I am trying to get into SEM and factor analysis. I understand a factor is a latent construct, say e.g. $intelligence$, user-defined by the (weighted) average of a set of indicators $x_1, x_2\dots x_n$. The $x_i$ load (i.e. correlate) on the factor differently and the loadings should be over $0.40$ at minimum. So far so good (presumed).
When I run a factor analysis in Stata with the factor
command, in the output there are a number of factors being displayed instead of only one which I define. The literature I found tells then, the factor with the highest eigenvalue is the best one.
Why there are a number of factors being displayed since with my indicators $x_i$ I intend to define just one? All the literature I found just says e.g. "the first factor has the strongest eigenvector" or "only one factor has an eigenvector greater than $1$". I'm missing the step where factors or their sets respectively are derived from their sets of indicators. I'm confused now, what factors actually are since in a defined set of indicators there could be more than one.
I am quite sure my question is very obvious to anybody being familiar with this stuff. I'd appreciate any clarification though.
Here is an example of a Stata output that should look familiar to anybody.
. factor x1-x9, pcf
(obs=1,625)
Factor analysis/correlation Number of obs = 1,625
Method: principal-component factors Retained factors = 1
Rotation: (unrotated) Number of params = 9
--------------------------------------------------------------------------
Factor | Eigenvalue Difference Proportion Cumulative
-------------+------------------------------------------------------------
Factor1 | 3.76124 2.80650 0.4179 0.4179
Factor2 | 0.95473 0.10627 0.1061 0.5240
Factor3 | 0.84847 0.10176 0.0943 0.6183
Factor4 | 0.74671 0.05561 0.0830 0.7012
Factor5 | 0.69110 0.07429 0.0768 0.7780
Factor6 | 0.61681 0.07780 0.0685 0.8466
Factor7 | 0.53900 0.09177 0.0599 0.9065
Factor8 | 0.44723 0.05252 0.0497 0.9561
Factor9 | 0.39471 . 0.0439 1.0000
--------------------------------------------------------------------------
LR test: independent vs. saturated: chi2(36) = 3863.18 Prob>chi2 = 0.0000
Factor loadings (pattern matrix) and unique variances
---------------------------------------
Variable | Factor1 | Uniqueness
-------------+----------+--------------
x1 | 0.6243 | 0.6103
x2 | 0.5883 | 0.6539
x3 | 0.7222 | 0.4785
x4 | 0.7131 | 0.4915
x5 | 0.5818 | 0.6615
x6 | 0.6197 | 0.6160
x7 | 0.6085 | 0.6297
x8 | 0.5968 | 0.6439
x9 | 0.7392 | 0.4535
---------------------------------------
factor
command in stata. It corresponds to what is sometimes called exploratory factor analysis (EFA). So basically, you explore the patterns/structure underlying observed variables. That is why it suggests several factors. In Confirmatory Factor Analysis (CFA, which is a form of SEM), you specify the factor and test a hypothesis on the factor structure. There are many questions on EFA and CFA on CV. I recommend checking them. $\endgroup$pcf
I actually used principal component analysis (PCA). So your point is, PCA recommends me factors, aha. I could not find any question that clarifies my problem more than your comment, why don't you consider to point me to one of these or put some effort in it and make an answer out of it, thanks. $\endgroup$pcf
) yield very similar results, but they are not the same. In fact, PCA and EFA are used for different tasks. Here are some links which contain detailed discussions: stats.stackexchange.com/questions/1576/… , stats.stackexchange.com/questions/311020/… , stats.stackexchange.com/questions/123063/… , $\endgroup$