Indeed, modeling moderation with continuous latent variables is a bit tricky. If you were using Mplus, there would be the relative simple solution of the LMS approach.

In R (e.g., the lavaan package), my current understanding is that you can use one of three approaches, which are compared here:

Article Three Approaches to Estimate Latent Interaction Effects: Int...

There might also be a Bayesian solution to this, but I am not sure.

You can try using SPSS process tool. If sample size is not a constraint you can surely use the tool.

I hope the following link would help.

https://groups.google.com/forum/m/#!topic/lavaan/Ytwt9lrUC5k

By the way, it is easy to solve your problem with Mplus.

Thank you Heiko Breitsohl Ramesh Kumar Taha Eser . Unfortunately, I cannot conduct analyses with Mplus Taha Eser

Sule,

attached please find an english translation of the relevant chapter from my book

Steinmetz, H. (2014). Lineare Strukturgleichungsmodelle: Eine Einführung mit R [Linear structural equation modeling: An introduction using R) (Sozialwissenschaftliche Forschungsmethoden). München: Hampp.

about latent interaction modeling. The chapter presents the R / lavaan syntax for the residual centering approach introduced in the paper which Heiko was so nice to recommend.

I hope that helps.

Best,

Holger

Beyond the traditional approaches, there are some discussions about the combination of factor score regression and interaction modeling. Although factor scores have still the problems of being manifest variables with some portion of error (and thus are regarded problematic for estimating interaction effects), some authors highlight the more recent approaches to create "covariance-informed" or "instrumental variable-informed" factor scores that subsequently can be incorporated in an interaction model.

I have not spend time on that and hence can not evaluate how viable these approaches are. But the time will come :)

Here is a page with the respective lavaan syntax:

https://www.rdocumentation.org/packages/lavaan/versions/0.6-1.1161/topics/fsr

Best,

Holger

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, Online first. https://doi.org/10.1080/10705511.2018.1473773

Curran, P.J., Cole, V.T., Bauer, D.J., Hussong, A.M., & Gottfredson, N. (2016). Improving Factor Score Estimation Through the Use of Observed Background Characteristics. Structural Equation Modeling, 23(6), 827–844. https://doi.org/10.1080/10705511.2016.1220839

Devlieger, I., & Rosseel, Y. (2017). Factor Score Path Analysis: An Alternative for SEM? Methodology, 13(Suppl. 1), 31–38. http://dx.doi.org/10.1027/1614-2241/a000130

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. https://doi.org/10.1177/0013164415607618

I thank you very much Dr. Holger Steinmetz .

Dear Sule,

Yes you can do that. The Croon approach Holger mentioned are consistent. Moreover you can use PLSc or a similar approach, the non-iterative method of moment estimator. All three do not require any distributional assumptions as long as you include only interaction terms. The advantage of these approaches including the MIIV approach of Bollen is that they are simple compared to LMS and QML and can deal with very complex models. While the latter are more efficient when their assumptions hold, the approaches based on factor scores are more robust to missspecification.

We have implemented PLSc and non-iterative method of moments estimator (Croon approach is almost implemented) in the cSEM package:

https://github.com/M-E-Rademaker/cSEM

In case of questions about the package please contact me directly.

Best regards,

Florian

Hi, I find this thread and the papers you suggested very helpful. However, my case is a bit more difficult, as my moderator variable has a second-order structure with 4 first order factors and the mediator is a latent variable with two correlated factors.

Moreover, I am analyzing experimental data, so my independent variable is dichotomous. My dependent variable is a continuous manifest variable.

My hypotheses suggest the following model:

W1 =~ w01 + w02+ w03+ w04+ w05

W2 =~ w06 + w07

W3 =~ w08 + w09

W4 =~ w10+ w11+ w12

W =~ W1 + W2 + W3+ W4

M1=~m01 + m02 + m03 + m04

M2=~m05 + m06 + m07+ m08

M1~~M2

DV ~ c*IV+ b1*M1+ b2*M2 + b1*W + b2*W +b2*W:M1 + b2*W:M2

M1 ~ a1*IV +a1*W + a1*W:IV

M2 ~ a2*IV +a2*W + a2*W:IV

I would appreciate advice for the appropriate approach to model this moderated mediation. Thank you!

Doris,

in principle, your situation does not seen to be more complex than the one initiating this thread. You have more variables, but I think the techniques required would probably be the same, with the exception of a different estimator (e.g., WLSMV) due to the dichotomous nature of the DV.

Aside from that, maybe this paper is helpful:

Article Integrating Moderation and Mediation: A Structural Equation ...

Thank your for your fast answer, Heiko.

I realized I forgot to ask my essential question: Following the residual centering approach, I would multiply the respective observed indicators of my latent variables and regress the results on all observed indicators, save the results and use them as the indicators of the interaction variable. Is the second order structure of the moderator irrelevant for this procedure?