How to get the structure matrix in R using the factanal() function?

Question

This is my first post on this forum. I am new to R and factor analysis and I think I have a very simple question. I'm following Andy Field's "Discovering Statistics Using R" book (2012) for general guidance. Field recommends providing both the pattern and structure matrices when performing factor analysis using oblique rotation.

While the pattern matrix is simply the table of loadings, I am having more difficulty obtaining the structure matrix in R using the factanal() function.

To obtain the structure matrix for a PCA using the principal() function, Field provides the following formula: fit$loadings %*% fit$Phi , which multiplies the factor loading matrix by the factor correlation matrix. Although factanal() does store the factor correlations somewhere—as it provides them in a section of the general output (under loadings)—I cannot find an object called "Phi", "Factor Correlations", or an equivalent, within the output of the fit model (within R Studio). Thus, I don't know what term to put in Field's formula to replace Phi in order to get the structure matrix.

I have seen here and here that earlier, factanal() did not provide the factor correlations—yet now it does provide it in the output; I just don't know the right terminology to access it. Thanks for any help on this!

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Edit: As per the book, I use the following formula with four factors and oblique rotation for PCA:

pc4 <- principal(raqData, nfactors = 4, rotate = "oblimin").

In that case when I double click on object "pc4" in R Studio there is an object called "Phi" and I have no trouble obtaining the structure matrix with pc4$loadings %*% pc4$Phi .

Next, I attempt to use EFA, instead of PCA, again with four factors and oblique rotation (promax). This step works and I can get the factor correlations with (cut some output for conciseness):

> fit <- factanal(mydata, 4, rotation="promax")
> fit
Call...
Uniquenesses...
Loadings...
SS loadings...
Factor Correlations:
        Factor1 Factor2 Factor3 Factor4
Factor1   1.000  -0.874   0.632   0.571
Factor2  -0.874   1.000  -0.118  -0.438
Factor3   0.632  -0.118   1.000   0.356
Factor4   0.571  -0.438   0.356   1.000
Test of the hypothesis that 4 factors are sufficient...

Next, I attempt the structure matrix formula and get the following error:

> fit$loadings %*% fit$Phi
Error in fit$loadings %*% fit$Phi : 
  requires numeric/complex matrix/vector arguments

But when I inspect the "fit" object obtained with factanal() , there is no object called 'Phi" (as there were for principal() ). Not sure how to interpret the error above either. I have seen this error discussed here and here but I'm not clear about how to resolve it here.

Answer 1

Though this is an old post but for helping other people out, the two components loadings and psi (not phi) in factanal output are fit$loadings and fit$uniqueness. You can access loadings matrix by writing fit$loadings[,1:q] where q is the factor you used. For example- I ran factanal on Harman23.cor data present in the R studio. And checked for loadings to get below output.

> Harman23.FA <- factanal(factors = 3, covmat = Harman23.cor)
> Harman23.FA$loadings

Loadings:
               Factor1 Factor2 Factor3
height          0.886   0.267  -0.130 
arm.span        0.937   0.195   0.280 
forearm         0.874   0.188         
lower.leg       0.877   0.230  -0.145 
weight          0.242   0.916  -0.106 
bitro.diameter  0.193   0.777         
chest.girth     0.137   0.755         
chest.width     0.261   0.646   0.159 

               Factor1 Factor2 Factor3
SS loadings      3.379   2.628   0.162
Proportion Var   0.422   0.329   0.020
Cumulative Var   0.422   0.751   0.771

Clearly $loadings is not returning a matrix. Instead you can use this:

> as.matrix(Harman23.FA$loadings[,1:3])
                 Factor1   Factor2      Factor3
height         0.8859687 0.2666293 -0.130079754
arm.span       0.9371891 0.1951310  0.280364451
forearm        0.8741374 0.1876181  0.089154498
lower.leg      0.8768750 0.2303982 -0.144815728
weight         0.2422827 0.9164637 -0.106482093
bitro.diameter 0.1925419 0.7766283 -0.020581101
chest.girth    0.1368022 0.7554074  0.003677235
chest.width    0.2605798 0.6458413  0.159110501

Note that the factor used here is 3 and hence while accessing loadings we have mentioned colums 1:3. Similarly for running factanal with a factor q, you should mention Harman23.FA$loadings[,1:q].

Also note that psi which is the uniquess matrix is a diagonal matrix. So you should access it like this

> psihat <- diag(Harman23.FA$uniquenesses)
> psihat
          [,1]  [,2]     [,3]      [,4]       [,5]      [,6]      [,7]      [,8]
[1,] 0.1270475 0.000 0.000000 0.0000000 0.00000000 0.0000000 0.0000000 0.0000000
[2,] 0.0000000 0.005 0.000000 0.0000000 0.00000000 0.0000000 0.0000000 0.0000000
[3,] 0.0000000 0.000 0.192735 0.0000000 0.00000000 0.0000000 0.0000000 0.0000000
[4,] 0.0000000 0.000 0.000000 0.1570353 0.00000000 0.0000000 0.0000000 0.0000000
[5,] 0.0000000 0.000 0.000000 0.0000000 0.09005497 0.0000000 0.0000000 0.0000000
[6,] 0.0000000 0.000 0.000000 0.0000000 0.00000000 0.3593523 0.0000000 0.0000000
[7,] 0.0000000 0.000 0.000000 0.0000000 0.00000000 0.0000000 0.4106308 0.0000000
[8,] 0.0000000 0.000 0.000000 0.0000000 0.00000000 0.0000000 0.0000000 0.4896708

Otherwise your matrix multiplication fit$loadings %*% fit$Phi will be non-conformal