T 8j XXk aik Y jk 142

where the constant Xk is the singular value of the kth multiplicative component that is ordered X1>X2>... >Xt; the aik elements are elements of the kth left singular vector of the true interaction and represent geno-typic sensitivity to hypothetical environmental factors represented by the kth right singular vector with elements Yjk. The aik and yjk elements satisfy the ortho-normalization constraints Xaikak = Y,K = 0 for k ^ k and Xa2k =i XYy =1. When Eqn 14.2 is i j saturated, the number of bilinear terms is t = min(T - 1, J - 1), and for any smaller value, the model is said to be truncated. The GE interaction parameters Xk, aik and yjk are estimated from the data.

Gabriel (1978) described the least squares fit of Eqn 14.2 and explained how the residual matrix of the GE term, Z = yy -yy -y.j + y , is subjected to singular value decomposition (SVD) after adjusting for the additive (linear) terms. The first two components can be displayed in a graph called a biplot. Zobel et al. (1988) and Gauch et al. (2008) named Eqn 14.2 the Additive Main Effects and Multiplicative Interaction (AMMI) model. Other types of linear-bilinear models, described by Cornelius et al. (1996), are:

the Sites (environments) Regression (SREG) model:

the Genotypes Regression (GREG) model:

the Completely Multiplicative Model (COMM):

and the Shifted Multiplicative Model (SHMM):

The SHMM was the first linear-bilinear model that, along with other statistical tools, was used for identifying subsets of genotypes or environments in which genotypic rank changes would be negligible (Cornelius et al., 1992, 1993; Crossa and Cornelius, 1993; Crossa et al., 1993, 1995). The SREG (Crossa and Cornelius, 1997) model is very useful in plant breeding because the bilinear terms contain both the main effects of genotypes (G) and GE. The SREG model has been preferred to SHMM for grouping environments without genotypic rank change (Crossa and Cornelius, 1997). The interaction parameters aik and yjk in the bilinear terms model the behaviour of genotypes and environments, and when ( , ) and (1,2) are plotted together in the biplot (Gabriel, 1978), useful interpretations of the relationships between genotypes, environments, and GE are obtained. In the biplot, the interaction between the ith genotype and the jth environment is obtained by projecting one vector on to the other. In the AMMI model, the composition of the two-

Table 14.1. Least squares estimates of genotypic and environmental effects for a two-way table of genotypes and environments with i = 1, 2, ..., I genotypes, j = 1, 2, ..., J environments and r replicates.

Environment 1

Environment 2

. . Environment J

Marginal mean of genotypes

Estimate of genotypic effect ()

Genotype 1

yii.

y,2

. . yij.

yi..

T = Yi.. - y...

Genotype 2

y2,.

y22

. . y2J.

y2..

T2 = Y2.. - y...

Genotype I

yii.

yi 2.

. yiJ.

Yl

Ti = YI.. - y...

Marginal mean of environments

y.i.

y.2

. . y.J.

A = y...

-

Estimate of environmental effect (5j)

Si = y.i. - y...

S2 = y.2. - y...

. . SJ = y.J. - y...

-

Table 14.2. Least squares estimate of the genotype x environment (GE) term in Eqn 14.1 is

(t8)s = zs = yf - y, - y.y. + y... with i = 1, 2, ..., I genotypes, j = 1, 2, ..., J environments and r replicates.

Table 14.2. Least squares estimate of the genotype x environment (GE) term in Eqn 14.1 is

(t8)s = zs = yf - y, - y.y. + y... with i = 1, 2, ..., I genotypes, j = 1, 2, ..., J environments and r replicates.

Environment 1

Environment 2 .

. Environment J

Genotype 1

Yn. - Y1.. - y.1. + y...

y12. - y,. - y.2. + y... .

7u. - y,. - y.j. + y...

Genotype 2

y21. - y2.. - y.1. + y...

y22. - y2.. - y.2. + y... .

Y2J. - y2.. - y.j. + y...

Genotype I y, 1. - y, - y.1. + y y,2. - y,.. - y.2. + y... yu. - y,.. - y.j. + y...

Genotype I y, 1. - y, - y.1. + y y,2. - y,.. - y.2. + y... yu. - y,.. - y.j. + y...

way I x J matrix to be subjected to singular value decomposition is shown in Table 14.2, where only the GE interaction (see Eqn 14.2) is modelled by the bilinear terms. In the SREG model, the bilinear term models the main effects of genotypes (G) plus GE interaction (usually called a GGE biplot), and the composition of the two-way I x J matrix to be subjected to singular value decomposition (see Eqn 14.3) is shown in Table 14.3.

Recently there was an ongoing debate examining the merits and demerits of AMMI versus GGE biplots for genotype and environment identification (Yan et al., 2007; Gauch et al., 2008). In a recent article, Yang et al. (2009) pointed out the advantages and disadvantages of these fixed effects linear-bilinear models and discussed relevant issues concerning the use of biplot analysis as a descriptive statistical tool. The authors pointed out that several issues affect the validity of such analysis but are generally ignored by the current biplot literature. Some of these issues are:

• What if genotypes or environments, or both, are random effects?

• Can biplot analysis contribute to detecting crossover interaction?

• How relevant is biplot analysis for understanding the nature and causes of interaction?

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