Steps done in factor analysis compared to steps done in PCA

Question

I know how to perform PCA (principal component analysis), but I would like to know steps that should be used for factor analysis.

To perform PCA, let us consider some matrix $A$, for instance:

         3     1    -1
         2     4     0
         4    -2    -5
        11    22    20

I have calculated its correlation matrix B = corr(A):

        1.0000    0.9087    0.9250
        0.9087    1.0000    0.9970
        0.9250    0.9970    1.0000

Then I have done eigenvalue decomposition [V,D] = eig(B), resulting in eigenvectors:

        0.5662    0.8209   -0.0740
        0.5812   -0.4613   -0.6703
        0.5844   -0.3366    0.7383

and eigenvalues:

        2.8877         0         0
             0    0.1101         0
             0         0    0.0022

General idea behind the PCA is to choose significant components, form new matrix which have columns eigenvectors, then we need to project original matrix (in PCA it is zero centered). But in factor analysis, for instance we should choose components that have higher that $1$ singular value, also then we are using rotation of factors, please tell me how it is done? For instance in this case.

Please help me understand factor analysis steps, as compared to the PCA steps.

Answer 1

This answer is to show concrete computational similarities and differences between PCA and Factor analysis. For general theoretical differences between them, see questions/answers 1, 2, 3, 4, 5.

Below I will do, step by step, Principal Component analysis (PCA) of iris data ("setosa" species only) and then will do Factor analysis of the same data. Factor analysis (FA) will be done by Iterative principal axis (PAF) method which is based on PCA approach and thus makes one able to compare PCA and FA step-by-step.

Iris data (setosa only):

  id  SLength   SWidth  PLength   PWidth species
1      5.1      3.5      1.4       .2 setosa 
   2      4.9      3.0      1.4       .2 setosa 
   3      4.7      3.2      1.3       .2 setosa 
   4      4.6      3.1      1.5       .2 setosa 
   5      5.0      3.6      1.4       .2 setosa 
   6      5.4      3.9      1.7       .4 setosa 
   7      4.6      3.4      1.4       .3 setosa 
   8      5.0      3.4      1.5       .2 setosa 
   9      4.4      2.9      1.4       .2 setosa 
  10      4.9      3.1      1.5       .1 setosa 
  11      5.4      3.7      1.5       .2 setosa 
  12      4.8      3.4      1.6       .2 setosa 
  13      4.8      3.0      1.4       .1 setosa 
  14      4.3      3.0      1.1       .1 setosa 
  15      5.8      4.0      1.2       .2 setosa 
  16      5.7      4.4      1.5       .4 setosa 
  17      5.4      3.9      1.3       .4 setosa 
  18      5.1      3.5      1.4       .3 setosa 
  19      5.7      3.8      1.7       .3 setosa 
  20      5.1      3.8      1.5       .3 setosa 
  21      5.4      3.4      1.7       .2 setosa 
  22      5.1      3.7      1.5       .4 setosa 
  23      4.6      3.6      1.0       .2 setosa 
  24      5.1      3.3      1.7       .5 setosa 
  25      4.8      3.4      1.9       .2 setosa 
  26      5.0      3.0      1.6       .2 setosa 
  27      5.0      3.4      1.6       .4 setosa 
  28      5.2      3.5      1.5       .2 setosa 
  29      5.2      3.4      1.4       .2 setosa 
  30      4.7      3.2      1.6       .2 setosa 
  31      4.8      3.1      1.6       .2 setosa 
  32      5.4      3.4      1.5       .4 setosa 
  33      5.2      4.1      1.5       .1 setosa 
  34      5.5      4.2      1.4       .2 setosa 
  35      4.9      3.1      1.5       .2 setosa 
  36      5.0      3.2      1.2       .2 setosa 
  37      5.5      3.5      1.3       .2 setosa 
  38      4.9      3.6      1.4       .1 setosa 
  39      4.4      3.0      1.3       .2 setosa 
  40      5.1      3.4      1.5       .2 setosa 
  41      5.0      3.5      1.3       .3 setosa 
  42      4.5      2.3      1.3       .3 setosa 
  43      4.4      3.2      1.3       .2 setosa 
  44      5.0      3.5      1.6       .6 setosa 
  45      5.1      3.8      1.9       .4 setosa 
  46      4.8      3.0      1.4       .3 setosa 
  47      5.1      3.8      1.6       .2 setosa 
  48      4.6      3.2      1.4       .2 setosa 
  49      5.3      3.7      1.5       .2 setosa 
  50      5.0      3.3      1.4       .2 setosa 

We have 4 numeric variables to include in our analyses: SLength SWidth PLength PWidth, and the analyses will be based on covariances, which is the same as to say that we analyse centered variables. (If we chose to analyse correlations that would be analysing standardized variables. Analysis based on correlations produce different results than analysis based on covariances.) I will not display the centered data. Let's call these data matrix X.

