Factor scores after SEM

[Alberto Anaya]

I'm working with EFA, CFA and SEM with a dataset about public transportation satisfaction. I have 3 latent variables that linked create another one called Quality. After I did the sem:

sem (Access -> p4 p5 p61 p62) ///
(Card -> p63 p64) ///
(Conf -> p71 p73 p74 p75) ///
(Quality -> Access Card Conf), ///
method(mlmv) vce(robust) latent(Access Card Conf Quality) ///
cov(Card*Conf Access*Card Access*Conf) ///
standardized
estat gof, stats(all)

predict fs_Access fs_Card fs_Conf fs_Quality, latent(Access Card Conf Quality)
summ fs_Access fs_Card fs_Conf

I would like to know: what do factor scores mean after SEM? I know it's the latent variable impact on each subject, but i dont get the scale even if it's standardized neither the meaning clearly.

Could I run a regression with the factor scores of each latent variable?

Any help, welcome.

[Clyde Schechter]

What do factor scores mean? So, SEM has constructed a latent variable that represents a "common source factor" that, in part, accounts for observed variables p4 p5 p61 and p62. It represents, in a loose sense, what these variables "have in common." Now, notice that without any additional constraints, this model is unidentified. If you imagine a different latent variable, call it Card2 whose values are all exactly 10 times the values of Card, you could fit this model equally well: you would just change the coefficients of p4 p5 p61 and p62, multiplying each of them by a factor of 1/10. To identify the model (i.e. select only one of the infinitely many possible variables that could equally well be substituted for Card) some constraint must be imposed. The convention used by Stata (and most SEM packages) is to constrain the coefficient of the first mentioned variable (p4 in this case) to be 1. This is, of course, an arbitrary choice. You could equally well have constrained the coeffiient of p62 to be 1. Or you could constrain the coefficient of p5 to be (e+pi)^-2.1. Or you could constrain something else: you could constrain the variance of Card to be 1 (i.e. you could require Card to be a standardized variable). The reason I'm going through all this is to point out that the actual values of the variable Card are, in part, the result of arbitrary constraints imposed. What is relevant about Card are the relative values of Card in different observations and how those relate to the observed variables p4 p5 p61 and p62.

The factor scores calculated after SEM are then just the values that Card would have to take on in your data in order to best fit the model and the arbitrary constraints imposed to identify the model. But any linear transformation of Card would also work for the model (but not for the arbitrary constraints). So the actual values of the factor scores are really not meaningful at all, but the relationships among them are. You may find that unsatisfying and difficult to understand; I certainly did when I first started working with factor analysis. But that is the truth. The factor scores are one instantiation of a possible variable that fits your model's relationships; but they are only one of infinitely many possible ways of doing that, the particular way having been fixed by an arbitrary constraint. Any linear transform of that variable would also work, and would correspond to different sets of constraints. So factor scores have no natural units and no natural interpretation except as a concrete example of a variable that will fit the model.

When you go to the second-level latent variable Quality, this is then an abstraction of what is common among Access, Card, and Conf.

Could I run a regression with the factor scores of each latent variable?

You could. I have seen it done. I have even done it myself. But it really goes against the whole idea of latent variables in a structural equations model, and it is not encouraged. Rather than using the factor scores as proxies for the latent variables, the better way to use a latent variable in a regression equation is to use the latent variable itself directly in the regression equation, and add that regression equation to the pre-existing -sem- command that defined the latent variable.

[Alberto Anaya]

Thank you so much, Clyde.

Yes, it's kinda unsatisfying that factor scores have no natural units and no natural interpretation.

The idea of the regression was to have a "measure" of this latent variable for each subject and I wanted to regress with some external info per subject too. So now you say that i can add a regression, where and how i can put it? with the tool within Builder?

Thanks again.

