sem models distribution of the latent variables and interpretation of the coefficients

[Sule Yaylaci]

Hi,
I am using Stata14 for fitting an sem model with a second-order latent variable. In my dataset, there are 9 exogenous variables (V1-V9), which I use to estimate three latent variables (L1, L2, L3), and then the three latent variables are used to estimate a second order order variable, L4. Beyond the measurement part, I also have a structural part in my model in which I try to estimate the effect of some demographic and attitudinal variables (Male Age2 Age3 Educ2 Educ3 Educ4 X1 X2 X3 E1 E2 E3) on L4. The exogenous variables are also on a scale from 1-4. My question is about the range of the estimated latent variables, and how I can go about interpreting the effects of the demographic and attitudinal variables on L4.
When I add the 'standardized' option, I assume the coefficients are in standard deviation, and the latent variables are centered around zero with an SD of 1. So, in that case, coefficient of the 'Male' variable would show the difference in L4 values in standard deviation between female and male respondents. I am not really sure what this means. It is not the most intuitive interpretation. I have three questions about this.
1) I understand the distribution of the latent variables are somewhat abstract anyway, but it would be great if I could get the L4 distributed on a scale from 0-1 instead of from -0.5 to 0.5 or -1 to +1. Is this something possible? Can I add an option to SEM model to get the latent variables to be on a 0-1 scale.
2) When I do not add 'standardized' option, what is the default distribution of the latent variables? And how can I see the distribution of latent variables after sem?
3) Also, I wanted to use margins to get some more meaningful effects, but I could not figure out how to use margins for latent variables. If you can provide any help, I would be grateful.

Data example and the commands I used are below.
I appreciate any help.
Thanks in advance for your time for reading.

