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

Rotated factor loadings (pattern matrix) and unique variancesg

    -----------------------------------------------------------
        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

Rotated factor loadings (pattern matrix) and unique variances

    -----------------------------------------------------------
        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  
    -----------------------------------------------------------
    (blanks represent abs(loading)<.3)

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

Rotated factor loadings (pattern matrix) and unique variancesi

    -----------------------------------------------------------
        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  
    -----------------------------------------------------------
    (blanks represent abs(loading)<.3)

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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