# "Two stage" factor Analysis: factoring saved factor scores

I have a model which consists of 19 questions, which are divided in three factors (factor 1 - nine questions, factor 2 - six questions, factor 3 - four questions). For this I did a factor analysis and saved factor scores (computed by regression method) as variables.

But these three factors measure one big construct, to me. Now I was wondering if I can put the three created scores variables in a factor analysis again and extract one factor this time and save its factor scores as a variable, - to measure the overall construct.

Is this correct/possible to do?

** additional information ** To give some aditional information: I extracted the factors based on 'maximum likelihood', with a promax rotation. Saved these scores as regression variables, put those three variables in the factor analysis (SPSS). With the same method and was wondering if that would be a good construct.

In this case I factored, innovativeness (9 items), pro-activeness (6 items) and risk taking behavior (4 items). And I wanted to measure the overall construct, Intrapreneurial Behavior (IB) in this way. What do you suggest, how to deal with this issue to create a construct for IB?

• I would agree with both @JeremyMiles and DLDahly answers presenting two different (exploratory and confirmatory) approaches. By the way, when you do, after your oblique rotation, second-order factor analysis, you don't need to use the saved factor scores (which are always approximate in FA) as input in order to perform the second factoring and obtain its loadings. Use just the correlation matrix between the oblique factors as the input. That matrix is always returned by FA under oblique rotation ant it is not necessarily identical to the correlations between the saved factor scores. Nov 28 '15 at 18:35

Of course, it's "possible" to do what you're asking. The question is whether or not this is the best way to deal with the issue. You have left out mention of a number of important considerations: first, did you rotate a PCA to create a CFA with 3 factors? That you've noted "cfa" as a keyword, suggests rotation. To me, this means "common factor analysis." Is that correct?

One thing that often gets ignored about unrotated PCA is that it results in a mathematically unique solution where the first factor has been called a "junk" factor by some academics insofar as everything loads on it. Rotation cancels uniqueness by adjusting the loadings across the retained factors to something called "simple structure." The goal of simple structure is that each variable load on a single factor only and be zero (or close to it) for the other factors. Given that, have you examined the first, unrotated PCA component for its value wrt your objective?

Next, factor analysis results in a set of linear combinations that recover a reduced percentage of the total variance. A second, higher-order factor analysis would reduce the recovered variance even more.

Finally, if you want to get really geeky, check out the literature on additive and ultrametric trees for a good discussion of second-order factor analysis. This is not an area that's seen much recent research that I'm aware of but there's a Sage book with this title by James Corter that dates back 25 years or so.

In my opinion, leveraging the first PC would be a safe, easy solution.

• 1) There is no rotation in confirmatory factor analysis, and 2) if the question really is about CFA, then factor analysis does not result in linear combinations that maximally describe total variance - see this thread for important distinctions between PCA and FA. stats.stackexchange.com/questions/123063/… Jun 3 '15 at 15:13
• There is sort of rotation - you just choose it yourself. Jun 3 '15 at 16:14
• To give some aditional information: I extracted the factors based on 'maximum likelihood', with a promax rotation. Saved these scores as regression variables, put those three variables in the factor analysis (SPSS). Some method and was wondering if that would be a good construct. In this case I factored, innovativeness (9 items), pro-activeness (6 items) and risk taking behavior (4 items). And I wanted to measure the overall construct, Intrapreneurial Behavior (IB) in this way. What do you suggest, how to deal with this issue to create a construct for IB?
– Timv
Jun 4 '15 at 11:41
• That's very important information and it would be better if you could edit your question to reflect it there. Jun 4 '15 at 11:45
• @Jeremy Miles I agree with your statement that, like everything else to do with PCA, CFA and, the option of rotating the factors input to SEM is a subjective choice of the analyst. Jun 5 '15 at 15:46

Yes, you can do this. It's called 'second order' or 'higher order (if you go more than two, although I'm not sure I've seen this) factor analysis.

It's got a (short) Wikipedia entry: http://en.wikipedia.org/wiki/Higher-order_factor_analysis

I interpret your question as wanting to estimate a latent construct with 3 indicators, and that each of these indicators is also a latent construct, each with some set of observed indicators. This kind of confirmatory factor analysis is certainly possible (in AMOS, Mplus, and probably any other software). If you go down this path, you want to estimate it as one big model, rather than estimating factor scores and then using those in a subsequent CFA. You also want to make sure your 3 "smaller" CFAs make sense before moving onto the full model.