# Steps done in factor analysis compared to steps done in PCA

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.

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)

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

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 = A*diag(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. (Usually some last eigenvalues will be negative.)

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.

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.

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 factors' variances.

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