PCA steps:

Step 0. Compute centered variables X and covariance matrix S.
Covariances S (= X'*X/(n-1) matrix: see https://stats.stackexchange.com/a/22520/3277)
.12424898   .09921633   .01635510   .01033061
.09921633   .14368980   .01169796   .00929796
.01635510   .01169796   .03015918   .00606939
.01033061   .00929796   .00606939   .01110612
Step 1.1. Decompose data X or matrix S to get eigenvalues and right eigenvectors.
          You may use svd or eigen decomposition (see https://stats.stackexchange.com/q/79043/3277)
Eigenvalues L (component variances) and the proportion of overall variance explained
           L            Prop
PC1   .2364556901   .7647237023 
PC2   .0369187324   .1193992401 
PC3   .0267963986   .0866624997 
PC4   .0090332606   .0292145579
Eigenvectors V (cosines of rotation of variables into components)
              PC1           PC2           PC3           PC4
SLength   .6690784044   .5978840102  -.4399627716  -.0360771206 
SWidth    .7341478283  -.6206734170   .2746074698  -.0195502716 
PLength   .0965438987   .4900555922   .8324494972  -.2399012853 
PWidth    .0635635941   .1309379098   .1950675055   .9699296890
Step 1.2. Decide on the number M of first PCs you want to retain.
          You may decide it now or later on - no difference, because in PCA values of components do not depend on M.
          Let's M=2. So, leave only 2 first eigenvalues and 2 first eigenvector columns.
Step 2. Compute loadings A. May skip if you don't need to interpret PCs anyhow.
Loadings are eigenvectors normalized to respective eigenvalues: A value = V value * sqrt(L value)
Loadings are the covariances between variables and components.
Loadings A
              PC1           PC2

SLength    .32535081     .11487892
SWidth     .35699193    -.11925773
PLength    .04694612     .09416050
PWidth     .03090888     .02515873
Sums of squares in columns of A are components' variances, the eigenvalues
Standardized (rescaled) loadings.
St. loading is Loading / sqrt(Variable's variance);
these loadings are computed if you analyse covariances, and are suitable for interpretation of PCs
(if you analyse correlations, A are already standardized).
              PC1           PC2

SLength    .92300804     .32590717
SWidth     .94177127    -.31461076
PLength    .27032731     .54219930
PWidth     .29329327     .23873031
Step 3. Compute component scores (values of PCs).
Regression coefficients B to compute Standardized component scores are: B = Adiag(1/L) = inv(S)A
B
              PC1           PC2

SLength   1.375948338   3.111670112 
SWidth    1.509762499  -3.230276923 
PLength    .198540883   2.550480216 
PWidth     .130717448    .681462580
Standardized component scores (having variances 1) = X*B
      PC1           PC2
  .219719506   -.129560000 
 -.810351411    .863244439 
 -.803442667   -.660192989 
-1.052305574   -.138236265 
  .233100923   -.763754703 
 1.322114762    .413266845 
 -.606159168  -1.294221106 
 -.048997489    .137348703 
  ...
Raw component scores (having variances = eigenvalues) can of course be computed from standardized ones.
In PCA, they are also computed directly as X*V
      PC1           PC2
  .106842367   -.024893980 
 -.394047228    .165865927 
 -.390687734   -.126851118 
 -.511701577   -.026561059 
  .113349309   -.146749722 
  .642900908    .079406116 
 -.294755259   -.248674852 
 -.023825867    .026390520 
  ...

FA (iterative principal axis extraction method) steps:

