# Variance Models

## Factor Analysis models

FAk, FACVk and XFAk are different parameterizations of the factor analytic model in which S is modelled as S= GG' + P where G is a matrix of k loadings on the covariance scale and P is a diagonal vector of specific variances. See Smith et al. (2001) and Thompson et al. (2003) for examples of factor analytic models in multi-environment trials.

The general limitations are
• that P may not include zeros except in the XFAk formulation
• constraints are required in G for k> 1 for identifiability. Typically, one zero is placed in the second column, two zeros in the third column, etc.
• The total number of parameters fitted (kw + w - k(k-1)/2) may not exceed w(w+1)/2.

#### Correlation form

FAk models the variance-covariance matrix S on the correlation scale as S= DCD, where
• D is diagonal such that DD = diag(S),
• C is a correlation matrix of the form FF' + E where F is a matrix of k loadings vectors on the correlation scale and E is diagonal and is defined by difference,
• the parameters are specified in the order: loadings for each factor (F) followed by the variances (diag(S); when k is greater than 1, constraints on the elements of F are required.

#### Covariance scale

FACVk models ( CV for covariance) are an alternative formulation of FA models in which S is modelled as S= GG' + P where G is a matrix of k loadings on the covariance scale and P is diagonal. The parameters in FACV
• are specified in the order: loadings (G) followed by specific variances P; when k is greater than 1, constraints on the elements of G are required,
• are related to those in FA by G= DF and P= DED,

#### Extended form

XFAk ( X for extended) is the third form of the factor analytic model and has the same parameterisation as for FACV, that is, S= GG' + P. However, XFA models
• have parameters specified in the order diag(P) and vec(G); when k is greater than 1, constraints on the elements of G are required,
• may not be used in R structures,
• are used in G structures in combination with the xfa(f,k) model term,
• return the factors as well as the effects.
• permit some elements of P to be fixed to zero,
• are computationally faster than the FACV formulation for large problems when k is much smaller than w,

Special consideration is required when using the XFAk model. The SSP must be expanded to have room to hold the k factors. This is achieved by using the xfa(f,k) model term in place of f in the model. For example,

```  y ~ site !r geno.xfa(site,2)
0 0 1
geno.xfa(site,2) 2
geno
xfa(site,2) 0 XFA2
```