This function rotates to an errors-in-variables representation. The
optimization is not iterative and does not use the GPA algorithm.
The function can be used directly or the
function name can be passed to factor analysis functions like factanal.
The loadings matrix is rotated so the \(k\) rows indicated by identity
form an identity matrix, and the remaining \(M-k\) rows are free parameters.
\(\Phi\) is also free. The default makes the first \(k\) rows
the identity. If inverting the matrix
of the rows indicated by identity fails, the rotation will fail and the
user needs to supply a different choice of rows.
Not all authors consider this representation to be a rotation.
Viewed as a rotation method, it is oblique, with an
explicit solution: given an initial loadings matrix \(L\) partitioned as
\(L = (L_1^T, L_2^T)^T\), then (for the default
identity) the new loadings matrix is
\((I, (L_2 L_1^{-1})^T)^T\)
and \(\Phi = L_1 L_1^T\), where \(I\) is the \(k\)
by \(k\) identity matrix. It is
assumed that \(\Phi = I\) for the initial loadings matrix.
One use of this parameterization is
for obtaining good starting values (so it looks a little strange
to rotate towards this solution afterwards). It has a few other purposes:
(1) It can be useful for comparison with
published results in this parameterization;
(2) The
S.E.s are more straightfoward to compute, because it is the solution
to an unconstrained
optimization (though not necessarily computed as such);
(3) One
may have an idea about which reference variables load on only one
factor, but not impose restrictive constraints on the other loadings,
so, in a nonrestrictive
way, it has similarities to CFA;
(4) For some purposes, only the subspace spanned by the factors
is important, not the specific parameterization within this subspace;
(5) The back-predicted indicators (explained portion of the indicators)
do not depend
on the rotation method. Combined with the greater ease to obtain
correct standard errors of this method, this allows easier and more
accurate prediction-standard errors.