Yes, you can have mediating and moderating variables in SEM. Some have suggested to put the moderating variables as control variables in a regression than in SEM. The SEM model will become too complicated to solve.

Dear Sara,

Sure you can have moderators in SEM. Various approaches have been developed for that purpose. The typical applied is latent moderated structural equations (LMS) implemented in MPlus. Others are 2SMM, PLSc, and QML...

As Florian Schuberth has been mentioned, you can have moderator(s) in a model. Moreover, a model can be developed for testing the Mediation , Moderation(Interaction) and Multi-group effects.

Sara,

in case to you don't have access to Mplus, you can apply the latent product term approach in every SEM software

see

Little, T. D., Bovaird, J. A., & Widaman, K. F. (2006). On the merits of orthogonalizing powered and product terms: Implications for modeling interactions among latent variables. Structural Equation Modeling, 13(4), 497-519.

Steinmetz, H., Davidov, E., & Schmidt, P. (2011). Three approaches to estimate latent interaction effects: Intention and perceived behavioral control in the theory of planned behavior. Methodological Innovations Online, 6(1), 95-110.

Moosbrugger, H., Schermelleh-Engel, K., & Klein, A. (1998). Methodological problems of estimating latent interaction effects. Psychological Research, 2(2), 95-108.

A second strategy could be to estimate factor scores. This idea was discussed by Yang-Wallentin et al. and Schumacker some time ago, however, with limited responses by the literature. Recently, there were some discussions on factor score path analysis, which could bring some dynamics into the field. The two problems, however, are that a) measurement error is not completely eliminated (but this can be improved by using covariates) and b) factor score estimation improves with increasing numbers of item--which, almost always implies a misspecification of the model.

Best,

Holger

Yang-Wallentin, F., Schmidt, P., Davidov, E., & Bamberg, S. (2004). Is there any interaction effect between intention and perceived behavioral control? Methods of Psychological Research Online, 8(2), 127-157.

Schumacker, R. E. (2002). Latent variable interaction modeling. Structural Equation Modeling, 9(1), 40-54.

Devlieger, I., & Rosseel, Y. (2017). Factor score path analysis: An alternative for SEM. Methodology, 13, 31-38. doi:10.1027/1614-2241/a000130

Beauducel, A. (2005). How to describe the difference between factors and corresponding factor-score estimates. Methodology, 1(4), 143-158. doi:10.1027/1614-2241.1.4.143

Curran, P. J., Cole, V., Bauer, D. J., Hussong, A. M., & Gottfredson, N. (2016). Improving factor score estimation through the use of observed background characteristics. Structural Equation Modeling: A Multidisciplinary Journal, 23(6), 827-844. doi:10.1080/10705511.2016.1220839

Curran, P. J., Cole, V. T., Bauer, D. J., Rothenberg, W. A., & Hussong, A. M. (2018). Recovering predictor–criterion relations using covariate-informed factor score estimates. Structural Equation Modeling: A Multidisciplinary Journal, 1-16. doi:10.1080/10705511.2018.1473773

Best,

Holger

Dear Sara,

from your question it is not clear whether your variables are latent or observable. My previous answer refers to a situation where your variables involved in the moderation are latent. If yuor variables are observable you can also use PROCESS, otherwise PROCESS will lead to biased estimates...

Holger Steinmetz: The idea of using factor scores in a quite general way was already introduced by Wall & Amemiya (2000) and has been investiagted by Brandt et al. (2014) showing that his approach performs quite good compared to others. I think its used is limited because lacking software implementation.

I am not familiar with all the literature you mentioned but refering to Devlieger & Rosseel (2017), particularly the Croon approach they mentioned, can you please elaborate on your two raised concerns:

a) measurement error is not completely eliminated (but this can be improved by using covariates)

b) factor score estimation improves with increasing numbers of item--which, almost always implies a misspecification of the model.

As far as I know, this approach is theoretically consistent (not only consistent at large as e.g., PLS) for any number of indicators. I say theoretically as the measurement models need to be identified and as typically each measurement model is estimated by its own, thus 3 indicators are required.

Best wishes,

Flo

References

Brandt, H., Kelava, A., & Klein, A. (2014). A simulation study comparing recent approaches for the estimation of nonlinear effects in SEM under the condition of nonnormality. Structural equation modeling: a multidisciplinary journal, 21(2), 181-195.

Wall, M. M., & Amemiya, Y. (2000). Estimation for polynomial structural equation models. Journal of the American statistical association, 95(451), 929-940.

Hi Florian Schuberth

thanks for the two references, I was completely unaware that Holger and Tino have a paper on that....I'll add these.

With regard to the mentioned two issues, I unfortunately can only speculate (as I haven't yet spend time on the factor scores topic) but I think the reason is the indeterminacy problem--that is that the factor score itself is only an estimate of the true factor. This could explain the "measurement error" comment which I found in Bispe et al. and the number-of-indicator issue by the reduced indeterminacy. For instance, in a recent paper, Devlieger et al. (2016) write: "The degree of indeterminacy is small if the relationship between the indicators and the latent variable is strong or if the number of indicators is high (Acito & Anderson, 1986)".

Bisbe, J., Coenders, G., Saris, W. E., & Batita-Foguet, J. M. (2006). Correcting measurement error bias in interaction models with small samples. Metodološki zvezki, 3(2), 267-287.

Devlieger, I., Mayer, A., & Rosseel, Y. (2016). Hypothesis testing using factor score regression: A comparison of four methods. Educational and Psychological Measurement, 76(5), 741-770. doi:10.1177/0013164415607618

An idea could be to add instruments for the IV and moderator (and product term).

