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 TD posted on Tuesday, July 31, 2007 - 9:33 am
Hello -

I have a latent factor with 2 indicators. I suspect that this factor may have a quadratic effect. However, I am unclear on how to model quadratic effects in Mplus. Thus far, my model for this effect looks as follows:

Model: sc by min (q1)
flun (q2);

Model Constraint:
NEW (min2 flun2);

However, I get the following error message:

Unknown parameter label in MODEL CONSTRAINT: Q1

I would appreciate any input you could offer.
 Bengt O. Muthen posted on Tuesday, July 31, 2007 - 6:58 pm
Model constraint pertains to parameters, not latent variables.

Ex 5.13 is similar. So say

sc by min flun;

scsq |sc XWITH sc;

min flun on scsq;
 Bengt O. Muthen posted on Tuesday, July 31, 2007 - 7:02 pm

But note that you need more than 2 indicators as usual with factor analysis.
 Gregory Smith posted on Monday, April 28, 2008 - 7:12 am
I gather there is disagreement in the literature concerning the best method for estimating interactions and, more generally, product terms of latent variables in SEM. I can see how to represent the quadratic term I have in mind using MPlus, but I don't know how to relate this method to the SEM literature on how to do it. Is the method in MPlus essentially what Jaccard and Wan (1995, Psych Bull) argued for, or does it have some other basis? Is there a literature demonstrating that the method used in MPlus produces relatively unbiased estimates (given assumptions)? Thanks.
 Bengt O. Muthen posted on Monday, April 28, 2008 - 7:30 am
Marsh has an article in Psych Methods a few years ago that compares different methods for latent variable interactions.
 Gregory Smith posted on Monday, April 28, 2008 - 7:52 am
Thank you. Marsh et al. advocated for an unconstrained approach. Their latent variable reflecting the interaction is given as the sum of three latent variables: the two latent variables plus their product. Is this the procedure that MPlus uses? Thanks again.
 Bengt O. Muthen posted on Monday, April 28, 2008 - 8:10 am
No, Mplus takes the maximum-likelihood approach. This is same estimator as referred to as the "LMS" method in Klein et al's Psychometrika article, albeit with a slightly different algorithm.
 Gregory Smith posted on Monday, April 28, 2008 - 12:12 pm
Thanks; that's great.
 Gregory Smith posted on Monday, April 28, 2008 - 5:11 pm
Here is what I would like to test, using three waves of data:

wave 1 predictor x1, controlled for wave 1 predictor x2 and wave 1 predictor y-squared, predicts wave 2 y-squared.

wave 2 y-squared, controlled for wave 2 predictor x2, predicts wave 3 variable x2.

wave 1 predictor x1, again controlled for wave 1 predictor x2, predicts wave 3 x2. But, this effect is mediated by wave 2 y-squared.

Can I test this with SEM? If so, how?
 Bengt O. Muthen posted on Thursday, May 01, 2008 - 10:27 am
You can use create the 2 y-square variables in DEFINE and then use them as usual in the MODEL part in line with the UG path analysis examples.
 Christoph Weber posted on Friday, November 27, 2009 - 7:31 am
Dear Dr. Muthen,
I'm modelling 2-wave-2-variable data using a parallel latent difference score model, my syntax is:

deltaX by Xt2@1;
Xt2 ON Xt1@1;
[deltaX xt1];

deltaY by Yt2@1;
Yt2 ON Yt1@1;
[deltaY Yt1];

Further I'm modelling a quadratic effect of X on Y and DeltaY using the LMS-approach.

xsq |xt1 xwith xt1;
Yt1 ON xt1 xsq;
DeltaY ON xt1 xsq deltaX;

I get an sign. xsq-effect at time 1. Now I'm worring about the robustness of the results. E.g. Coenders et al (2008) show, that LMS yields biased estimates when the indicators are nonnormal distributed (this is actually the case in my example). Now I tried to replicate the crossectional results at time1 using different constrained (Coenders et al 2008) and unconstrained (Marsh et al 2004)approaches. The results show similar t-values, but plotting the results lead to different interpretations (U-shaped vs. exp-shaped). What would you suggest in this situation.
Might this be the consequence of different scale of the X-variables due to centering the indicators?
 Linda K. Muthen posted on Friday, November 27, 2009 - 2:59 pm
From what you show, xt1 is an observed variable. The XWITH option cannot be used for two observed variables. You need to use the DEFINE command in this case.
 Christoph Weber posted on Saturday, November 28, 2009 - 2:21 am
sorry, I didn't mention the measurement part. X and Y are latent variablen. X is measured by 5 indicators and Y is a second order factor measured by 4 first order factor, where each first order factor is measured by 3 items.
 Bengt O. Muthen posted on Sunday, November 29, 2009 - 5:10 pm
Regarding your initial question, I think the issue is not if the items are nonnormal but if the latent variables are nonnormal. You expect latent variable interactions to generate nonnormal items.

