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?
I would like to join myself to the request done above by some people. For many different reasons, one may be interested in obtaining the unrotated solution. So please, I think it should be interesting to be able to have this in the output. I have tried with the solution proposed by Bengt on August 31st, 2012, but it does not give the result I expected (that the first factor explains the largest possible amount of common variance).
The unrotated Mplus solution does not order the factors in terms of common variance. It is formed based on what I stated in my Aug 31 post. The ordering of factors due to size of common variance would seem more of a principal component aim than a factor analysis aim.
Dear Bengt, thank you so much for your response. Definitely I was confused about what the unrotated solution would be. However, I have tried with other pieces of software, e.g., the psych package for R, and the output of the unrotated solution gives the result I expected: the explained common variance of the first factor equals the first eigenvalue of the reduced correlation matrix. (Unfortunately, I have some other needs that this package does not cover, e.g. giving estimation errors for the estimates).
May I ask if the unrotated solution has something to do with the factor extraction method, e.g. that a principal axes method would give the solution I expected? If so, could it be the case that the EFA analysis (instead of the ESEM) would give the most explained common variance in the first factor?
It sounds like that psych package is a bit outdated in that it uses a principal component analysis (PCA) estimator of the factor model. That is not the best estimator because it doesn't consider measurement error (which factor analysis does). The default ML estimator of Mplus is better.
If you use an orthogonal rotation method (which is like PCA) - where the factors are uncorrelated - you can compute the variance contribution by each factor by summing its square loadings. In your results table you can then re-arrange the factor columns of loadings in the factor order of decreasing variance contribution. The order of the factors is arbitrary in the sense that different orders give the same model fit. Mplus allows orthogonal rotation.
Thank you so much for your response again. I took a look at the help file for this function, and this is what it says about the procedure:
"Perhaps the most conventional technique is principal axes (PAF). An eigen value decomposition of a correlation matrix is done and then the communalities for each variable are estimated by the first n factors. These communalities are entered onto the diagonal and the procedure is repeated until the [trace of the reduced correlation matrix] does not vary. Yet another estimate procedure is maximum likelihood. For well behaved matrices, maximum likelihood factor analysis (either in the fa or in the factanal function) is probably preferred."
I guess being that the case, the procedure used by this package is substantially different from the one done in Mplus. Actually, I tried with the ML procedure in the psych package and, effectively, the explained common variance of the first factor does not match the first eigenvalue, as you say. In any case, it doesn't seem to do a PCA but PAF instead (which I understand are closely related but are not the same, being PAF appropriate for exploratory factor analysis), as explained in the help file.
Anyways, I understand that Mplus does PAF, nor there are plans for implementing such an extraction method, is it so? Thank you so much.
PAF is a common but largely outdated method. Mplus does not plan to include it. Apart from ML we also offer ULS. If you want factors ordered by most variance explained (limiting to the uncorrelated case) you can re-order the factors in the tables you present without altering the model.
Hi again. Thank you so much for your responses. I understand now that PAF is not a viable alternative, so I only have the option of finding a rotation method that gives me the maximum of explained common variance for the general factor.
My final conclusion is that, based on the fact that most rotation methods are based on (variations of) maximizing/minimizing some criterion to find a simple structure, there does not exist up to now such a rotation method that can adapt to my needs. I have tried the only two options I find plausible: a target, and a bi-geomin(orthogonal) rotation. Only the second option seemed to give an acceptable result.