The default est method and quartimin rotation give parameters
using standard
(quasi) ML factor analysis (on the correlation matrix and then scaled back).
The function factanal with no rotation is used to find the initial
(orthogonal) solution. Rotation is then done
(by default with quartimin using GPFoblq optimization).
factanal always uses the correlation matrix, so standardizing does
not affect the solution.
If rotation is "none" the result of the factanal
estimation is not rotated. In this case, to avoid confusion with a rotated
solution, the factor covariance matrix Phi is returned as NULL.
Another possibility for its value would be the identity matrix, but this is
not calculated so NULL avoids confusion.
The arguments rotation, rotationArgs are used for rotation.
The quartimin default uses GPArotation and its default
normalize=TRUE, eps=1e-5, maxit=1000, and Tmat=I
are passed through the rotation method to GPFoblq. The estimated loadings, Bartlett predictor matrix, etc.,
are put in the returned FAmodel (see below).
The Bartlett factor score coefficient matrix can be calculated as
$$(B' \Omega^{-1} B)^{-1} B' \Omega^{-1} x$$
or equivalently as
$$(B' \Sigma^{-1} B)^{-1} B' \Sigma^{-1} x,$$
The first is simpler because $\Omega$ is diagonal, but breaks down
with a Heywood case, because $\Omega$ is then singular (one or
more of its diagonal elements are zero). The second only requires
nonsingularity of $\Sigma$. Typically, $\Sigma$ is not singular
even if $\Omega$ is singular.
$\Sigma$ is calculated from $B \Phi B' + \Omega$,
where $B, \Phi,$ and $\Omega$ are the
estimated values returned from factanal and rotated.
The data covariance could also be used for $\Sigma$.
(It returns the same result with this estimation method.)
The returned FAmodel object is a list containing
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]