[Joseph Coveney]

Maybe going through this will help.

version 15.1

clear *

set seed `=strreverse("1434478")'
quietly set obs 500

generate int pid = _n
generate double latent = rnormal()
label variable latent "Actual Latent Factor's Value"

forvalues i = 1/3 {
    generate double manifest`i' = `i' * latent + rnormal()
}

sem (manifest1 manifest2 manifest3 <- Latent), nocnsreport nolog

/* I would like to know: what do factor scores mean after SEM? */
predict double latent_hat, latent(Latent)
graph twoway scatter latent_hat latent, msize(small) mcolor(black) ///
    ylabel( , angle(horizontal) nogrid)
pause on
pause

/* Could I run a regression with the factor scores of each latent variable? */
generate double predictor = runiform()
sem (manifest1 manifest2 manifest3 <- Latent) (Latent <- predictor), nocnsreport nolog

exit

[Joseph Coveney]

. . . it's kinda unsatisfying that factor scores have no natural units and no natural interpretation.

I sorta disagree with that, by the way. With default constraints that Stata (and Mplus) give you, the latent factor is scaled by the first-loading indicator variable, and so it does have units as natural as whatever you choose. And at least if you've dealt with random- or mixed-effects (hierarchical) models, the interpretation isn't so unnatural (or at least nonintuitive), either.

[Alberto Anaya]

I sorta disagree with that, by the way. With default constraints that Stata (and Mplus) give you, the latent factor is scaled by the first-loading indicator variable, and so it does have units as natural as whatever you choose. And at least if you've dealt with random- or mixed-effects (hierarchical) models, the interpretation isn't so unnatural (or at least nonintuitive), either.

So, what is your opinion about using the factor scores in a regression as a dependent variable with some other external variables?

[Joseph Coveney]

So, what is your opinion about using the factor scores in a regression as a dependent variable with some other external variables?

Don't.

Instead, incorporate your regression model directly into the SEM as a submodel (structural component), just as Clyde recommended and I demonstrated.

[Alberto Anaya]

Alright, instead of the predictor you used, could I use another construct?

[Joseph Coveney]

Sure, it would basically be like an errors-in-variables regression. But if your manifest predictor variables are measured without much error (compared to the latent response variable), then I'm not sure how much it would buy you over just using the manifest predictor variables themselves.

[Herman vandeWerfhorst]

I sorta disagree with that, by the way. With default constraints that Stata (and Mplus) give you, the latent factor is scaled by the first-loading indicator variable, and so it does have units as natural as whatever you choose. And at least if you've dealt with random- or mixed-effects (hierarchical) models, the interpretation isn't so unnatural (or at least nonintuitive), either.

Thanks. I am struggling with the interpretation of my SEM model. But are you saying that the latent factor that I have is scaled on the likert-style manifest variables with range 1-5? As in: one unit change in that interpretation? Then I am not sure if my effect sizes make sense. Thanks.

[Joseph Coveney]

. . . are you saying that the latent factor that I have is scaled on the likert-style manifest variables with range 1-5? As in: one unit change in that interpretation?

Yes.

You can see that in the output below: when I increase the values of the first indicator variable, the variance of the latent factor is correspondingly increased.

Would fitting a generalized SEM that has a distribution family and link function for ordered-categorical manifest variables be closer to what you're looking for? Output from an example is also shown below.

Then I am not sure if my effect sizes make sense.

I'm afraid that I don't follow you here, and so I cannot comment.

.ÿ
.ÿversionÿ17.0

.ÿ
.ÿclearÿ*

.ÿ
.ÿ//ÿseedem
.ÿsetÿseedÿ1124860975

.ÿ
.ÿtempnameÿCorr

.ÿmatrixÿdefineÿ`Corr'ÿ=ÿJ(4,ÿ4,ÿ0.5)ÿ+ÿI(4)ÿ*ÿ0.5

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ÿÿ4.ÿÿÿÿÿÿÿÿÿquietlyÿreplaceÿ`manifest'ÿ=ÿ`manifest'ÿ+ÿ1
ÿÿ5.ÿ}

.ÿtabulateÿy1

ÿÿÿÿÿÿÿÿÿy1ÿ|ÿÿÿÿÿÿFreq.ÿÿÿÿÿPercentÿÿÿÿÿÿÿÿCum.
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ÿÿÿÿÿÿÿÿÿÿ4ÿ|ÿÿÿÿÿÿÿÿÿ50ÿÿÿÿÿÿÿ20.00ÿÿÿÿÿÿÿ80.00
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------------+-----------------------------------
ÿÿÿÿÿÿTotalÿ|ÿÿÿÿÿÿÿÿ250ÿÿÿÿÿÿ100.00

