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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); min2=q1**2; flun2=q2**2; However, I get the following error message: *** ERROR Unknown parameter label in MODEL CONSTRAINT: Q1 I would appreciate any input you could offer. 


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; 


P.S. But note that you need more than 2 indicators as usual with factor analysis. 


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. 


Marsh has an article in Psych Methods a few years ago that compares different methods for latent variable interactions. 


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. 


No, Mplus takes the maximumlikelihood 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. 


Thanks; that's great. 


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 ysquared, predicts wave 2 ysquared. wave 2 ysquared, 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 ysquared. Can I test this with SEM? If so, how? 


You can use create the 2 ysquare variables in DEFINE and then use them as usual in the MODEL part in line with the UG path analysis examples. 


Dear Dr. Muthen, I'm modelling 2wave2variable data using a parallel latent difference score model, my syntax is: deltaX by Xt2@1; Xt2 ON Xt1@1; Xt2@0; [deltaX xt1]; deltaY by Yt2@1; Yt2 ON Yt1@1; Yt2@0; [deltaY Yt1]; Further I'm modelling a quadratic effect of X on Y and DeltaY using the LMSapproach. xsq xt1 xwith xt1; Yt1 ON xt1 xsq; DeltaY ON xt1 xsq deltaX; I get an sign. xsqeffect 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 tvalues, but plotting the results lead to different interpretations (Ushaped vs. expshaped). What would you suggest in this situation. Might this be the consequence of different scale of the Xvariables due to centering the indicators? 


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. 


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. 


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. 


Thanks, Regarding the matter of normality: I thought that the LMS and also the QMLapproach 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 QMLapproach (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 LMSApproach 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)? thanks Christoph Weber 


Sorry I'm wrong, of course LMS and QML require normal latent variables, with exception of the interaction variable. 


The normality issues are a little obtuse in the literature. Marsh et al and KleinMoosbrugger 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 firstorder 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. 


Dear Dr. Muthen, one more question: Does the LMSappoach assume, that the measurement errors are uncorrelated? regards Christoph Weber 


I don't believe there is such an assumption but in Mplus the default is that these residual covariances are zero. 


Thanks, 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? thanks 


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. 


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(x²). 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. Thanks Christoph WEber Ps. Do you know about any application using LMS with correlated residuals? 


No. Others? 


Just to clarify your answer: Do you mean that my argument regarding the effect of the error correlation on the LMSresults is "not" valid, or do you mean that you don't know other applications using LMS with correlated residuals. Thanks Christoph Weber 


It is the answer to your question about applications using LMS with correlated residuals. 


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 


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. 


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 


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 logtransformed exogenous(LOGX1) variables. All variables are observed and continuous. The output where I explicitly stated correlation between the exogenous variables (second model) said:THE STANDARD ERRORS OF THE MODEL PARAMETER ESTIMATES MAY NOT BE TRUSTWORTHY FOR SOME PARAMETERS DUE TO A NONPOSITIVE DEFINITE FIRSTORDER DERIVATIVE PRODUCT MATRIX. THIS MAY BE DUE TO THE STARTING VALUES BUT MAY ALSO BE AN INDICATION OF MODEL NONIDENTIFICATION. THE CONDITION NUMBER IS 0.410D12. PROBLEM INVOLVING PARAMETER PSI(LOGX1). Could you please help me understand what might be happening here? First model: X1 WITH X2 Y1 ON X1 X2 LOGY2 ON LOGX1 X2 Y1 LOGY3 ON LOGX1 X2 Y1 Y2 Second model: X1 WITH X2 LOGX1 WITH X2 LOGX1 WITH X1 Y2 WITH X1 Y2 WITH X2 Y2 WITH LOGX1 Y1 ON X1 X2 LOGY2 ON LOGX1 X2 Y1 LOGY3 ON LOGX1 X2 Y1 Y2 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! 


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! 


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? Thanks. 


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. 


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 Ahmed, 


I Googled and found Professor Andreas Klein, GoetheUniversity, 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 nonlinear CFA (Ex. 5.7) with two quadratic factors. Then compare the linear and nonlinear 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. 


Items may be nonlinearly 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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