Step 0.1. Compute centered variables X and covariance matrix S.
Step 0.2. Decide on the number of factors M to extract.
          (There exist several well-known methods in help to decide, let's omit mentioning them. Most of them require that you do PCA first.)
          Note that you have to select M before you proceed further because, unlike in PCA, in FA loadings and factor values depend on M.
          Let's M=2.
Step 0.3. Set initial communalities on the diagonal of S.
          Most often quantities called "images" are used as initial communalities (see https://stats.stackexchange.com/a/43224/3277).
          Images are diagonal elements of matrix S-D, where D is diagonal matrix with diagonal = 1 / diagonal of inv(S).
          (If S is correlation matrix, images are the squared multiple correlation coefficients.)
With covariance matrix, image is the squared multiple correlation multiplied by the variable variance.
S with images as initial communalities on the diagonal
.07146025  .09921633  .01635510  .01033061
.09921633  .07946595  .01169796  .00929796
.01635510  .01169796  .00437017  .00606939
.01033061  .00929796  .00606939  .00167624
Step 1. Decompose that modified S to get eigenvalues and right eigenvectors.
        Use eigen decomposition, not svd.
        (Some last eigenvalues may be negative. This is because a reduced covariance matrix can be not positive semidefinite.)
Eigenvalues L
F1   .1782099114
F2   .0062074477
    -.0030958623
    -.0243488794
Eigenvectors V
               F1            F2 
SLength   .6875564132   .0145988554   .0466389510   .7244845480
SWidth    .7122191394   .1808121121  -.0560070806  -.6759542030
PLength   .1154657746  -.7640573143   .6203992617  -.1341224497
PWidth    .0817173855  -.6191205651  -.7808922917  -.0148062006
Leave the first M=2 values in L and columns in V.
Step 2.1. Compute loadings A.
Loadings are eigenvectors normalized to respective eigenvalues: A value = V value * sqrt(L value)
               F1            F2 
SLength   .2902513607   .0011502052
SWidth    .3006627098   .0142457085
PLength   .0487437795  -.0601980567
PWidth    .0344969255  -.0487788732
Step 2.2. Compute row sums of squared loadings. These are updated communalities.
          Reset the diagonal of S to them
S with updated communalities on the diagonal
.08424718  .09921633  .01635510  .01033061
.09921633  .09060101  .01169796  .00929796
.01635510  .01169796  .00599976  .00606939
.01033061  .00929796  .00606939  .00356942
REPEAT Steps 1-2 many times (iterations, say, 25)
Extraction of factors is done.
Let us look at the final eigenvalues of the reduced covariance 
matrix after iterations:
Eigenvalues L
F1   .2026316056
F2   .0137096989
     .0005000572
    -.0005882867
The eigenvalues are the factors' (F1 and F2) variances. The overall common variance is .2026316056 + .0137096989 = .2163413036,
so F1, for example, explains .2026316056/.2163413036 = 93.7% of the common variance.
That 93.7% of the common variance amounts to .2026316056/.3092040816 = 65.5% of the total variability 
(.3092040816, the total variance, is the trace of the initial, non-reduced covariance matrix).
[Note. The .0005000572 + -.0005882867 do not count a common variance; these "dross" eigenvalues are nonzero due to the fact
the 2-factor model does not predict the covariances without any error.]
Final loadings A and communalities (row sums of squares in A).
Loadings are the covariances between variables and factors.
Communality is the degree to what the factors load a variable, it is the "common variance" in the variable.
               F1            F2                        Comm
SLength   .3125767362   .0128306509                .0978688416
SWidth    .3187577564  -.0323523347                .1026531808
PLength   .0476237419   .1034495601                .0129698323
PWidth    .0324478281   .0423861795                .0028494498
Sums of squares in columns of A are the factors' variances: .2026316056 and .0137096989.
The main goal of factor analysis is to explain correlations or covariances by means of the loadings.
A*t(A) is the restored covariances:
.0978688416   .0992211576   .0162133990   .0106862785 
.0992211576   .1026531808   .0118336023   .0089717050 
.0162133990   .0118336023   .0129698323   .0059301186 
.0106862785   .0089717050   .0059301186   .0028494498
See that off-diagonal elements above are quite close to those of the input covariance matrix:
S
.1242489796   .0992163265   .0163551020   .0103306122

.0992163265   .1436897959   .0116979592   .0092979592

.0163551020   .0116979592   .0301591837   .0060693878

.0103306122   .0092979592   .0060693878   .0111061224
Standardized (rescaled) loadings and communalities.
St. loading is Loading / sqrt(Variable's variance);
these loadings are computed if you analyse covariances, and are suitable for interpretation of Fs
(if you analyse correlations, A are already standardized).
               F1            F2                        Comm
SLength   .8867684574   .0364000747                .7876832626
SWidth    .8409066701  -.0853478652                .7144082859
PLength   .2742292179   .5956880078                .4300458666
PWidth    .3078962532   .4022009053                .2565656710
Step 3. Compute factor scores (values of Fs).
        Unlike component scores in PCA, factor scores are not exact, they are reasonable approximations.
        Several methods of computation exist (https://stats.stackexchange.com/q/126885/3277).
        Here is regressional method which is the same as the one used in PCA.
Regression coefficients B to compute Standardized factor scores are: B = inv(S)*A (original S is used)
B
              F1           F2

SLength  1.597852081   -.023604439
SWidth   1.070410719   -.637149341
PLength   .212220217   3.157497050
PWidth    .423222047   2.646300951
Standardized factor scores = X*B
These "Standardized factor scores" have variance not 1; the variance of a factor is SSregression of the factor by variables / (n-1).
      F1           F2
  .194641800   -.365588231
 -.660133976   -.042292672
 -.786844270   -.480751358
-1.011226507    .216823430
  .141897664   -.426942721
 1.250472186    .848980006
 -.669003108   -.025440982
 -.050962459    .016236852
  ...
Factors are extracted as orthogonal. And they are.
However, regressionally computed factor scores are not fully uncorrelated.
Covariance matrix between computed factor scores.
      F1      F2
F1   .864   .026
F2   .026   .459
Factor variances are their squared loadings.
You can easily recompute the above "standardized" factor scores to "raw" factor scores having those variances:
raw score = st. score * sqrt(factor variance / st. scores variance).

After the extraction (shown above), optional rotation may take place. Rotation is frequently done in FA. Sometimes it is done in PCA exactly the same way. Rotation rotates loading matrix A into some form of "simple structure" which facilitates interpretation of factors greatly (then rotated scores can be recomputed). Since rotation is not what differentiates FA from PCA mathematically and because it is a separate large topic, I won't touch it.