The function estTSF.ML is a wrapper to estTSFmodel.
The function estTSF.ML estimates 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, if specified, is then done
with GPFoblq.
factanal always uses the correlation matrix, so standardizing does
not affect the solution.
If diff. is TRUE (the default) the indicator data is differenced
before it is passed to factanal. This is necessary if the data is not
stationary. The resulting Bartlett factor score coefficient matrix (rotated)
is applied to the undifferenced data. See Gilbert and Meijer (2005) for a
discussion of this approach.
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, methodArgs, normalize,
eps, maxit, and Tmat are passed to
GPFoblq.
The estimated loadings, Bartlett factor score coefficient matrix and
predicted factor scores
are put in a TSFmodel which is part of the returned object.
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 TSFestModel object is a list containing
- model
the estimated TSFmodel.
- data
the indicator data used in the estimation.
- estimates
a list of
- estimation
- a character string indicating the name of the
estimation function.
- diff.
- the setting of the argument
diff.
- rotation
- the setting of the argument
rotation.
- uniquenesses
- the estimated uniquenesses.
- BpermuteTarget
- the setting of the argument
BpermuteTarget.