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st: Re: Oblique Rotation returns no correlation between factors?


From   "Joseph Coveney" <stajc2@gmail.com>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: Re: Oblique Rotation returns no correlation between factors?
Date   Wed, 13 Mar 2013 22:52:08 +0900

Ivan Kuznetsov wrote:

I doing some research using factor analysis (principle axis factoring) 
using categorical data. I have used polychoric correlations to correct for 
the categorical variables problem. But now when I try to rotate obliquely 
using "oblimin" and use "estat common" to find the correlation between 
factors, Stata returns that there is 0 correlation, which is very strange. 
I have tried different extraction methods and different rotation methods 
and yet I still find no correlation. Any suggestions?

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

It seems to work for me, at least in the toy example below.  Even if you set the
correlation matrix such that the factors are uncorrelated (change the 0.3's
below to zeroes) there is still a small positive value reported by -estat
common-.  That is, there doesn't appear to be any set-to-zero behavior if the
correlation estimate falls below some threshold.  Negative correlations (change
the sign of the 0.3's below) are also faithfully reported.

Is there something unusual about your dataset, for example, heavy skewness?

(-polychoric- is from SSC.)

Joseph Coveney


. clear *

. set more off

. set seed `=date("2013-03-12", "YMD")'

. tempname Correlation

. matrix define `Correlation' = (1, 0.3 \ 0.3, 1) # (1, 0.5, 0.5 \ ///
>   0.5, 1, 0.5 \ 0.5, 0.5, 1)

. drawnorm M1 M2 M3 M4 M5 M6, double corr(`Correlation') n(200)
(obs 200)

. 
. *
. * Original continuous values
. *
. factor M?, pf factors(2)
(obs=200)

Factor analysis/correlation                        Number of obs    =      200
    Method: principal factors                      Retained factors =        2
    Rotation: (unrotated)                          Number of params =       11

    --------------------------------------------------------------------------
         Factor  |   Eigenvalue   Difference        Proportion   Cumulative
    -------------+------------------------------------------------------------
        Factor1  |      1.97500      1.31624            0.8824       0.8824
        Factor2  |      0.65876      0.54130            0.2943       1.1767
        Factor3  |      0.11746      0.10758            0.0525       1.2292
        Factor4  |      0.00988      0.25203            0.0044       1.2336
        Factor5  |     -0.24215      0.03858           -0.1082       1.1254
        Factor6  |     -0.28074            .           -0.1254       1.0000
    --------------------------------------------------------------------------
    LR test: independent vs. saturated:  chi2(15) =  308.16 Prob>chi2 = 0.0000

Factor loadings (pattern matrix) and unique variances

    -------------------------------------------------
        Variable |  Factor1   Factor2 |   Uniqueness 
    -------------+--------------------+--------------
              M1 |   0.6467   -0.2908 |      0.4972  
              M2 |   0.5735   -0.2938 |      0.5848  
              M3 |   0.5516   -0.3699 |      0.5589  
              M4 |   0.6091    0.3126 |      0.5312  
              M5 |   0.4666    0.3695 |      0.6457  
              M6 |   0.5786    0.3417 |      0.5484  
    -------------------------------------------------

. rotate , oblique oblimin

Factor analysis/correlation                        Number of obs    =      200
    Method: principal factors                      Retained factors =        2
    Rotation: oblique oblimin (Kaiser off)         Number of params =       11

    --------------------------------------------------------------------------
         Factor  |     Variance   Proportion    Rotated factors are correlated
    -------------+------------------------------------------------------------
        Factor1  |      1.66759       0.7451
        Factor2  |      1.60701       0.7180
    --------------------------------------------------------------------------
    LR test: independent vs. saturated:  chi2(15) =  308.16 Prob>chi2 = 0.0000

Rotated factor loadings (pattern matrix) and unique variances

    -------------------------------------------------
        Variable |  Factor1   Factor2 |   Uniqueness 
    -------------+--------------------+--------------
              M1 |   0.6743    0.0664 |      0.4972  
              M2 |   0.6330    0.0228 |      0.5848  
              M3 |   0.6936   -0.0655 |      0.5589  
              M4 |   0.0650    0.6506 |      0.5312  
              M5 |  -0.0767    0.6288 |      0.6457  
              M6 |   0.0183    0.6629 |      0.5484  
    -------------------------------------------------

