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

 From "Joseph Coveney" To 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.

(-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

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

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

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

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