If you were charged with conducting a hierarchical factor analysis, then there is likely a presumed higher order (or hierarchical factor).  Additionally, since you are fitting the scale to another country/culture I would first advise the use of exploratory factor analysis.  You can do this very easily in SPSS or even R Code.  (You can do CFA using both as well).  I would also posited the eigenvalue > 1 is a poor choice for factor extraction decision-making.  You should perhaps use Horn's Parallel Analysis and/or the MAP procedure (Velicer).  Now to your question: If you want second order factors, then simply take the resulting first order corr matrix and do a second factor analysis on the six factors. I am assuming the lower order factors are fairly highly correlated so perhaps the EFA procedure of principal axis factoring with an oblique rotation is appropriate (i.e., promax or oblimin).   Better yet, you might consider using the Schmid-Leiman Procedure or exploratory bifactor analysis (both available in R) if there is a presumed higher order factor and if you find lower order factors fairly highly correlated. Check out some of my research posted here or that of Watkins or Canivez for further guidance. Best wishes to you.

Second order model is much easy  when you are using Amos software. Because you can handle the multiple variable in one analysis.

Dear Dilyana,

Firstly, I'd recommend the statstutor worksheet: http://statstutor.ac.uk/resources/uploaded/factoranalysis.pdf.

Secondly, I'd recommend Andy Field's text book on discovering statistics using SPSS.

Thank you all

If you were charged with conducting a hierarchical factor analysis, then there is likely a presumed higher order (or hierarchical factor).  Additionally, since you are fitting the scale to another country/culture I would first advise the use of exploratory factor analysis.  You can do this very easily in SPSS or even R Code.  (You can do CFA using both as well).  I would also posited the eigenvalue > 1 is a poor choice for factor extraction decision-making.  You should perhaps use Horn's Parallel Analysis and/or the MAP procedure (Velicer).  Now to your question: If you want second order factors, then simply take the resulting first order corr matrix and do a second factor analysis on the six factors. I am assuming the lower order factors are fairly highly correlated so perhaps the EFA procedure of principal axis factoring with an oblique rotation is appropriate (i.e., promax or oblimin).   Better yet, you might consider using the Schmid-Leiman Procedure or exploratory bifactor analysis (both available in R) if there is a presumed higher order factor and if you find lower order factors fairly highly correlated. Check out some of my research posted here or that of Watkins or Canivez for further guidance. Best wishes to you.

Dear Dilyana,

as you can see so far there is no possibility to perform direct hierarchical factor analysis in SPSS. Especially when you need to perform confirmatory factor analysis. Exploratory hierarchical factor analysis is possible to perform in Statistica for Windows, while confirmatory, as all said requires AMOS, R,...

But if you have (or use) just SPSS there is a way to perform exploratory hierarchical factor analysis.

First, after first exploratory factor analysis you should save your factors.

Then in second step, you choose those extracted factors (first order factors) as manifest variables (number of second order factors depends on your hypothesis or criteria that you have chose).

And finally, I would recommend to check it with Goldberg's hierarchical factor analysis from the top down http://projects.ori.org/lrg/PDFs_papers/Goldberg_2006_BassAckwards_JRP.pdf

Of course, this procedure can not replace the CFA in testing models but it can give you very useful information about structure of examined construct.

All the best

In your Spss, you run an EFA for every construct. If you EFA procedure produce more than one dimensions, then that particular construct might be a second order construct (normally termed as higher order construct).

To confirm the second order construct, you need to employ Spss-Amos where you draw the main construct linking to the dimensions (or in Cfa, they called the sub-constructs). So we have one main construct and a few sub-constructs. From here you can execute the Amos software for a second order Cfa. I give you one chapter from my book to read. Its very easy to understand.

12/21/15

See articles by Bruce Thompson at Texas A&M University, Texas and Larry Daniel at the University of North Florida.  See articles in Educational and Psychological Measurement by William L. Johnson &  Annabel M. Johnson for both SPSS and SAS  syntax codes.  Also use Horn's parallel analysis for the number of factors to extract.  Or, as Dr. Thompson notes, if you use the eigenvalue criterion (that usually over extracts the number of factors ), then use another method or two to add additional information for your exploratory analysis.  Such research is valuable building a research base in the area your are studying.

Here is a link to IBM's SPSS Help page providing information on how to perform a second-order factor analysis in SPSS.

Process is similar to Đorđe Čekrlija's explanation provided earlier.

Reference:

Can SPSS do a second-order factor analysis? http://www-01.ibm.com/support/docview.wss?uid=swg21479182

Hopefully someone else can verify the workability of IBM's syntax

You may have read of the recent death of Richard Gorsuch (the great author of the seminal book on factor analysis). In his last article, he did use the eigenvalue criterion in his solution.  I just mention this to note his solution.  I routinely check "Horn's Parallel Analysis" in solutions; however, I find it very interesting that Gorsuch did use the scree plot (and eigenvalue > 1) in his last article.  Please see my comment above for Drs. Thompson's and Daniel's  statements about this matter.

