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