Factor rotation matrix

    --------------------------------
                 | Factor1  Factor2 
    -------------+------------------
         Factor1 |  0.8755   0.8488 
         Factor2 | -0.4833   0.5288 
    --------------------------------

. estat common

Correlation matrix of the oblimin(0) rotated common factors

    ----------------------------------
         Factors |  Factor1   Factor2 
    -------------+--------------------
         Factor1 |        1           
         Factor2 |    .4875         1 
    ----------------------------------

. 
. *
. * Discretized values
. *
. foreach var of varlist M? {
  2.     generate byte C`var' = 0
  3.     forvalues cut = 0.25(0.25)0.75 {
  4.         quietly replace C`var' = C`var' + 1 if normal(`var') > `cut'
  5.     }
  6. }

. 
. polychoric C*

Polychoric correlation matrix

           CM1        CM2        CM3        CM4        CM5        CM6
CM1          1
CM2  .48408458          1
CM3  .47677285  .46502406          1
CM4  .40142969  .23287674  .14154395          1
CM5  .12098617  .22584329  .08295264  .41714915          1
CM6  .25471765  .07127948  .20458655  .49077708  .43353546          1

. factormat r(R), n(`r(N)') pf factors(2)
(obs=200)

Factor analysis/correlation                        Number of obs    =      200
    Method: principal factors                      Retained factors =        2
    Rotation: (unrotated)                          Number of params =       11

    --------------------------------------------------------------------------
         Factor  |   Eigenvalue   Difference        Proportion   Cumulative
    -------------+------------------------------------------------------------
        Factor1  |      1.87721      1.16953            0.8843       0.8843
        Factor2  |      0.70768      0.65029            0.3334       1.2177
        Factor3  |      0.05739      0.06274            0.0270       1.2447
        Factor4  |     -0.00535      0.23165           -0.0025       1.2422
        Factor5  |     -0.23700      0.04016           -0.1116       1.1306
        Factor6  |     -0.27716            .           -0.1306       1.0000
    --------------------------------------------------------------------------
    LR test: independent vs. saturated:  chi2(15) =  289.74 Prob>chi2 = 0.0000

Factor loadings (pattern matrix) and unique variances

    -------------------------------------------------
        Variable |  Factor1   Factor2 |   Uniqueness 
    -------------+--------------------+--------------
             CM1 |   0.6511   -0.2714 |      0.5024  
             CM2 |   0.5550   -0.3507 |      0.5690  
             CM3 |   0.5182   -0.3685 |      0.5956  
             CM4 |   0.6151    0.3051 |      0.5286  
             CM5 |   0.4628    0.3641 |      0.6533  
             CM6 |   0.5332    0.3867 |      0.5662  
    -------------------------------------------------

. rotate , oblique oblimin

Factor analysis/correlation                        Number of obs    =      200
    Method: principal factors                      Retained factors =        2
    Rotation: oblique oblimin (Kaiser off)         Number of params =       11

    --------------------------------------------------------------------------
         Factor  |     Variance   Proportion    Rotated factors are correlated
    -------------+------------------------------------------------------------
        Factor1  |      1.57736       0.7431
        Factor2  |      1.51799       0.7151
    --------------------------------------------------------------------------
    LR test: independent vs. saturated:  chi2(15) =  289.74 Prob>chi2 = 0.0000

Rotated factor loadings (pattern matrix) and unique variances

    -------------------------------------------------
        Variable |  Factor1   Factor2 |   Uniqueness 
    -------------+--------------------+--------------
             CM1 |   0.6524    0.1063 |      0.5024  
             CM2 |   0.6666   -0.0239 |      0.5690  
             CM3 |   0.6604   -0.0615 |      0.5956  
             CM4 |   0.0967    0.6388 |      0.5286  
             CM5 |  -0.0518    0.6096 |      0.6533  
             CM6 |  -0.0294    0.6710 |      0.5662  
    -------------------------------------------------

Factor rotation matrix

    --------------------------------
                 | Factor1  Factor2 
    -------------+------------------
         Factor1 |  0.8623   0.8324 
         Factor2 | -0.5063   0.5542 
    --------------------------------

. estat common

Correlation matrix of the oblimin(0) rotated common factors

    ----------------------------------
         Factors |  Factor1   Factor2 
    -------------+--------------------
         Factor1 |        1           
         Factor2 |    .4372         1 
    ----------------------------------

. 
. exit

end of do-file

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