Re: st: sem multiple correlations and factor weights

[kmacdonald@stata.com (Kristin MacDonald, StataCorp LP)]

Dave Garson <garson@ncsu.edu> asked about obtaining multiple correlations and
factor loadings with -sem-:

> For SEM, both SAS and SPSS print out a table of multiple correlations and a
> table of factor weights of indicator variable loadings on latent variables.
> I could not find these in Stata's sem, nor in the reference guide (though
> the guide calls indicator paths "loadings" but that is a different meaning
> from SAS or SPSS). Tips on getting this output in sem would be appreciated. 

The multiple correlations and squared multiple correlations can be
obtained by typing -estat eqgof- after -sem-.  For example, type

  . webuse sem_2fmm, clear
  . sem (Affective -> a1 a2 a3 a4 a5) (Cognitive -> c1 c2 c3 c4 c5)
  . estat eqgof

The full output is given below my signature.


With regards to factor loadings, in my experience, the coefficients on the
paths from the latent variables to the observed indicators are equivalent to
the factor loadings reported by SAS Proc Calis and by Amos when the same
estimation method is used and the same set of identifying constraints are
imposed.  If Dave is making comparisons to standardized results reported by
other packages, he will need to specify the -standardized- option with -sem-
or replay the results by typing 

  . sem, standardized

If Dave is comparing the unstandardized factor loadings, he will need to make
sure that the same constraints are used to set the scale of the latent
variable.  By default, -sem- constrains the coefficient on the path to the
first observed variable to 1.  Another common way to set the scale of the
latent variable is to constrain the variance of the latent variable to 1. If
the other packages are setting constraints in this way, Dave can use the
-variance()- option to set the same constraints with -sem-.  For the model
above, we could type

  . sem (Affective -> a1 a2 a3 a4 a5) (Cognitive -> c1 c2 c3 c4 c5), ///
    variance(Affective@1 Cognitive@1)  


--Kristin
kmacdonald@stata.com


. webuse sem_2fmm, clear
(Affective and cognitive arousal)

. 
.   sem (Affective -> a1 a2 a3 a4 a5) (Cognitive -> c1 c2 c3 c4 c5)

Endogenous variables

Measurement:  a1 a2 a3 a4 a5 c1 c2 c3 c4 c5

Exogenous variables

Latent:       Affective Cognitive

Fitting target model:

Iteration 0:   log likelihood = -9542.8803  
Iteration 1:   log likelihood = -9539.5505  
Iteration 2:   log likelihood = -9539.3856  
Iteration 3:   log likelihood = -9539.3851  

Structural equation model                       Number of obs      =       216
Estimation method  = ml
Log likelihood     = -9539.3851

 ( 1)  [a1]Affective = 1
 ( 2)  [c1]Cognitive = 1
------------------------------------------------------------------------------
             |                 OIM
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Measurement  |
  a1 <-      |
   Affective |          1  (constrained)
  -----------+----------------------------------------------------------------
  a2 <-      |
   Affective |   .9758098   .0460752    21.18   0.000      .885504    1.066116
  -----------+----------------------------------------------------------------
  a3 <-      |
   Affective |   .8372599   .0355086    23.58   0.000     .7676643    .9068556
  -----------+----------------------------------------------------------------
  a4 <-      |
   Affective |   .9640461   .0499203    19.31   0.000      .866204    1.061888
  -----------+----------------------------------------------------------------
  a5 <-      |
   Affective |   1.063701   .0435751    24.41   0.000     .9782951    1.149107
  -----------+----------------------------------------------------------------
  c1 <-      |
   Cognitive |          1  (constrained)
  -----------+----------------------------------------------------------------
  c2 <-      |
   Cognitive |   1.114702   .0655687    17.00   0.000     .9861901    1.243215
  -----------+----------------------------------------------------------------
  c3 <-      |
   Cognitive |   1.329882   .0791968    16.79   0.000     1.174659    1.485105
  -----------+----------------------------------------------------------------
  c4 <-      |
   Cognitive |   1.172792   .0711692    16.48   0.000     1.033303    1.312281
  -----------+----------------------------------------------------------------
  c5 <-      |
   Cognitive |   1.126356   .0644475    17.48   0.000     1.000041    1.252671
-------------+----------------------------------------------------------------
    var(e.a1)|   384.1359   43.79119                      307.2194    480.3095
    var(e.a2)|   357.3524   41.00499                      285.3805    447.4755
    var(e.a3)|   154.9507   20.09026                      120.1795    199.7822
    var(e.a4)|   496.4594   54.16323                      400.8838    614.8214
    var(e.a5)|   191.6857   28.07212                      143.8574    255.4154
    var(e.c1)|   171.6638   19.82327                       136.894    215.2649
    var(e.c2)|   171.8055   20.53479                      135.9247    217.1579
    var(e.c3)|   276.0144   32.33535                      219.3879    347.2569
    var(e.c4)|   224.1994   25.93412                      178.7197    281.2527
    var(e.c5)|   146.8655    18.5756                      114.6198    188.1829
var(Affect~e)|   1644.463   193.1032                      1306.383    2070.034
var(Cognit~e)|   455.9349   59.11245                      353.6255    587.8439
-------------+----------------------------------------------------------------
 cov(Affec~e,|
   Cognitive)|   702.0736   85.72272     8.19   0.000     534.0601     870.087
------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(34)  =     88.88, Prob > chi2 = 0.0000

. 
.   estat eqgof

Equation-level goodness of fit

------------------------------------------------------------------------------
             |             Variance            |
     depvars |    fitted  predicted   residual | R-squared        mc      mc2
-------------+---------------------------------+------------------------------
observed     |                                 |
          a1 |  2028.598   1644.463   384.1359 |  .8106398  .9003553  .8106398
          a2 |  1923.217   1565.865   357.3524 |  .8141903  .9023249  .8141903
          a3 |  1307.726   1152.775   154.9507 |  .8815113  .9388883  .8815113
          a4 |  2024.798   1528.339   496.4594 |  .7548104  .8687982  .7548104
          a5 |  2052.328   1860.643   191.6857 |  .9066009  .9521559  .9066009
          c1 |  627.5987   455.9349   171.6638 |  .7264752  .8523351  .7264752
          c2 |  738.3325    566.527   171.8055 |  .7673061  .8759601  .7673061
          c3 |  1082.374   806.3598   276.0144 |  .7449917   .863129  .7449917
          c4 |   851.311   627.1116   224.1994 |  .7366422  .8582786  .7366422
          c5 |  725.3002   578.4346   146.8655 |  .7975107  .8930346  .7975107
-------------+---------------------------------+------------------------------
     overall |                                 |  .9949997
------------------------------------------------------------------------------
mc  = correlation between depvar and its prediction
mc2 = mc^2 is the Bentler-Raykov squared multiple correlation coefficient


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