# SEM with categorical one-indicator measurement models

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### Lilia Heinrich

Apr 3, 2019, 8:34:28 AM4/3/19
to lavaan
Hi,

Hello, I would like to know if a SEM with an endogenous latent variable - D (measurement model consists of a catigorical manifest variable -y4) is possible?

I think that does not work... And this true, so R gives errors and warnings out. The problem is D...I think so.
I would be very happy if I could exchange information here. In this case, can I do something for D or y4 to make it work?
(y4~~0.1 *y4 : To fix residual variance for y4 dosen't work)

Thank you all!

My syntax:

model <-'

A =~   x1 + x2

B =~   y1 + y2

C =~ y3

D=~ y4

B~ A
C~ A
D~ B + C

y3~~0.1 *y3

'
fit <- cfa(model,data=data, estimator="WLSMV", ordered=c( "x1", "x2", y4 ))
summary(fit, fit.measure=T,standardized=T,rsquare= T, modindices=T)

### Terrence Jorgensen

Apr 4, 2019, 9:43:08 AM4/4/19
to lavaan
In this case, can I do something for D or y4 to make it work?

You need to put "y4" in quotation marks for the ordered= argument, like the other variables are.

(y4~~0.1 *y4 : To fix residual variance for y4 dosen't work)

You need to set parameterization = "theta" for residual variances to be model parameters.

Terrence D. Jorgensen
Assistant Professor, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam

### Lilia Heinrich

Apr 7, 2019, 9:33:23 AM4/7/19
to lavaan

четверг, 4 апреля 2019 г., 15:43:08 UTC+2 пользователь Terrence Jorgensen написал:
In this case, can I do something for D or y4 to make it work?

You need to put "y4" in quotation marks for the ordered= argument, like the other variables are.
It was just a typo...

(y4~~0.1 *y4 : To fix residual variance for y4 dosen't work)

You need to set parameterization = "theta" for residual variances to be model parameters.
Thank you...but it doesn't help. Maybe because y4 isn't continuous and alone for D?!

### Terrence Jorgensen

Apr 8, 2019, 7:21:24 AM4/8/19
to lavaan
(y4~~0.1 *y4 : To fix residual variance for y4 dosen't work)

You need to set parameterization = "theta" for residual variances to be model parameters.
Thank you...but it doesn't help. Maybe because y4 isn't continuous and alone for D?!

Perhaps it is not allowed then.  I can see why, because the residual variance is already fixed for identification even for multiple categorical indicators.  So fixing a single categorical indicator's residual variance does not resolve the additional identifiability issue -- i.e., it will not imply that a certain proportion of the variance is (un)reliable.  Ultimately, you would have to fix the factor D's variance or loading to a value that would sum the explained variance (effect of C on D, and indirect effects of A and B on D via C) and D's residual variance, and make that sum equal the proportion of reliable variance (easy if you use the default delta parameterization, which conveniently fixes the total variance of the latent item response to 1).

I suspect it is simpler to simply model y4 itself.  If it's reliability is really 0.9, then the effect of C on it won't be very attenuated.

### Lilia Heinrich

Apr 9, 2019, 6:50:54 AM4/9/19
to lavaan

понедельник, 8 апреля 2019 г., 13:21:24 UTC+2 пользователь Terrence Jorgensen написал:
(y4~~0.1 *y4 : To fix residual variance for y4 dosen't work)

You need to set parameterization = "theta" for residual variances to be model parameters.
Thank you...but it doesn't help. Maybe because y4 isn't continuous and alone for D?!

Perhaps it is not allowed then.  I can see why, because the residual variance is already fixed for identification even for multiple categorical indicators.  So fixing a single categorical indicator's residual variance does not resolve the additional identifiability issue -- i.e., it will not imply that a certain proportion of the variance is (un)reliable.  Ultimately, you would have to fix the factor D's variance or loading to a value that would sum the explained variance (effect of C on D, and indirect effects of A and B on D via C) and D's residual variance, and make that sum equal the proportion of reliable variance (easy if you use the default delta parameterization, which conveniently fixes the total variance of the latent item response to 1).

I suspect it is simpler to simply model y4 itself.  If it's reliability is really 0.9, then the effect of C on it won't be very attenuated.

Thank you a lot!  "to simply model y4 itself"- do you mean- don't model the latent var. D with y4, rather put y4 independently, just like a simple var. in regression, without a measurement model for y4(D)?
I think is a good idea!