08 can be attributed to the strong effect of the GE interaction in the supportive years of 2017 and 2018, whereas low yields were noted under less favorable weather conditions in 2016. Low yield stability is a considerable defect in locations characterized by varied weather and environmental conditions [13,18,38]. The genotype 14CS0887 is definitely not suitable for cultivation in north-eastern Poland.…”

“…AMMI models are widely used in plant breeding. In recent years (2017-2019), they have been applied to evaluate various plant species, from the most popular crops such as wheat [18][19][20][21], oilseed rape [17,22], maize [23][24][25], rice [26][27][28][29], and sugar beet [30] to cassava [31], peanuts [32], and non-edible crops such as cotton [33] and willow [16].…”

An analysis of simulated yield data for pepper shows how genotype × environment interaction in yield can be understood in terms of yield components and their QTLs

“…(2014, 2016), Assis et al. (2018), and Paderewski and Rodrigues (2018). The GGE model is whereas the AMMI model can be written as where is the yield of genotype i in environment j , μ is the grand mean, are the genotype mean deviations (genotype means minus the grand mean), are the environment mean deviations, and are the genotypic and environmental parameters (scores) for the n th multiplicative interaction term (i.e., the genotype and environment principal component scores and loadings for PCA axis n ), N is the number of interaction principal component (IPC) axes retained, and is a residual.…”

“…Well-known representatives of this class of models are (a) the model underlying principal components analysis (PCA) of the genotype × environment table, also called GGE biplot model (see Yan and Kang, 2003), and (b) the AMMI model, which is a combination of ANOVA for the genotypic and environmental main effects and PCA for the residuals from additivity (Gauch, 1988(Gauch, , 1992Gollob, 1968;Mandel, 1969). Useful generalizations of the AMMI model have also been proposed by Rodrigues et al (2014Rodrigues et al ( , 2016, Assis et al (2018), and Paderewski and Rodrigues (2018). The GGE model is…”

Genotype by year interaction for selected quantitative traits in hybrid lines of Triticum aestivum L. with Aegilops kotschyi Boiss. and Ae. variabilis Eig. using the additive main effects and multiplicative interaction model

“…The additive main effects and multiplicative interaction (AMMI) model (Zobel et al 1988;Gauch and Zobel 1990) has been used extensively for analyses of multi-environment yield trials in order to understand complex genotype (G), environment/year (E/Y) and genotype-by-year interactions (GYI). The AMMI model has commonly been used to evaluate the genotype 9 environment interactions (Ghaed-Rahimi et al 2015;Golkari et al 2016;Paderewski et al 2016;Shahriari 2018;Rodrigues and Paderewski 2018;Bocianowski et al 2019bBocianowski et al , 2020Singh et al 2019).…”