## Mixedeffect Linear Bilinear Models

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

A mixed-model analogue of biplot analysis has been developed using the factor analytic (FA) model for approximating the variance-covariance GE structure (Piepho, 1998; Smith et al., 2002). Research conducted by Crossa et al. (2006) and Burgueno et al. (2008) described how to model variance-covariance GE and GGE using the FA model and how to incorporate the additive (relationship A) matrix and the additive x additive covariance matrix into the FA model based on pedigree information. Burgueno et al. (2008) also described the equivalence between SREG2 and FA(2) for finding

Table 14.3. Least squares estimates of the combined effects of genotype (G) plus the genotype x environment (GE) [x, + (x8)ij ] = zj = yij - y j with i = 1, 2, ..., I genotypes, j = 1, 2, ..., J environments and r replicates.

### Environment 1 Environment 2 . . Environment J

Table 14.3. Least squares estimates of the combined effects of genotype (G) plus the genotype x environment (GE) [x, + (x8)ij ] = zj = yij - y j with i = 1, 2, ..., I genotypes, j = 1, 2, ..., J environments and r replicates.

Environment 1 Environment 2 . . Environment J

Genotype 1 |
YH. |
- y.1. |
y12. |
- y.2. |
yu. |
- y.j. |

Genotype 2 |
y21. |
- y.1. |
y22. |
- y.2. |
Iij. |
- y.j. |

Genotype I |
y 1 |
- y.1. |
y, 2. |
- y.2. |
yu. |
- Yj . |

subsets of genotypes and environments without COI.

Factor analytic and sites regression models for assessing crossover genotype x environment interaction

In the FA model, the random effect of the ith genotype in the jth environment (glj) is expressed as a linear function of latent variables xik with coefficients 8jk for k = 1, 2, ... tt plus a residual, n, i.e.

gij = Ij + Z x 8k + n, so that the jth k=1 'k cell mean can be written as yij = glj +elj. With only the first two latent factors being retained, gij is approximated by g.. « . + xi.18n + xi2 8J2 + . Therefore, SREG2 (Eqn 14.3) can be perceived as consisting of a set of multiple regression equations (one for each environment), each regression equation consisting of an environmental mean or environmental effect as intercept plus two terms for regression on two genotypic regressor variables, af1and ai2 (either observed or latent), with yj1 and yj2 as the regression coefficients. Thus there is a clear connection between the SREG2 and the FA(2) models, as described by Burgueno et al. (2008). A similar connection between the AMMI2 and FA(2) models was also established by Smith et al. (2002).

Under principal component rotation, the directions and projections of the vectors of FA(2) and SREG2 in the biplot are the same. Therefore, the property of SREG by which the first principal component of SREG2 accounts for non-crossover interaction (non-COI) and the second principal component of SREG2 is due to COI variability should hold for FA(2) as well. However, the absolute values of genotypic and environmental scores under the FA(2) and SREG2 models may not necessarily be the same because shrinkage is involved in Best Linear Unbiased Predictions (BLUPs) (Henderson, 1984) of random effects in the FA(2) model but not in least squares estimates of fixed effects in the SREG2 model. Other important differences between SREG and FA are: (i) the standard errors of the estimable functions of fixed effects under SREG differ from those of predictable functions of a mixture of fixed and random effects under FA; and (ii) FA models are more flexible in handling unbalanced data (the SREG model does not handle missing data).

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