I can't answer your question about comparing the 3 methods. It seems that if you center variables for one approach you want to do the same in the other approaches just to be on the safe side.
 Christoph Weber posted on Monday, November 30, 2009 - 8:13 am

Regarding the matter of normality: I thought that the LMS and also the QML-approach have little problems in dealing with nonnormal latent variables (esp. the interaction variable), but that both approaches assume normal distributed items. As I understand Marsh et al (2004), they show that in case of the violation of this assumption the QML-approach (and maybe also the LMS) leads to positively biased estimates. Am I right?

and I have a further question:
In a next stept I included a second independent variable "M".
I'm interested in the following model:

Y ON X M Xsq M*X M*Xsq.

I used the LMS-Approach and got the expected M*XSq effect. Is it possible to "replicate" the effect using an interaction approach with product indicators. I didn't find any analogous application in the literature. Or would it be better to use a more simple approach (eg. Schuhmacker 2002)?

Christoph Weber
 Christoph Weber posted on Monday, November 30, 2009 - 8:50 am
Sorry I'm wrong, of course LMS and QML require normal latent variables, with exception of the interaction variable.
 Bengt O. Muthen posted on Monday, November 30, 2009 - 9:30 am
The normality issues are a little obtuse in the literature. Marsh et al and Klein-Moosbrugger consider an exogenous measurement part for a vector of indicators x and its latent variables interact in influencing y. This means that x can be assumed to be normal, but the interactions of the latents imply that y is nonnormal. So when Marsh on page 289 talks about indicators of the first-order factors being normal, he's talking about x, not y.

The real issue is the nonnormality of the latent variables and that's the focus of Marsh's "Study 4". "LMS" (that is "ML" in Mplus terms) has a bit of bias if the latents are not normal. QML is a little better in this regard.

I don't know the answer to your last question.
 Christoph Weber posted on Sunday, December 06, 2009 - 12:07 pm
Dear Dr. Muthen,
one more question:

Does the LMS-appoach assume, that the measurement errors are uncorrelated?


Christoph Weber
 Linda K. Muthen posted on Monday, December 07, 2009 - 11:07 am
I don't believe there is such an assumption but in Mplus the default is that these residual covariances are zero.
 Christoph Weber posted on Monday, December 07, 2009 - 12:05 pm
my actual problem is:

I'm comparing the LMS approach with the Marsh et al unconstrained approach and the Coenders et al approach (see previous posts).
And LMS yields an estimate for the squared term (.65), that is three times higher than the estimates for the other approaches (.18).Items were for all approaches mean centered. My independet variable is measured by 5 items. The error correlation between the first two items were set free and for the Marhs and Coenders approach also I estimate an error correlation between the two squared indicators. So I explicitly take the correlated errors into acount.

When I exclude the error correlation, all approaches yield similar results.

In a next step I used the fully constrained approach, where it is assumed that all error covariations are 0. Especially the constraints on the error variances of the squared term lead to an increase of the estimate (.35).
So I thought that there would be an problem with error correlation in the LMS approach.

Do you have any suggestions, how to deal with the situation?

 Bengt O. Muthen posted on Monday, December 07, 2009 - 5:29 pm
Perhaps the Marsh and Coenders approaches are affected by the residual correlation? For example, are the constraints for interaction variables correct with residual correlations?

If you can't see a theoretical reason for your findings, maybe you need to simulate data within Mplus to see this clearly.
 Christoph Weber posted on Tuesday, December 08, 2009 - 4:53 am
I think, that the Marsh et al approach will not be affected by the residual correlations, because all parameters are free estimated (I didn't set var(x)=mean(x˛). And when I did the full constrained approach (Aligna, Moulder), the constraints on the measurement errors, which are actually not corrected for the correlated residuals, lead to an convergence of the squared estimate to the LMS results. So I thought this would be an argument, that LMS is also affected by the residual correlation.

But I will also do a simulation.


Christoph WEber

Ps. Do you know about any application using LMS with correlated residuals?
 Bengt O. Muthen posted on Tuesday, December 08, 2009 - 8:16 am
No. Others?
 Christoph Weber posted on Wednesday, December 09, 2009 - 5:37 am
Just to clarify your answer:
Do you mean that my argument regarding the effect of the error correlation on the LMS-results is "not" valid, or do you mean that you don't know other applications using LMS with correlated residuals.

Thanks Christoph Weber
 Linda K. Muthen posted on Wednesday, December 09, 2009 - 6:33 am
It is the answer to your question about applications using LMS with correlated residuals.
 michela addis posted on Friday, March 12, 2010 - 5:10 am
Dear Prof. Muthen,

I am testing a longitudinal path analysis with only observed variables, but I want to test the nonlinear effects among them. I have found many references regarding how to do that with latent variables, but none when observed variables are included. It might be a silly topic, but from my understanding I should:
1) standardize the original variables
2) compute their quadratic terms
3) include the variables and their quadratic terms into the model through the expected links
4) imposing that the quadratic terms are not linked to any other variables not already considered.

