# Variance Models

## Factor Analysis models

FA`k`, FACV`k`
and XFA`k` 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 XFA`k`
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

FA`k`
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

FACV`k`
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

XFA`k`
( 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 XFA`k`
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

## See Also

**Return to start**