The SPSS code for higher order factor analysis used SPSS 20.0.  The article with the code and examples of its utility was published in Educational and Psychological Measurement.  One will have to check if the program is still applicable to the latest SPSS release.  If not, let me know.  We can work around the problem.

Dr. Johnson,

I ran the code provided in the Johnson and Johnson (2000) paper: Johnson, W. L., & Johnson, A. M. (2000). Using SPSS/PC for higher order factoring. Educational and Psychological Measurement, 60(4), 648-649.

I got several error messages running the program/code. Please see attached document.  

I'd appreciate your feedback on resolving these issues.

Tobi.

Hi Everyone,

I have described below the steps which I believe should be followed in assessing higher order constructs.

I will appreciate it if someone can verify whether my listed steps are correct. If not, please recommend changes where necessary:

1. Step 1: run EFA of individual factors (rotation: Varimax; extraction: PCA or ML)

2. Step 2: after combining all items for all factors re-run EFA (rotation: Direct Oblimin; extraction: PCA) to verify that items load into individual respective factors I need to  and .

3. Step 3: run Horn's parallel analysis using either Principal components Analysis or Principal Axis Factoring to determine number of higher order factor to extract .

4. Step 4: run Schmid-Leiman analysis in SPSS or SEM package (e.g. R) to assess higher order construct 

5. Step 5: run omega to estimate test's general factor saturation  .

6. Step 6: review estimates of individual items. Retain factor items with estimates >/= 0.5.

7. Step 7: review steps 1-6 for accuracy.

Thanks!

Tobi Popoola,

  Sure, I'll be happy to check this.  Question: you ran this on SPSS  23.0?  The published program was run in 2000 on an earlier version.  The version I use now is 20.0.  It looks like the newer version will not run the older SPSS version used in 2000.  This is unfortunate but a problem at times when new versions come out.  Let me take a look at your program, and I'll get back with you.

Tobi, from what I can assess you have attempted is to rotate the first order factor loadings.  This is not how you do a second order analysis. Following is what you do for a second order solution...then you use a program (like the printout you sent) to rotate the final second order solution.   Let's suppose you run a first order factor or component analysis on 25 questions  (doing an OBLIQUE rotation) and obtain a 25 x 4 PROMAX first order solution.  You then have to find the interfactor correlation matrix in your pages of the first order solution printout..  Factor that interfactor correlation matrix. Suppose you get a 4 x 2 matrix. Rotate that matrix to varimax (varimax will stop your solution at the second order).  Then take your 25 x 4 PROMAX rotated first order solution and postmultiply it  by the 4 x 2 VARIMAX rotated matrix.  This will give you a 25 x 2 second order solution. Then use the rotation program you sent and rotate the 25 x 2 matrix to the VARIMAX criterion.  This will complete your second order analysis.  You will have to decide what will be your salient items  in your final pattern-structure solution (like l0.40l, l0.50l, etc) for your interpretation of your 2nd order analysis. Also, the decision to conduct an orthogonal rotation at any order terminates the higher order sequence. Had you wanted to proceed to a third order solution, there are matrix rotations and some other checks to be sure you can proceed to another level of solution. 

Tobi, oh yes one more thing.  I run Horn's parallel analysis after each factor extraction to check how many factors to retain. Interesting, Richard Gorsuch (who wrote the classic book on factor analysis) died recently, but in his last article, he used the Scree test to determine the number of factors to retain). This just shows there is not only one way to solve problems....although most researchers may not use the Scree test.  Both solutions might agree.  I don't run Schmid-Lieman since the solution I use is another way to solve for higher-order solutions.   I believe it is conceptually easier to follow, and it does the same thing as the Schmid-Lieman solution.  

Dr. Johnson,

Thanks for the detailed information on running higher order factors.

Note on SPSS Version

It might be possible that the code is not running due to variations in the SPSS versions. However, I cannot conclude for a fact. Moreover, from your explanation my procedure requires amendment.

Web link to compute Horn's parallel analysis

In case you are interested, this link by Patil, Singh, Mishra, and Donovan (2007) provides a means to determine number of factors to retain. It is based on Horn's parallel analysis: http://ires.ku.edu/~smishra/parallelengine.htm. Users simply need to include 2 details (see below) before clicking the submit button to generate the result:

1. Number of Variables in your Dataset to be Factor Analyzed

2.  Sample Size of Your dataset.

Selection of Method to conduct a higher order factor analysis

 I dd observe earlier that the Johnson and Johnson approach differed from the S-L approach. For instance, I observed that their earlier paper ((Johnson, Johnson, and Heimberg, 1999) made no reference to S-L.

Although the S-L method is cited more (based on articles & authors who provide info on higher-order construct analysis), I presume that since the J&J approach was published in a reputable journal and has been used severally, researchers can justify using this method in lieu of the S-L approach. Are these presumptions correct? Or, are there more appropriate ways to justify the use of the J&J method?

Tobi, check Richard Gorsuch's book:  Factor Analysis.  He explains this approach.  See  chapter 11 of his 2nd edition book.  The first page of chapter 11 outlines the procedure I have used.