* Example generated by -dataex-. For more info, type help dataex
clear
input float(V1 V2 V3 V4 V5 V6 V7 V8 V9) double Male byte(Age2 Age3 Educ2 Educ3 Educ4 X1 X2 X3 E1 E2 E3)
4 2 2 3 1 2 2 2 2 0 0 0 1 0 0 0 1 0 1 0 0
2 1 1 1 3 2 2 1 1 1 0 0 1 0 0 0 0 0 1 0 0
4 4 1 3 4 3 1 1 3 1 1 0 1 0 0 0 1 0 0 0 0
4 3 3 3 3 3 3 2 4 1 1 0 1 0 0 0 0 1 0 1 0
3 2 2 3 3 3 2 1 3 1 1 0 0 0 0 0 1 0 0 1 0
3 2 1 3 3 3 2 1 4 1 0 1 0 1 0 0 1 0 0 0 0
4 2 2 3 4 3 1 2 3 1 1 0 1 0 0 0 1 0 1 0 0
3 2 4 3 3 3 1 1 2 0 0 1 1 0 0 0 0 1 0 0 0
3 2 3 3 3 3 3 2 3 0 0 1 1 0 0 0 1 0 0 1 0
3 1 2 2 1 1 4 3 4 1 0 1 0 1 0 0 1 0 0 1 0
2 1 3 3 3 4 3 1 2 0 1 0 0 1 0 0 1 0 0 0 0
3 2 2 1 3 3 2 2 3 1 0 1 1 0 0 0 1 0 0 1 0
4 2 3 3 3 3 3 3 4 1 1 0 1 0 0 0 1 0 1 0 0
3 2 2 3 2 3 2 1 2 0 1 0 1 0 0 0 1 0 0 0 0
3 3 2 2 3 3 3 2 3 0 1 0 1 0 0 0 1 0 1 0 0
3 2 3 4 3 4 3 2 3 1 1 0 1 0 0 0 1 0 1 0 0
2 2 2 3 2 2 1 1 3 1 0 1 0 0 0 0 0 1 1 0 0
1 1 1 1 2 3 1 1 1 1 0 1 0 0 1 0 0 0 1 0 0
3 3 3 3 4 3 2 2 2 1 1 0 0 1 0 0 1 0 1 0 0
3 3 3 3 3 3 2 1 2 0 0 1 0 0 0 1 0 0 0 0 1
3 1 3 3 2 3 3 3 4 0 1 0 0 1 0 1 0 0 0 1 0
3 1 3 3 4 3 1 2 1 0 0 0 0 0 1 0 0 1 0 0 0
3 3 2 3 3 3 2 1 3 0 1 0 0 0 0 1 0 0 0 0 0
3 4 2 4 3 3 1 2 4 1 1 0 0 1 0 0 0 0 0 1 0
3 2 2 3 3 3 2 1 3 0 1 0 0 1 0 0 1 0 1 0 0
3 3 1 3 3 3 4 3 3 0 1 0 1 0 0 0 1 0 0 1 0
4 1 1 3 3 3 1 1 4 1 0 0 0 0 1 0 0 1 0 1 0
3 2 1 4 3 3 1 1 2 0 0 0 1 0 0 0 0 0 1 0 0
1 1 2 1 3 2 1 1 4 1 0 1 0 1 0 0 0 0 0 1 0
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3 4 2 3 4 4 2 1 3 1 0 1 0 0 0 0 1 0 0 0 1
2 4 2 3 1 3 3 4 3 0 1 0 1 0 0 1 0 0 0 0 1
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3 3 2 2 3 2 1 2 2 1 1 0 0 1 0 0 0 0 0 0 0
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4 3 4 1 3 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 1
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3 2 2 2 3 3 3 2 2 1 1 0 0 0 1 0 1 0 1 0 0
4 4 3 3 4 3 4 2 4 1 0 1 1 0 0 0 1 0 0 0 1
3 3 2 2 3 3 4 2 4 0 0 1 0 1 0 0 1 0 1 0 0
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3 3 3 3 3 3 3 1 3 1 1 0 0 1 0 0 0 1 1 0 0
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4 4 3 4 4 3 1 1 1 1 0 1 0 0 0 1 0 0 1 0 0
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3 1 1 2 4 3 2 2 3 0 1 0 0 0 0 0 1 0 0 1 0
2 2 3 2 3 2 3 2 2 1 0 0 1 0 0 0 1 0 1 0 0
3 2 2 3 3 3 3 2 3 0 0 0 0 1 0 0 0 1 1 0 0
3 1 4 3 4 4 4 1 4 1 1 0 0 1 0 0 1 0 1 0 0
3 1 3 3 3 2 1 1 2 1 0 1 0 1 0 0 0 0 0 1 0
4 3 4 2 3 4 4 3 3 0 0 1 0 0 0 0 1 0 0 1 0
3 2 3 2 4 4 1 1 2 1 0 1 1 0 0 0 1 0 1 0 0
3 3 4 2 4 3 3 2 4 0 0 1 1 0 0 0 1 0 0 1 0
3 2 2 3 1 3 2 2 2 1 0 1 0 1 0 0 1 0 0 1 0
1 1 3 1 4 4 2 1 2 1 1 0 0 0 0 0 0 0 0 0 0
3 2 3 3 3 3 2 2 3 0 0 0 1 0 0 0 1 0 0 0 0
3 3 3 3 3 3 2 1 2 1 1 0 0 0 0 1 0 0 0 0 0
3 1 1 4 4 2 2 1 3 0 1 0 0 1 0 0 0 0 0 0 0
1 1 1 1 2 1 2 2 2 1 1 0 0 1 0 0 1 0 1 0 0
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3 3 3 2 4 4 3 1 3 1 0 1 1 0 0 0 1 0 1 0 0
1 1 1 2 2 3 2 2 3 1 1 0 0 0 0 0 1 0 1 0 0
3 2 2 2 3 3 3 1 2 1 0 1 0 0 0 0 0 0 1 0 0
3 3 2 2 3 2 2 3 4 1 1 0 0 0 0 0 1 0 0 1 0
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3 2 2 3 3 1 2 1 3 1 0 1 1 0 0 0 1 0 1 0 0
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3 2 1 3 3 3 2 1 3 0 0 1 1 0 0 0 1 0 1 0 0
4 2 1 4 4 3 2 2 1 0 0 1 0 0 1 0 0 0 0 1 0
3 1 2 3 3 3 2 1 3 1 0 1 0 0 0 0 1 0 1 0 0
4 2 2 1 3 3 1 2 1 1 0 0 1 0 0 0 1 0 1 0 0
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3 3 4 3 3 3 2 1 4 1 0 1 0 1 0 0 0 0 1 0 0
2 1 2 3 3 3 2 2 2 0 0 0 0 1 0 0 1 0 0 0 0
2 1 2 3 3 3 2 1 2 0 0 1 0 1 0 0 0 1 1 0 0
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3 2 3 3 3 3 4 2 3 1 0 1 1 0 0 0 1 0 1 0 0
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1 1 2 3 4 3 2 4 4 0 0 1 1 0 0 0 1 0 0 1 0

