How to obtain the same varimax-rotated PCA results in MATLAB and SPSS?

Question

I'm trying to perform a PCA Extraction + Varimax Rotation in MATLAB and obtain the same results as in SPSS.

My data is the following matrix A:

var1  var2  var3
----------------
10    7     3
3     10    8
8     2     6

This is the syntax I used in SPSS:

FACTOR
/VARIABLES var1 var2 var3
/MISSING LISTWISE 
/ANALYSIS var1 var2 var3
/PRINT EXTRACTION ROTATION
/CRITERIA FACTORS(3) ITERATE(100)
/EXTRACTION PC
/CRITERIA ITERATE(25)
/ROTATION VARIMAX
/METHOD=CORRELATION.

I have no problems to find the eigeinvalues and the variances using:

[V,D] = eig(corr(A));
eig_total = diag(D); % Eigeinvalues
var_exp = D./sum(D); % Percentage of variance
cum_var = cumsum(var_exp); % Cumulative variance
% is there a way to obtain the same resuluts using the pca function?

However, I don't know how to obtain these tables:

I've tried the following code, but none of them give me the same results:

rotated_solution1 = rotatefactors(V)
rotated_solution2 = rotatefactors(pca(A))

Did you google `matlab factor analysis`? http://www.mathworks.com/help/stats/factoran.html — ttnphns, Sep 08 '15 at 19:25
@ttnphns Yes I did, however this function is based on maximum likelihood estimate, it is not perfoming a PCA — mat, Sep 08 '15 at 19:27

score 2 · Accepted Answer · edited Apr 13 '17 at 12:44

Factor analysis rotates loadings, not eigenvectors; see my answer here for a lengthy discussion: Is PCA followed by a rotation (such as varimax) still PCA?

Your V are eigenvectors, and loadings are given by V*sqrt(D), so what you need to do is rotatefactors(V*sqrt(D)). But it's better to make sure that the zero column of V is kicked out, otherwise Matlab seems to run into some numerical issues.

Here is the code:

%// PCA
[V,D] = eig(corr(A));

%// Sorting the eigenvectors/eigenvalues
[~, ind] = sort(diag(D), 'descend');
D = D(ind, ind);
V = V(:, ind);

%// Keeping only the first two (non-zero ones)
D = D(1:2,1:2);
V = V(:,1:2);

%// Computing loadings
L = V*sqrt(D);

%// Rotating loadings
L_rotated = rotatefactors(L);

Now L is

    0.9950   -0.1001
   -0.6617   -0.7497
   -0.8970    0.4420

and L_rotated is

    0.9072    0.4208
   -0.1874   -0.9823
   -0.9971   -0.0767

the same as in SPSS. Note that the column signs are arbitrary, see here: Does the sign of scores or of loadings in PCA or FA have a meaning? May I reverse the sign?

P.S. Answering your question from the comments: I don't think there is a built-in function in Matlab that would do all of that.

One should also take care of the option "Kaiser normalization" in SPSS as default for Varimax. (which should be indifferent only if all pc's are used for rotation). I don't know how matlab does handle this... — Gottfried Helms, Mar 25 '17 at 22:18

How to obtain the same varimax-rotated PCA results in MATLAB and SPSS?

1 Answers1