.ÿ
.ÿsemÿ(y?ÿ<-ÿF),ÿnocnsreportÿnodescribeÿnofootnoteÿnolog

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Estimationÿmethod:ÿml

Logÿlikelihoodÿ=ÿ-1633.3708

------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿOIM
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ÿÿÿÿÿÿÿÿÿÿÿFÿ|ÿÿÿÿÿÿÿÿÿÿ1ÿÿ(constrained)
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.ÿtempnameÿV

.ÿscalarÿdefineÿ`V'ÿ=ÿr(table)["b",ÿ"/:var(F)"]

.ÿ
.ÿ//ÿLatentÿfactorÿisÿscaledÿtoÿtheÿfirstÿindicatorÿvariableÿwhoseÿSDÿisÿnowÿ5×
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ÿÿÿÿÿÿÿÿÿy1ÿ|ÿÿÿÿÿÿFreq.ÿÿÿÿÿPercentÿÿÿÿÿÿÿÿCum.
------------+-----------------------------------
ÿÿÿÿÿÿÿÿÿÿ5ÿ|ÿÿÿÿÿÿÿÿÿ50ÿÿÿÿÿÿÿ20.00ÿÿÿÿÿÿÿ20.00
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------------+-----------------------------------
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.ÿsemÿ(y?ÿ<-ÿF),ÿnocnsreportÿnodescribeÿnofootnoteÿnolog

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Estimationÿmethod:ÿml

Logÿlikelihoodÿ=ÿ-2035.7302

------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿOIM
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Measurementÿÿ|
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ÿÿÿÿÿÿÿÿÿÿÿFÿ|ÿÿÿÿÿÿÿÿÿÿ1ÿÿ(constrained)
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ÿÿ-----------+----------------------------------------------------------------
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------------------------------------------------------------------------------

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.ÿ//ÿYouÿmightÿwantÿtoÿconsiderÿsomethingÿalongÿtheÿfollowingÿlines
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Logÿlikelihoodÿ=ÿ-1480.4946

------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿCoefficientÿÿStd.ÿerr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿconf.ÿinterval]
-------------+----------------------------------------------------------------
y1ÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿFÿ|ÿÿÿÿÿÿÿÿÿÿ1ÿÿ(constrained)
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y4ÿÿÿÿÿÿÿÿÿÿÿ|
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ÿÿÿÿÿÿÿÿcut1ÿ|ÿÿ-1.156936ÿÿÿ.1325353ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-1.416701ÿÿÿÿ-.897172
ÿÿÿÿÿÿÿÿcut2ÿ|ÿÿ-.3427787ÿÿÿ.1095733ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-.5575384ÿÿÿ-.1280191
ÿÿÿÿÿÿÿÿcut3ÿ|ÿÿÿ.3633023ÿÿÿ.1098783ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.1479448ÿÿÿÿ.5786598
ÿÿÿÿÿÿÿÿcut4ÿ|ÿÿÿ1.155807ÿÿÿ.1313199ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.8984249ÿÿÿÿÿ1.41319
-------------+----------------------------------------------------------------
/y2ÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿcut1ÿ|ÿÿ-1.188479ÿÿÿ.1385542ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-1.46004ÿÿÿÿ-.916918
ÿÿÿÿÿÿÿÿcut2ÿ|ÿÿ-.3445468ÿÿÿ.1129976ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-.566018ÿÿÿ-.1230756
ÿÿÿÿÿÿÿÿcut3ÿ|ÿÿÿ.3739916ÿÿÿ.1131765ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.1521698ÿÿÿÿ.5958135
ÿÿÿÿÿÿÿÿcut4ÿ|ÿÿÿ1.181701ÿÿÿ.1368353ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.9135089ÿÿÿÿ1.449894
-------------+----------------------------------------------------------------
/y3ÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿcut1ÿ|ÿÿ-1.220329ÿÿÿ.1451672ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-1.504852ÿÿÿ-.9358069
ÿÿÿÿÿÿÿÿcut2ÿ|ÿÿ-.3627333ÿÿÿ.1165009ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-.5910708ÿÿÿ-.1343958
ÿÿÿÿÿÿÿÿcut3ÿ|ÿÿÿ.3683094ÿÿÿ.1165825ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.139812ÿÿÿÿ.5968068
ÿÿÿÿÿÿÿÿcut4ÿ|ÿÿÿ1.220783ÿÿÿ.1450807ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.9364305ÿÿÿÿ1.505136
-------------+----------------------------------------------------------------
/y4ÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿcut1ÿ|ÿÿ-1.342824ÿÿÿ.1694428ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-1.674926ÿÿÿ-1.010722
ÿÿÿÿÿÿÿÿcut2ÿ|ÿÿ-.4027732ÿÿÿ.1296094ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ-.656803ÿÿÿ-.1487434
ÿÿÿÿÿÿÿÿcut3ÿ|ÿÿÿ.4273181ÿÿÿ.1302345ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.1720631ÿÿÿÿÿ.682573
ÿÿÿÿÿÿÿÿcut4ÿ|ÿÿÿ1.348792ÿÿÿ.1694757ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1.016625ÿÿÿÿ1.680958
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿvar(F)|ÿÿÿ.8569842ÿÿÿÿ.232904ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.5030839ÿÿÿÿÿ1.45984
------------------------------------------------------------------------------

