### Stata Annotated Output Factor Analysis

This page shows an example factor analysis with footnotes explaining the output. We will do an iterated principal axes (ipf option) with SMC as initial communalities retaining three factors (factor(3) option) followed by varimax and promax rotations.

These data were collected on 1428 college students (complete data on 1365 observations) and are responses to items on a survey.  We will use item13 through item24 in our analysis.

use http://www.ats.ucla.edu/stat/stata/output/m255, clear

factor item13-item24, ipf factor(3)
(obs=1365)

(iterated principal factors; 3 factors retained)
Factor     Eigenvaluea    Differenceb   Proportionc   Cumulatived
------------------------------------------------------------------
1        5.85150         5.04464      0.8336         0.8336
2        0.80687         0.44540      0.1149         0.9485
3        0.36146         0.23001      0.0515         1.0000
4        0.13146         0.07619      0.0187         1.0187
5        0.05527         0.02362      0.0079         1.0266
6        0.03164         0.02946      0.0045         1.0311
7        0.00218         0.00658      0.0003         1.0314
8       -0.00440         0.01466     -0.0006         1.0308
9       -0.01906         0.02688     -0.0027         1.0281
10       -0.04594         0.01440     -0.0065         1.0215
11       -0.06035         0.03050     -0.0086         1.0129
12       -0.09084               .     -0.0129         1.0000

a. Eigenvalue: An eigenvalue is the variance of the factor. In the initial factor solution, the first factor will account for the most variance, the second will account for the next highest amount of variance, and so on. Some of the eigenvalues are negative because the matrix is not of full rank, that is, although there are 12 variables the dimensionality of the factor space is much less There are at most seven factors possible.

b. Difference: Gives the differences between the current and following eigenvalue.

c. Proportion: Gives the proportion of variance accounted for by the factor.

d. Cumulative: Gives the cumulative proportion of variance accounted for by this factor plus all of the previous ones.

               Factor Loadingse
Variable |      1          2          3    Uniquenessf
-------------+-------------------------------------------
item13 |   0.71339   -0.39873    0.09231    0.32356
item14 |   0.70320   -0.33908    0.09782    0.38097
item15 |   0.72122   -0.24499    0.10575    0.40864
item16 |   0.64779   -0.18905    0.11144    0.53220
item17 |   0.78307   -0.07337    0.06670    0.37698
item18 |   0.73947    0.34478    0.11291    0.32157
item19 |   0.61655    0.41588    0.15515    0.42284
item20 |   0.55009    0.23916    0.09318    0.63152
item21 |   0.73173    0.11683    0.00067    0.45093
item22 |   0.61281    0.26089   -0.02282    0.55588
item23 |   0.81937   -0.02620   -0.34543    0.20863
item24 |   0.69515    0.01825   -0.38727    0.36646

e. Factor Loadings: The factor loadings for this orthogonal solution represent both how the variables are weighted for each factor but also the correlation between the variables and the factor.

f. Uniqueness: Gives the proportion of the common variance of the variable not associated with the factors. Uniqueness is equal to 1 - communality.

rotate, varimax horst

Factor analysis/correlation                        Number of obs    =     1365
Method: iterated principal factors             Retained factors =        3
Rotation: orthogonal varimax (Horst on)        Number of params =       33

--------------------------------------------------------------------------
Factor  |     Variance   Difference        Proportion   Cumulative
-------------+------------------------------------------------------------
Factor1  |      2.94943      0.29428            0.4202       0.4202
Factor2  |      2.65516      1.23992            0.3782       0.7984
Factor3  |      1.41524            .            0.2016       1.0000
--------------------------------------------------------------------------
LR test: independent vs. saturated:  chi2(66) = 8683.10 Prob>chi2 = 0.0000

-----------------------------------------------------------
Variable |  Factor1   Factor2   Factor3 |   Uniquenessh
-------------+------------------------------+--------------
item13 |   0.7714    0.1740    0.2260 |      0.3236
item14 |   0.7256    0.2130    0.2171 |      0.3810
item15 |   0.6756    0.2950    0.2187 |      0.4086
item16 |   0.5908    0.2926    0.1820 |      0.5322
item17 |   0.5867    0.4461    0.2825 |      0.3770
item18 |   0.2865    0.7386    0.2255 |      0.3216
item19 |   0.1702    0.7281    0.1343 |      0.4228
item20 |   0.2278    0.5396    0.1594 |      0.6315
item21 |   0.4020    0.5333    0.3210 |      0.4509
item22 |   0.2178    0.5584    0.2913 |      0.5559
item23 |   0.4488    0.3769    0.6692 |      0.2086
item24 |   0.3235    0.3205    0.6528 |      0.3665
-----------------------------------------------------------

Factor rotation matrix

-----------------------------------------
| Factor1  Factor2  Factor3
-------------+---------------------------
Factor1 |  0.6584   0.6121   0.4381
Factor2 | -0.6840   0.7294   0.0088
Factor3 |  0.3141   0.3055  -0.8989
-----------------------------------------

g. Rotated Factor Loadings: The factor loadings for the varimax orthogonal rotation represent both how the variables are weighted for each factor but also the correlation between the variables and the factor. A varimax rotation attempts to maximize the squared loadings of the columns.

h. Uniqueness: Same values as in e. above because it is still a three factor solution.