Is this procedure right?
My doubt refers mainly to the following issues:
1) Shall I standardize the variables before using them?
2) Shall I impose that (let's say):
X1 with X1Squared@0?
and do that for any other possible relationships not explicitely considered (otherwise they are estimated by Mplus)?

Thank you very much for your support.

Michela Addis
 Linda K. Muthen posted on Friday, March 12, 2010 - 8:08 am
I would not standardize the variables. I would center the covariates. You can use the DEFINE command to create variables squared or the interaction between variables and then model them as follows:

y ON x1 x2 x1x1 x1x2;

Nothing else is necessary.
 michela addis posted on Friday, March 12, 2010 - 9:54 am
Thank you very much.
I am taking your suggestion, but I have a new question please:
I would like to test not only the nonlinear effects of the covariates on Y (i.e. Y1 on X1 X1X1 X2 X2X2) but also the nonlinear effects among dependent variables (i.e. Y2 on Y1 Y1Y1 X1 X1X1). Does it change something?
My problem is that I get very good fit indexes for the linear relationships and for the nonlinear relationships involving the covariates, but as soon as I include the nonlinear relationships among the dependent variables (which I already know they exist) I get very bad fit indexes, and the modification indexes tell me that I should include Y1 with Y1Y1. Should I do that? And I am wondering whether I am neglecting something... for instance: should I center also Y1? Many thanks! michela
 Linda K. Muthen posted on Saturday, March 13, 2010 - 9:49 am
You can use the same approach.
 naT posted on Wednesday, March 02, 2011 - 6:00 pm
I have a question about the difference between the explicit use of WITH and implicit WITH.I am working with path model (specified below). It has 3 endogenous(Y1, LOGY2, LOGY3), 3 exogenous(X1, X2, Y2) and a log-transformed exogenous(LOGX1) variables. All variables are observed and continuous.

Could you please help me understand what might be happening here?

First model:
Y1 ON X1 X2

Second model:
Y1 ON X1 X2

My understanding was that the correlation between non linear effects (Log variables) and original variable, or the correlation between observed independent variables need not to be stated unless it is interested. Please help me with this issue. Thank you!
 Linda K. Muthen posted on Thursday, March 03, 2011 - 6:51 am
When you explicitly mention the covariances among the observed exogenous variables, they are brought into model estimation and distributional assumptions are made about them. When you don't, the model is estimated conditioned on these variables. The identification message most likely comes about because the mean and variance of a binary covariate are not orthogonal. If this is the case, the message can be ignored.
 naT posted on Thursday, March 03, 2011 - 11:52 am
I see the difference now. Thank you!
 Xiaowan Lin posted on Tuesday, July 03, 2012 - 2:08 am
Linda, you replied that "I would not standardize the variables. I would center the covariates. You can use the DEFINE command to create variables squared or the interaction between variables and then model them as follows:

y ON x1 x2 x1x1 x1x2; "

Here, are x1 and x2 latent variables or observed variables?
If they are latent variable, how can we center them first before we create new interaction term?

 Linda K. Muthen posted on Tuesday, July 03, 2012 - 10:50 am
Latent variables don't need to be centered. They most often have means of zero. You can use the XWITH option to create the interactions.
 Ahmed Khalil Ben Ayed posted on Tuesday, May 05, 2015 - 11:33 am
Dear Prof. Muthen,

I am currently working on a comparative simulation study using different approaches for latent nonlinear effects including LMS and QML and I couldn’t find the updated contact informations of Prof. A. G. Klein or the QML software anywhere.

Could you please help me on this matter?

Many thanks

 Linda K. Muthen posted on Tuesday, May 05, 2015 - 11:49 am
I Googled and found

Professor Andreas Klein, Goethe-University, Frankfurt, Germany
 Djangou C posted on Thursday, October 10, 2019 - 11:03 pm
I am turning to you for help. I ran a two factor CFA model followed with an SEM . All indicators are continuous. One reviewer is asking to investigate nonlinear relationship. My question to you is how could I test the following hypothesis under the structural equation modelling framework:

1.It is reasonable to assume that the items are linearly related to each other

2.The factors are linearly related to the outcomes variables

To test hypothesis 1. I was thinking of using non-linear CFA (Ex. 5.7) with two quadratic factors. Then compare the linear and non-linear CFA through the IC indices. Is there something else I could do? Is it for instance possible to consider the square of the items in the CFA?

For hypothesis 2. I was thinking of using SEM with interaction between two latent variables. Any other option?

Many thanks for your invaluable help.
 Bengt O. Muthen posted on Friday, October 11, 2019 - 2:35 pm
Items may be non-linearly related to factors which can be checked out by using

fxf | f XWITH f;
y ON fxf;

where y is a factor indicator of f.

Similarly, with the help of XWITH, you can use f1xf2, f1xf1, and f2xf2 as predictors.

And, ICs are good to evaluate the models.
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