.ÿ*ÿSameÿvarianceÿforÿlatentÿfactorÿasÿoriginallyÿscaledÿy1
.ÿdisplayÿinÿsmclÿasÿtextÿ"Ratioÿofÿlatentÿfactors'ÿvariances:ÿ"ÿ///
>ÿÿÿÿÿÿÿÿÿasÿresultÿ%4.2fÿr(table)["b",ÿ"/:var(F)"]ÿ/ÿ`V'
Ratioÿofÿlatentÿfactors'ÿvariances:ÿ1.00

.ÿ
.ÿ*ÿy1ÿnowÿarbitaryÿvaluesÿforÿorderedÿcategories
.ÿquietlyÿrecodeÿy1ÿ(5ÿ=ÿ-1)ÿ(10ÿ=ÿ0)ÿ(15ÿ=ÿ100)ÿ(20ÿ=ÿ101)ÿ(25ÿ=ÿ102)

.ÿtabulateÿy1

ÿÿÿÿÿÿÿÿÿy1ÿ|ÿÿÿÿÿÿFreq.ÿÿÿÿÿPercentÿÿÿÿÿÿÿÿCum.
------------+-----------------------------------
ÿÿÿÿÿÿÿÿÿ-1ÿ|ÿÿÿÿÿÿÿÿÿ50ÿÿÿÿÿÿÿ20.00ÿÿÿÿÿÿÿ20.00
ÿÿÿÿÿÿÿÿÿÿ0ÿ|ÿÿÿÿÿÿÿÿÿ50ÿÿÿÿÿÿÿ20.00ÿÿÿÿÿÿÿ40.00
ÿÿÿÿÿÿÿÿ100ÿ|ÿÿÿÿÿÿÿÿÿ50ÿÿÿÿÿÿÿ20.00ÿÿÿÿÿÿÿ60.00
ÿÿÿÿÿÿÿÿ101ÿ|ÿÿÿÿÿÿÿÿÿ50ÿÿÿÿÿÿÿ20.00ÿÿÿÿÿÿÿ80.00
ÿÿÿÿÿÿÿÿ102ÿ|ÿÿÿÿÿÿÿÿÿ50ÿÿÿÿÿÿÿ20.00ÿÿÿÿÿÿ100.00
------------+-----------------------------------
ÿÿÿÿÿÿTotalÿ|ÿÿÿÿÿÿÿÿ250ÿÿÿÿÿÿ100.00

.ÿ*ÿButÿsameÿvarianceÿforÿlatentÿfactor
.ÿquietlyÿgsemÿ(y?ÿ<-ÿF,ÿoprobit)

.ÿdisplayÿinÿsmclÿasÿtextÿ"Ratioÿofÿlatentÿfactors'ÿvariances:ÿ"ÿ///
>ÿÿÿÿÿÿÿÿÿasÿresultÿ%4.2fÿr(table)["b",ÿ"/:var(F)"]ÿ/ÿ`V'
Ratioÿofÿlatentÿfactors'ÿvariances:ÿ1.00

.ÿ
.ÿexit

endÿofÿdo-file


.