SEM and the predicted latent variable

[Emma Zandi]

Hello

I am doing a confirmatory factor analysis, using SEM, to construct three latent variables for subsequent use in a regression analysis.

I use the following command

sem (Visual -> v1, ) (Visual -> v2, ) (Visual -> v3, ) (Verbal -> v4, ) ///
(Verbal -> v5, ) (Verbal -> v6, ) (Speed -> v7, )(Speed -> v8, ) ///
(Speed -> v9, ), covstruct(_lexogenous, diagonal) vce(sbentler) ///
standardized latent(Visual Verbal Speed) ///
cov( Visual*Verbal Visual*Speed Verbal*Speed) nocapslatent

foreach v in Visual Verbal Speed {
    predict `v', latent(`v')
    egen std_`v'= std(`v') // standardized values
    }

sum Visual Verbal Speed std_Visual std_Verbal std_Speed

and I get the following output

Endogenous variables

Measurement:  v1 v2 v3 v4 v5 v6 v7 v8 v9

Exogenous variables

Latent:       Visual Verbal Speed

Fitting target model:

Iteration 0:   log pseudolikelihood = -4554.5345  
Iteration 1:   log pseudolikelihood = -4549.0267  
Iteration 2:   log pseudolikelihood = -4548.3346  
Iteration 3:   log pseudolikelihood = -4548.3322  
Iteration 4:   log pseudolikelihood = -4548.3322  

Structural equation model                       Number of obs     =        145
Estimation method    = ml
Log pseudolikelihood = -4548.3322

 ( 1)  [v1]Visual = 1
 ( 2)  [v4]Verbal = 1
 ( 3)  [v7]Speed = 1
-----------------------------------------------------------------------------------
                  |           Satorra-Bentler
     Standardized |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
------------------+----------------------------------------------------------------
Measurement       |
  v1              |
           Visual |     .67665   .0860549     7.86   0.000     .5079856    .8453145
            _cons |   4.293115   .2497337    17.19   0.000     3.803646    4.782584
  ----------------+----------------------------------------------------------------
  v2              |
           Visual |   .5165186   .0684681     7.54   0.000     .3823235    .6507137
            _cons |   5.598598   .3782855    14.80   0.000     4.857172    6.340024
  ----------------+----------------------------------------------------------------
  v3              |
           Visual |   .6935859    .069044    10.05   0.000     .5582622    .8289096
            _cons |   1.926279   .0893687    21.55   0.000      1.75112    2.101439
  ----------------+----------------------------------------------------------------
  v4              |
           Verbal |    .865565   .0332882    26.00   0.000     .8003213    .9308087
            _cons |   2.958508   .1669357    17.72   0.000      2.63132    3.285696
  ----------------+----------------------------------------------------------------
  v5              |
           Verbal |   .8293273   .0308301    26.90   0.000     .7689013    .8897533
            _cons |   4.068081   .2698725    15.07   0.000     3.539141    4.597022
  ----------------+----------------------------------------------------------------
  v6              |
           Verbal |   .8263318   .0355554    23.24   0.000     .7566445     .896019
            _cons |   2.182157   .1137125    19.19   0.000     1.959285    2.405029
  ----------------+----------------------------------------------------------------
  v7              |
            Speed |   .6591327   .0578271    11.40   0.000     .5457937    .7724717
            _cons |   3.805027   .2013456    18.90   0.000     3.410397    4.199658
  ----------------+----------------------------------------------------------------
  v8              |
            Speed |   .7958731   .0501476    15.87   0.000     .6975856    .8941605
            _cons |   5.246285   .3948288    13.29   0.000     4.472435    6.020136
  ----------------+----------------------------------------------------------------
  v9              |
            Speed |    .700846   .0517622    13.54   0.000     .5993938    .8022981
            _cons |   5.196225   .3564262    14.58   0.000     4.497643    5.894808
------------------+----------------------------------------------------------------
         var(e.v1)|   .5421447   .1164581                      .3558514    .8259653
         var(e.v2)|   .7332085   .0707301                       .606897    .8858089
         var(e.v3)|   .5189386   .0957759                      .3614243    .7451002
         var(e.v4)|   .2507973   .0576262                      .1598602    .3934642
         var(e.v5)|   .3122162   .0511366                      .2264857    .4303979
         var(e.v6)|   .3171758   .0587611                       .220599    .4560333
         var(e.v7)|   .5655441   .0762315                      .4342405    .7365506
         var(e.v8)|   .3665861   .0798222                      .2392386    .5617211
         var(e.v9)|   .5088149   .0725547                      .3847533    .6728796
       var(Visual)|          1          .                             .           .
       var(Verbal)|          1          .                             .           .
        var(Speed)|          1          .                             .           .
------------------+----------------------------------------------------------------
cov(Visual,Verbal)|   .5406683   .0896913     6.03   0.000     .3648765    .7164601
 cov(Visual,Speed)|   .5233425   .0942191     5.55   0.000     .3386766    .7080085
 cov(Verbal,Speed)|   .3361288   .1131726     2.97   0.003     .1143146     .557943
-----------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(24)  =     51.54, Prob > chi2 = 0.0009
Satorra-Bentler scaled test:    chi2(24)  =     50.29, Prob > chi2 = 0.0013