end
label values Male Male
label def Male 0 "0 Female", modify
label def Male 1 "1 Male", modify

sem (L1 -> V1 V2 V3) ///
(L2 -> V4 V5 V6) ///
 (L3 -> V7 V8 V9) /// 
 (L4 -> L1 L2 L3) //////
 (L4<- Male Age2 Age3 Educ2 Educ3 Educ4  ///
X1 X2 X3 E1 E2 E3 ///
 , difficult latent(L1 L2 L3 L4) nocapslatent

sem (L1 -> V1 V2 V3) ///
(L2 -> V4 V5 V6) ///
 (L3 -> V7 V8 V9) /// 
 (L4 -> L1 L2 L3) //////
 (L4<- Male Age2 Age3 Educ2 Educ3 Educ4  ///
X1 X2 X3 E1 E2 E3 ///
 , difficult latent(L1 L2 L3 L4) nocapslatent standardized

[Erik Ruzek]

Your questions are good ones, however I think you need to back up a bit. Start with your basic first-order measurement model, which has three factors composed of three items each (total of 9 items). When you fit that model model, a few things stand out - the factors are not highly correlated (rs of .68 (L1 with L2), .28 (L1 with L3), and .28 (L2 with L3). Factor correlations of that magnitude would not generally be viewed as strong evidence for moving to a higher-order factor model. When you examine model fit of the three-factor first-order model, it is pretty good.

. estat gof, stats(all)

----------------------------------------------------------------------------
Fit statistic        |      Value   Description
---------------------+------------------------------------------------------
Likelihood ratio     |
         chi2_ms(24) |     43.562   model vs. saturated
            p > chi2 |      0.009
         chi2_bs(36) |    318.652   baseline vs. saturated
            p > chi2 |      0.000
---------------------+------------------------------------------------------
Population error     |
               RMSEA |      0.064   Root mean squared error of approximation
 90% CI, lower bound |      0.032
         upper bound |      0.094
              pclose |      0.207   Probability RMSEA <= 0.05
---------------------+------------------------------------------------------
Information criteria |
                 AIC |   4362.301   Akaike's information criterion
                 BIC |   4461.250   Bayesian information criterion
---------------------+------------------------------------------------------
Baseline comparison  |
                 CFI |      0.931   Comparative fit index
                 TLI |      0.896   Tucker-Lewis index
---------------------+------------------------------------------------------
Size of residuals    |
                SRMR |      0.065   Standardized root mean squared residual
                  CD |      0.946   Coefficient of determination
----------------------------------------------------------------------------

Based on this, I see little support for a higher order factor (L4) that explains the variation and covariation in L1-L3. Ask yourself this - how would you interpret that factor given that the lower-order factors supporting it are themselves largely unrelated (specifically, L3 seems unrelated to L1 and L2)? I don't know the content of these items or the supposed constructs, so keep that in mind. But even so, it is perfectly viable to have a SEM where L1-L3 are predicted by the exogenous variables
.

1) I understand the distribution of the latent variables are somewhat abstract anyway, but it would be great if I could get the L4 distributed on a scale from 0-1 instead of from -0.5 to 0.5 or -1 to +1. Is this something possible? Can I add an option to SEM model to get the latent variables to be on a 0-1 scale.

Latent variables within a SEM are continuous variables with a distribution centered at 0. You can scale them such that they have a standard deviation of 1 by setting their variance to 1. This is done by default when you use the standardize option. To get them on any other scale, you would have to do something like predict them and then rescale the predicted variable. But that is generally not advisable because you then take all the uncertainty baked into the latent variable and treat it as a known value.

2) When I do not add 'standardized' option, what is the default distribution of the latent variables? And how can I see the distribution of latent variables after sem?

You get a normally-distributed variable that is centered (has a mean of 0) but has an estimated variance that you can discern from the model output.

3) Also, I wanted to use margins to get some more meaningful effects, but I could not figure out how to use margins for latent variables. If you can provide any help, I would be grateful.