The blanks option displays only factor loading greater than a specific value (say 0.3).

rotate, varimax horst blanks(.3)

Factor analysis/correlation                        Number of obs    =     1365
Method: iterated principal factors             Retained factors =        3
Rotation: orthogonal varimax (Horst on)        Number of params =       33

--------------------------------------------------------------------------
Factor  |     Variance   Difference        Proportion   Cumulative
-------------+------------------------------------------------------------
Factor1  |      2.94943      0.29428            0.4202       0.4202
Factor2  |      2.65516      1.23992            0.3782       0.7984
Factor3  |      1.41524            .            0.2016       1.0000
--------------------------------------------------------------------------
LR test: independent vs. saturated:  chi2(66) = 8683.10 Prob>chi2 = 0.0000

-----------------------------------------------------------
Variable |  Factor1   Factor2   Factor3 |   Uniqueness
-------------+------------------------------+--------------
item13 |   0.7714                     |      0.3236
item14 |   0.7256                     |      0.3810
item15 |   0.6756                     |      0.4086
item16 |   0.5908                     |      0.5322
item17 |   0.5867    0.4461           |      0.3770
item18 |             0.7386           |      0.3216
item19 |             0.7281           |      0.4228
item20 |             0.5396           |      0.6315
item21 |   0.4020    0.5333    0.3210 |      0.4509
item22 |             0.5584           |      0.5559
item23 |   0.4488    0.3769    0.6692 |      0.2086
item24 |   0.3235    0.3205    0.6528 |      0.3665
-----------------------------------------------------------

Factor rotation matrix

-----------------------------------------
| Factor1  Factor2  Factor3
-------------+---------------------------
Factor1 |  0.6584   0.6121   0.4381
Factor2 | -0.6840   0.7294   0.0088
Factor3 |  0.3141   0.3055  -0.8989
-----------------------------------------

rotate, promax horst blanks(.3)

Factor analysis/correlation                        Number of obs    =     1365
Method: iterated principal factors             Retained factors =        3
Rotation: oblique promax (Horst on)            Number of params =       33

--------------------------------------------------------------------------
Factor  |     Variance   Proportion    Rotated factors are correlated
-------------+------------------------------------------------------------
Factor1  |      4.86265       0.6927
Factor2  |      4.52052       0.6440
Factor3  |      4.30842       0.6138
--------------------------------------------------------------------------
LR test: independent vs. saturated:  chi2(66) = 8683.10 Prob>chi2 = 0.0000

-----------------------------------------------------------
Variable |  Factor1   Factor2   Factor3 |   Uniquenessj
-------------+------------------------------+--------------
item13 |   0.8518                     |      0.3236
item14 |   0.7855                     |      0.3810
item15 |   0.6969                     |      0.4086
item16 |   0.6044                     |      0.5322
item17 |   0.5087                     |      0.3770
item18 |             0.7626           |      0.3216
item19 |             0.8200           |      0.4228
item20 |             0.5541           |      0.6315
item21 |             0.4298           |      0.4509
item22 |             0.5265           |      0.5559
item23 |                       0.7187 |      0.2086
item24 |                       0.7502 |      0.3665
-----------------------------------------------------------

Factor rotation matrix

-----------------------------------------
| Factor1  Factor2  Factor3
-------------+---------------------------
Factor1 |  0.8977   0.8593   0.8479
Factor2 | -0.4157   0.4864   0.0071
Factor3 |  0.1462   0.1581  -0.5301
-----------------------------------------

i. Rotated Factor Loadings: The factor loadings for the promax oblique rotation represent how the each of the variables are weighted for each factor. Note: these are not correlations between variables and factors. The promax rotation allows the factors to be correlated in an attempt to better approximate simple structure.

i. Uniqueness: Same values as in e. and h. above because it is still a three factor solution.

The estat common command is a postestimation command that displays the correlation among the factors of an oblique rotation.

estat common

Correlation matrix of the promax(3) rotated common factors

--------------------------------------------
Factors |  Factor1   Factor2   Factor3
-------------+------------------------------
Factor1 |        1
Factor2 |    .5923         1
Factor3 |    .6807     .6482         1
--------------------------------------------

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