. 
. foreach v in Visual Verbal Speed {
  2.         predict `v', latent(`v')
  3.         egen std_`v'= std(`v') // standardized values
  4.         }

. 
. sum Visual Verbal Speed std_Visual std_Verbal std_Speed

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
      Visual |        145   -4.32e-09    4.000018  -9.592437   12.42246
      Verbal |        145   -1.23e-09    2.746986  -7.786823   7.084102
       Speed |        145    3.21e-08    13.91826   -40.7785   43.87231
  std_Visual |        145    2.20e-09           1  -2.398098   3.105601
  std_Verbal |        145   -1.70e-10           1  -2.834679   2.578864
-------------+---------------------------------------------------------
   std_Speed |        145   -1.02e-09           1  -2.929857   3.152141 

Now my question is that:
1- Although I specified the standardized option, I still get the predicted variables with standard deviations more than 1, and I Why?
2- The mean of the predicted latent variables is always close to zero? If that is so, are we assuming the kappa; associated intercept in estimating the latent KSI, in the formula is equal to zero? Or in other words what kind of distribution for the latents Ksi in this model is assumed?

Bests,
Emma

[Weiwen Ng]

Emma,

I'm not 100% sure why you get a predicted latent variable SD > 1 when you use the standardize option, but my guess is that the option only affects how the coefficients are reported, not how the underlying mathematics are calculated.

You can constrain the variance of the latent trait to one, and you'll get an approximately standard normal distribution when you predict the latent trait, e.g.

use http://www.stata-press.com/data/r15/sem_1fmm
sem (x1 x2 x3 x4 <- X), variance(X@1)
predict p_x_std, latent
sum p_x_std
    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
     p_std_x |        500   -1.87e-09    .9872921  -2.959064   2.869629

I omitted model output, but if you recorded the log likelihood (-8487.2337), then you went back and fit the same model without the constraint on the variance, you'd see the same log-likelihood, so it's fundamentally the same model (and you'd see the same if you fit the SEM model with the standardize option).

If you plotted a histogram of the predicted latent trait after any model, you'll see it's approximately normal. I'm pretty sure that SEM in general assumes the latent traits are normally distributed.

[Emma Zandi]

Thank you very much Weiwen.
I think that as you said the standardized option only affects how the factor loadings are reported, and to standardize the latents one need to constriant vairance to one, or standardize the variables manually.

However, I did not get the answer to my second question:

2- The mean of the predicted latent variables is always close to zero? If that is so, are we assuming the kappa; associated intercept in estimating the latent KSI, in the formula is equal to zero? Or in other words what kind of distribution for the latents Ksi in this model is assumed?

Best,
Emma

[Weiwen Ng]

Thank you very much Weiwen.
I think that as you said the standardized option only affects how the factor loadings are reported, and to standardize the latents one need to constriant vairance to one, or standardize the variables manually.

However, I did not get the answer to my second question:

2- The mean of the predicted latent variables is always close to zero? If that is so, are we assuming the kappa; associated intercept in estimating the latent KSI, in the formula is equal to zero? Or in other words what kind of distribution for the latents Ksi in this model is assumed?

Best,
Emma

I'm not familiar with the latent variable notation you're referencing. However, my understanding is that the predicted mean of the latent variable should be close to 0 - as if it were following a normal distribution centered at 0.

[Emma Zandi]

Thank you Weiwen.

Emma