kgt 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 !GP
10*0.1 # Psi (Specific variances, assuming 10 sites)
10*0.3 # First loadings
10*0.1 # Second loadings
In ASReml 3 if no loadings are fixed (i.e. !GP), ASReml will rotate the loadings
to orthogonality, and hold the leading loadings of lower
factors fixed. They are however updated in the orthogonalization
process which occurs at the beginning of each iteration (so
the final returned values have not been formally rotated).
Finding the REML solutions for multifactor Factor Analytic models
can be difficult.
The first problem is specifying initial values.
When using !CONTINUE and
progressing XFA(k)
to XFA(k+1),
ASReml3 initialises the next factor at SQRT(P*0.4) and changing the sign
of the (relatively) largest loading to negative.
One strategy which sometimes works in this context is to hold the previously estimated
factor loadings fixed for one round of iterations so that the
next factor aims at explaining variation previously
incorporated in Psi. Then allow all loadings to be updated for next round.
A second problem, at present unresolved,
is that sometimes the LogL rises to a relatively high value and then drifts away.
In an attempt to make the process easier, these two processes
have been linked as an additional meaning for the
!AILOADING qualifier.
For the first !AILOADING iterations,
the loading coefficients for all but the last factor are
held fixed. After that, loadings are rotated to orthogonal
and updated. If !AILOADING is not set by the user
and the model is an upgrade from a lower order XFA, !AILOADING
is set to 4.
is the coding for a large job tying to estimate
factors.
!WORK 1 !NOGRAPH !continue
Title: ALBUS2tage.
#trial,year,region,variety,yield,rep,weight,ems
#KFA02BURU,2002,NSW,KIEV-MUTANT,0.873,3,2136.562,0.0010000
trial !A
year !I
region !A
variety !A
yield
rep *
weight !*0.025
ems
!CYCLE 11 1 2 3 4
!DOPART $I
ALBUS2tage.csv !SKIP 1 !MAXIT 40 !AILOAD 20
!PART 11
!MAXIT 25
yield !wt=weight ~ mu trial !r trial.variety
1 1 1
0 !S2==0.025
trial.variety 2
trial 0 CORUH .1
87*.1
variety
!PART 1 2 3 4
yield !wt=weight ~ mu trial !r xfa(trial,$I).var
1 1 1
0 !S2==0.025
xfa(trial,$I).var 2
xfa(trial 0 XFA$I !GP
87*.01
87*.07 87*.07 87*.07 87*.07
variety
A previous set of analyses using these five models gave LogL values
for the models CORUH, XFA1, XFA2, XFA3 and XFA4 respectively of
2782, 2910, 3021, 3109 and 3200 using the strategies listed above
in separate runs. Running this job using the integrated strategy
produced LogL values of 2783, 2911, 3048, 3153 and 3206. However,
for models XFA3 and XFA4, the LogL drifted away again.
The XFA display reported in the .res file has been revised.
The current output from a small example with 9 environments
and 2 factors is %Ontario
DISPLAY of variance partitioning for XFA structure in xfa(Env,2).Geno
Lvl |----+----+----+----+----+----+----+----+----+----| TotalVar %expl PsiVar Loadings
1 | 1 | 0.3339 79.7 0.0679 0.5147 0.0335
2 | 1 2 0.1666 100.0 0.0000 0.4003 0.0797
3 | 1 2 | 0.2475 67.8 0.0798 0.3805 0.1514
4 | 1 2 0.1475 100.0 0.0000 0.3625 0.1269
5 | 1 2 0.4496 100.0 0.0000 0.6104 -0.278
6 | 1 2 0.1210 100.0 0.0000 0.2287 0.2622
7 | 1 2 | 0.4106 54.4 0.1872 0.4152 -0.226
8 | 1 2 0.0901 100.0 0.0000 0.0922 0.2857
9 | 1 2 0.1422 100.0 0.0000 0.2819 0.2506
0 |----+----+----+----+----+----+----+----+-- Average 0.2343 89.1 0.0372 0.3651 0.0763
In the figure, 1 indicates the proportion of TotalVar explained
by the first loading, 2 indicates the proportion explained by
first and second (provided it plots right of 1. Consequently,
the distance from 2 to the right margin represents PsiVar.
%expl reports the percentage
of TotalVar explained by all
loadings. The last row contains column averages.
See Also
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