Discussion on the J&J Method

First, thanks for bringing up the discussion on assessing higher order factors greater than 2. Is Gorsuch (1983) a good source to learn more?

Second, I reviewed the SPSS print outt. SPSS produces following matrices: factor correlation matrix, factor score coefficient matrix, factor score covariance matrix, structure matrix, pattern matrix,  factor matrix, and correlation matrix. Although the inter-factor correlation matrix is not conspicuously displayed, is the factor correlation matrix same as inter-factor correlation matrix? Alternatively, is it either the factor matrix or correlation matrix?

Lastly, not proud to say I don't understand procedures below. I'll refer to Gorsuch (1983) to better understand the following steps you described in your hypothetical example:

1. Factoring that interfactor correlation matrix. Suppose you get a 4 x 2 matrix. Rotate that matrix to varimax (varimax will stop your solution at the second order).  

2. Taking your 25 x 4 PROMAX rotated first order solution and post multiplying it  by the 4 x 2 VARIMAX rotated matrix.

Thanks again for providing the Gorsuch (1983) reference.

 Tobi, when the original correlation matrix  (Rvv) is factored, the results are said to be the primary or first-order factors.  Then you factor the correlations among the factors (Rff).  This is the factor correlation matrix.  I designate it as the interfactor correlation matrix. Your then proceed with the steps I noted before.  Gorsuch's chapter 11 goes through all this.  The common factor model allows one to use the Schmid-Leiman  or the procedure Gorsuch discusses.   All this brings to bear what we all learning in our doctoral statistics classes and our matrix theory.  I would recommend your getting Gorsuch's book.

Quite informative. Thanks Dr. Johnson.

I already ordered the book on Amazon yesterday.

Tobi, your earlier question.  In the 1999 Johnson, Johnson & Heimberg article, I didn't include any of the details for the second order analysis because of the type of journal the article went to.  That readership would not have been interested in the statistical analysis detailed explanations; thus, the main focus was the findings of the research and the pertinent applications.

Dr. Johnson, thanks for clarifying.

"You then have to find the interfactor correlation matrix in your pages of the first order solution printout..  Factor that interfactor correlation matrix."

Could you please clarify what you mean by Factoring the correlation matrix? I have an 8-factor solution and have located the interfactor correlation matrix, but am unsure what you mean by this step. 

I have conducted Parallel Analysis on the factor scores produced from my 8 factors, suggesting a 2-factor higher-order solution. I have then instructed SPSS to enact MLE with Varimax rotation which has produced a relatively clean (in terms of loading) 2-factor solution. However, I am pretty sure I am missing the point here.

I am confused as to how I post-multiply by a correlation matrix, given that my correlation matrix contains 64 (8 x 8) numbers - therefore what is being multiplied by what?

Thanks.

 when there is a 2nd order factor in an EFA, the subdimensions of that factor will all load together, instead of in separate factors. In such cases, I recommend doing a separate EFA for the items of that 2nd order factor. Then, if that EFA results in removing some items to achieve discriminant validity, you can try putting the EFA back together with the remaining items (although it still might not work). Then, during the CFA, be sure to properly model the 2nd order factor with an additional latent variable connected to its sub-factors. statwik  for further analysis 

If you run a second order analysis, you should have strong correlations among the variables in the first order correlation matrix.  If this is not the case, then one should not do a higher order analysis.  As most of us do, also run Horn's parallel analysis (access the program by doing an internet search). The program is easy to use.  It will tell you how many factors you should have.  Or, you can run Velicer's MAP program.   Review the answers posted here by several persons.   Also, note the criticisms of using the eigenvalue > 1 solution to determine the number of first order factors.  However, if you are doing an exploratory analysis, this is your decision.  This would build up the research base for others to review in their work; thus, some of these decisions depend on the purpose of your study.  The difference in the two solutions (principal components and principal factors)  is what is put in the diagonal of the original correlation matrix. Principal components has ones in the main diagonal, but in principal factor analysis the communalities are estimated. With a large matrix, the two solutions are similar.  Fred Stevens gives  reasons (justifications) for doing a principal component analysis if one is doing an exploratory analysis.  I hope this helps.

Hi ? I ran CFA in AMOS and fit indices were poor . I ran EFA in SPSS (etracted six factors) and delete some Items that were cross-loading , then run CFA again and now fit indices are within acceptable range ( only TLI failed =087). SRMR =055, CFI=093, RMSEA=065 and CMIN/DF

Dr. Bruce Thompson at Texas A&M University has published a book:

Exploratory and Confirmatory Factor Analysis, American Psychological

Association, Washington, DC, 2004. In chapter six of his book, he covers

oblique rotations and higher-order factoring. He gives an example of

the Schmid-Leiman solution for higher-order analysis using actual

data. This would be a great resource explaining the Schmid-Leiman

solution for higher-order factor analysis. ISBN 1-59147-093-5.

Actually there is no such thing as second order factor analysis - only second order construct which consists of few first order constructs. sinoniim to first order constructs are components, dimensiond or factors. One conduct the CFA procedure to validate the second order constructs for unidimensionality, validity and reliability prior to modelling structural model and SEM