This would work for manifest scores (although this would ideally require tested measurement models not ambiguous composites) or factors scores.

I have made a small simulation which I attached in a separate post. I did that with simple variables but trying that with factor scores would be interesting ( Sara Hoseingholizade please take a look into the file). Perhaps a nice paper idea to compare that with latent interaction models ;)

Best,

Holger

Dear Holger Steinmetz ,

Sara Hoseingholizade : Sorry in advacne for the off-topic

I think I misunderstood your first comment as for me factor score regression automatically involves a kind of correction, sorry for that.

I fully agree, if you use the plain factor scores without any correction, they most likely will lead to biased estimates. Your statement about the diminishing bias for an increasing number of items reminds me of PLS which is "consistent at large", i.e., besides increasing the number of observations to infinity, also the numbers of items needs to be increased to infinity to obtain the population parameter in probability. I guess similar is true if the scores are obtained by other methods. So in general, if one is interested in obtaining consistent estimates, regressing factor scores without any correction is not recommended as the scores are not reliable and thus the estimates suffer from attenuation.

However, if you correct for the reliability of the factor scores, this is what Devlieger et al (2016; Section "Bias correcting Method" ) or Wall & Amemiya (2000) suggest (and what I understood as factor score regression) your estimates are consistent. The general idea is to seperately estimate the factor models by MLE and then you can determine the reliability of your scores depending on the weights you used to build the scores. Btw, this is also what PLSc does, but without using MLE to estimate the loadings. This concept can be extended to non-linear models as well, see for example a presentation I gave in Ghent or the Wall & Amamiya (2000) paper: http://florianschuberth.com/wp-content/uploads/2019/02/Presentation-MoMpoly-Ghent.pdf

Your instrumental variable approach reminds me a little bit what Bollen (1995) suggests for non-linear factor models. In general, the MIIV-SEM approach of Bollen is quite powerful and can handle very complex models. And the nice thing, it is implemented in the R package MIIVsem :)

Best regards,

Flo

Bollen, K. A. (1995). Structural equation models that are nonlinear in latent variables: A least-squares estimator. Sociological methodology, 25, 223-252.

Hi Florian Schuberth ,

I, too, thought that factor scores would solve the error-problem. I have some mixed feelings towards factor scores. On one hand, it reduces the complexity of the model and the chisquare test can focus on causally essential implications. On the other hand, the Devlieger/Rosseel appraoch specifies factor models in isoloation which reduces the test of the measurement model. But I would assume that it still detects misspecified models.

With regard to the "bias correcting method", yes this could work. Exciting and definitely do-able. However, my last try with lavaan did not end well. The function (I guess it is "fcr( )" ) is still under construction.

Finally, yes the MIIV approach.... I would love to know how that works but cannot afford to spend much time on it. Do you know of any tutorial?

Best,

Holger

Yes, you can have mediating and moderating variables in SEM. Some have suggested to put the moderating variables as control variables in a regression than in SEM. The SEM model will become too complicated to solve.

Hey Holger Steinmetz ,

yes the factor scores function of lavaan is still under development.

We have implemented an adjusted version of the 2SMM estimator in the cSEM package (https://github.com/M-E-Rademaker/cSEM). In contrast to the original 2SMM, we fix the variance of the latent variable to 1 and we scale the scores to have unit variance. The rest is the same.

Considering the MIIV approach, I don't know about a tutorial, but a presentation by Bollen where he explains his approach:

https://timms.uni-tuebingen.de/Player/PlayerFlow/UT_20170920_001_fgme2017_0001

Best regards,

Flo

Florian Schuberth thanks for all the comments,they were so helpful and since then I am reviewing the sources you've mentioned. My moderator variable is observable.

Thanks again for all the help.:)

Alireza shabani shojaei That's a relief. Thanks

Hello Holger Steinmetz

Your suggestions are pretty helpful. I am on it and may confront so many other questions. I will again ask a question if confronted any problem. The sources are so helpful. Thanks a lot. :)

Dear Wioleta Kucharska I have checked afhayes once before but didn't deep into it. Upon your suggestion, I am checking it again. Thanks a lot.

Hi Chung Tin Fah

I guess that is why my supervisor is insisting on regression. :) Thanks a lot

Sara Hoseingholizade - Welcome, interesting discussion. Hope your Phd journey is smooth.

Yes. We can.

We do it with Gender as moderator here:

Article Influences of Role Models and Gender on Saudi Arabian Freshm...

We performed for that a two-group SEM analysis.

Any feedback. I'm following your comments.

Wassim J. ALOULOU thanks a lot. My moderator variable is also gender. I will deeply read your article to see if I can find how to do it.

You might want to consider a multi-group test to see if the overall model as well as specific paths vary by gender. You will have to conduct an invariance test(s) to ensure the validity of your findings.

The SEM-PLS model should be taken into account.

  Also a free program to take into account is jamovi that through the TOSTER option that can be downloaded from the JAMOVI library allows you to assess whether there is equivalence between factors, groups or sex if there is none, you can predict that there is a moderation between specific groups.

Yes, If moderating variable is a continuous variable Residual Centering approach can be used for moderation analysis.

Please refer the following paper.

Unconstrained Structural Equation Models of Latent Interactions: Contrasting Residual- and Mean-Centered Approaches

https://www.youtube.com/watch?v=9Lk3ObsmZho&t=504s

https://www.youtube.com/watch?v=8CjJosnbEU0&t=977s

These tutorial videos explain how to do a moderate mediation in AMOS or PROCESS.