I have not seen this done and my poking around with various margins options (predict especially) was not successful. Jeff Pitblado (StataCorp) might have more insight into this.

[Jeff Pitblado (StataCorp)]

margins after sem does not support latent variable predictions.

[Sule Yaylaci]

Thank you, Erik. The correlation is much stronger in the full dataset but I take your point about whether it is warranted to have a second-order latent variable.Thanks for pointing it out and taking the time to write a thoughtful and thorough response.

I also appreciate the response about the distribution of the latent variables. You wrote that 'You get a normally-distributed variable that is centered (has a mean of 0) but has an estimated variance that you can discern from the model output.' Do you mean the variance information at the bottom of the output (below measurement models) where a coefficient and a standard error is listed for the latent variables? When it is standardized, the coef for Var(e.L4) is .558 and the SE is .027 and when not standardized the coef for Var(e.L4) is .059 and se is .008. How do I go about finding out the range from here? I realize this may be a very simple calculation but I am a beginner in SEM so guidance would be useful.


When the estimated coefficients are standardized, what exactly does a coef of -0.076(SE:0.017) mean for the effect of the variable Male on L4? My interpretation is that Men's L4 value is 0.076 standard deviation lower than women's L4 value. But it just is not very meaningful to speak in terms of SD as unit change. When I predict L4, its mean is 0.52 and se is 0.17.Is there any way to translate the 0.076 SD decrease to an actual value of the L4?

In the unstandardized model, the coefficient of Male for the L4 as outcome is -.049 (SE:0.017). Can we interpret this as being Male is associated with a 5% decrease in the estimated L4 score as compared to L4 scores of women?

[Erik Ruzek]

With predictors of the latent variable into the model, the variance of the latent variable is a residual variance (hence the "e" in e.L4) To find the range of the variance of the latent variable, you would need to go back to a SEM without any predictors of the latent variables (i.e., the sem command containing only the measurement model). This would give you the unconditional variance of L4. From there, you could convert that to a standard deviation to determine the size the coefficients in the full SEM relative to that standard deviation.

In the standardized output, the latent variable has been scaled such that it has a total variance of 1 and thus a standard deviation of 1. Note that you can scale your measurement model in the full sem so that it has the latent variable in the standardized form (with a variance of 1). To do so, you need to look at how Stata does it in the standardized version and read up a bit on second order measurement models. There's a nice post on how you might get sem to estimate the model with the higher-order factor variance constrained to 1 from Jeff Pitblado (StataCorp) here. Jeff's shortcut version in post #15 does not work with your dataex (it might on your full dataset) but the longer version in which you create the matrix in post #13 does work on the data you posted, above.

The standardized output puts everything in a z-score type metric where both the predictors and outcomes have been standardized. So if that doesn't make sense to you, you need to roll your own solution.

In the unstandardized model, the coefficient of Male for the L4 as outcome is -.049 (SE:0.017). Can we interpret this as being Male is associated with a 5% decrease in the estimated L4 score as compared to L4 scores of women?

No, it is a .05 decrease in the value of the latent variable. As a percentage of the variance in L4, you could divide .05 by the variance of L4 in the unstandardized measurement model. Or convert that variance to a standard deviation and divide by the standard deviation.

[Sule Yaylaci]

Hi, Erik. Thanks again for your thorough answers. I have one more question. I predicted some of the latent variables to use them in some of the models to reduce the parameters. In my understanding, predict command was going to produce variables that are normalized (mean=0, std. dev. 1), but none of the predicted variables have a standard deviation of 1 (almost all of the versions have a standard deviation of 0.2-0.3). And before using the predict command, I did add the 'standardized' option to the model although not sure if it matters. Do you or anyone on this list by any chance know how predicted variables are scaled?

[Erik Ruzek]

With the default sem syntax, in which you let Stata determine how the latent variables are scaled, the variances (and thus, standard deviations) will be whatever they are estimated to be. In order to get them closer to 1, you need to use the version of the model syntax where you specify the variances of the latent variables as being @1. Even then, it will not be exactly 1. The larger your sample size, the more likely it will be 1, but not exactly. Note also that prediction of the latent variables ends up treating a latent, unobserved variable with a fair degree of uncertainty as a known quantity with much less uncertainty. None when you treat the predicted score as observed.