The methods are the same as for continuous outcomes and are described in
Joreskog, K.G. (1977). Factor analysis by least-squares and maximum-likelihood methods. In Statistical Methods for Digital Computers, K. Enslein, A. Ralston, and H.S. Wilf (Eds.). New York: John Wiley & Sons, pp. 125-153.
This means that the factor covariance matrix is an identity matrix and the remaining restrictions needed are imposed to make the computations simple, such as restricting
Lambda' * Theta_inverse * Lambda
to be diagonal.
Hervé CACI posted on Sunday, February 23, 2003 - 11:46 am
Would it be interesting in the near future that Mplus:
1. Output the unrotated factor loadings, so that users can run any rotation (other than VARIMAX and PROMAX) with another statistical package ?
2. Allow the exponent of the PROMAX rotation be modifiable (this was my wish some months ago) ?
bmuthen posted on Sunday, February 23, 2003 - 12:55 pm
The Mplus team has noted your interest in this.
Regarding rotations, PROMAX is obtained starting from a VARIMAX rotation. Let me ask you which rotations that you are interested in cannot start from a VARIMAX rotation?
Regarding the PROMAX rotation, do you have an example where different exponent values give substantively important differences of interpretation?
Hervé CACI posted on Monday, February 24, 2003 - 1:59 am
I was essentially refering to the OBLIMIN rotation. Raymond B. Cattell among others (see also Paul Kline) argued that simple structure (i.e. replicable solutions) is more easily reached with OBLIMIN than with other rotations. See his 1978 book (The Scientific Use of Factor Analysis) or Kline's "The New Psychometrics".
To do this and to take advantage of MPlus categorical variables analysis capabilities, the user has to have access to the unrotated factor loading matrix, unless there is a method to "unrotated" the VARIMAX solution ? I'm vaguely thinking at Kaiser's approach of factor similarity (Psychometrika, 198?), but I'm not a mathematician...
Regarding the PROMAX exponent values, my own (little) experience shows that augmenting the obliquity of the factors can help uncover a "simpler" solution although I haven't ever reached a simple structure in Cattell's word. Gorsuch suggest in his book ("Factor Analysis") that the researcher varies the exponent values to find the "better" solution.
By the way, let me add a further question. How can I compute the uniqueness vector from the VARIMAX and/or PROMAX outputs ?
bmuthen posted on Tuesday, February 25, 2003 - 9:11 am
Will take a look at the Oblimin when I get a chance if you give me the full Kline ref. Unrotated loadings means different things with different estimation procedures (see Joreskog's writings).
PROMAX gives simpler loadings but higher factor correlations when increasing the exponent. Mplus uses the value recommended in the Lawley-Maxwell book; this doesn't seem to be in need of change.
Uniquenesses are printed as the residuals in Mplus.
Anonymous posted on Monday, August 04, 2003 - 9:33 am
How do I put linear constraints on factor loadings (both in EFA and the measurement submodel in SEM)?
bmuthen posted on Monday, August 04, 2003 - 9:45 am
EFA does not allow constraints to be placed on the loadings - the rotation algorithms find simple structure instead. In CFA and SEM, currently the only linear restrictions are fixed and equal constraints, e.g.
f by y1 (1) y2 (1);
says that the y1 and y2 loadings are equal.
Saul Cohn posted on Saturday, February 25, 2006 - 12:17 pm
My unroated, orthogonal, and oblique factor loadings all look very similar. Which one then is ideal?
Factor 1 eigenvalue 2.064 Factor 2= 1.134
bmuthen posted on Saturday, February 25, 2006 - 12:29 pm
It is good that you wouldn't make different conclusions based on rotation method. Many people find the oblique solution natural to focus on, unless you have specific reasons to consider uncorrelated factors.
Ann Haas posted on Friday, August 31, 2012 - 12:47 pm
Is there a method in the current version of Mplus to output unrotated factor loadings using EFA?