How to integrate a Rasch measure into a SEM?

[Tobias Ludwig]

I´m using lavaan for a model consisting of 4 endogenous latent variables (enLV) and 4 exogenous latent variables (exLV). All LVs are measured by a 5-step Likert scale except one exLV, which is a dichotomous knowledge test. The raw scores of this test were used to fit a Rasch-Model using TAM in R. So I extended my data frame with the Rasch-measure and and the error variances for each case. How can I integrate these Rasch-measures into my SEM? Maybe as a single-item indicator?

Any help is highly appreciated!

PS: There´s some freaky literature, e.g. here: http://www2.wu.ac.at/marketing/mbc/download/Rasch_SEM.pdf

[Sunthud Pornprasertmanit]

When I see the Mplus scripts from the link, I think that it is the single-indicator factor with the fixed factor loading, fixed error variance, and fixed factor variance. You can translate the Mplus script to lavaan. However, it is very weird to me that all factor loading, error variance, and factor variance are fixed, which leads to df of 1.

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[Tobias Ludwig]

I don´t really understand the paper, it seems pretty magical to me. Is there any other chance to integrate a Rasch measure into SEM?

[Tobias Ludwig]

I just read, that lavaan can handle a LV with a single indicator. Can I specify the residual variance of the single indicator? (Since I already got that from the Rasch-Model...

[Terrence Jorgensen]

I just read, that lavaan can handle a LV with a single indicator. Can I specify the residual variance of the single indicator? (Since I already got that from the Rasch-Model...

Yes, if you have a factor "F" defined by a single indicator "x", the sem() and cfa() functions automatically fix the residual variance to zero, but you can specify the error variance (e.g., as 0.34) of an observed variable as the variable's covariance with itself:

modelsyntax <- '
F =~ x
x ~~ 0.34*x
'

Set the argument std.lv = TRUE or FALSE to control whether you freely estimate the factor loading or factor variance, fixing the other to 1.

Terry

[Tobias Ludwig]

Thanks, sure, but the Rasch-model gives me a variance for every case. Is there a chance to specify this column in the model?

[yrosseel]

On 08/10/2014 03:35 PM, Tobias Ludwig wrote:
> Thanks, sure, but the Rasch-model gives me a variance for every case. Is
> there a chance to specify this column in the model?

No. You have two options:

1) you retrieve the original dichotomous scores, and you just define
your exLV as a latent variable measured by these binary items

(and if you must be 'Rasch', you should also constrain the factor
loadings to be equal, by giving them the same label)

2) like Terry suggested, you specify a single-indicator latent variable,
and you specify the (residual) variance of the single variable to a
constant, defined as

(1 - rho) * var(x)

and 'rho' is a measure of reliability, which you can retrieve from the
RUMM output, and var(x) is the observed variance of the rash scores.

Yves.

[Tobias Ludwig]

Thanks a lot for this answer, Yves. In Science Education it´s quite common to use Rasch-models to evaluate assessments. (Would take me to long to explain that Rasch and a measurement model in the SEM framework with constrained factor loadings would be equal)...

Regarding your suggestions 2): 

So that means, I simply take my model based reliability and the variance of the PersonAbility (which is actually close to 1) to calculate my single-indicator residual variance based on the formula (=.40) Then I constrain the residual variance of the single-item-indicator fw like this, correct?

fw~~fw.resvar*fw

fw.resvar==0.4055417

Is there a chance to access R objects inside the model syntax? 

[yrosseel]

On 09/04/2014 06:31 PM, Tobias Ludwig wrote:
> Is there a chance to access R objects inside the model syntax?

No. And this will also not happen in 0.5-17. But I do find the idea
intriguing, and I will try to see if this is possible somehow.

Yves.

[Tobias Ludwig]

Sorry for digging on this old thread. Sunthud,  I just figured out, that it actually does´t matter: When I follow the link posted above I yield a model with 1 df, where actually nothing is estimated. When I let the factor loading or the factor variance to be estimated free (whilst the other is fixed to 1) I either yield a factor variance (which surprisingly corresponds exactly to the latent factor variance of the Rasch model) or I yield a factor loading which surprisingly corresponds to the regression calculated from the link above. 

Am Donnerstag, 7. August 2014 20:14:57 UTC+2 schrieb Sunthud Pornprasertmanit: