I am confused by the descriptions about how Mplus (version 8.2) centers variables that are not mentioned on neither "WITHIN" nor "BETWEEN".
In the User Book (Version 8, pp. 274-275) the following explanations are provided:
"When a covariate is not mentioned on the WITHIN statement, it is decomposed into two uncorrelated latent variable... This decomposition can be viewed as an implicit, latent group-mean centering of the latent within-level covariate"
After reading this description, I am very confused. What is the difference between this "latent group-mean centering" and group-mean centering Mplus does by using "DEFINE: center x y m(groupmean)"?
When writing the Method part in my paper, is it correct to equal this "latent group-mean centering" with normal group-mean centering, and simply say, all these variables are group-mean centered?
No, the two approaches are different and usually the centering using the latent variable decomposition is better. This is described in the 2 papers:
Lüdtke, O., Marsh, H.W., Robitzsch, A., Trautwein, U., Asparouhov, T., & Muthén, B. (2008). The multilevel latent covariate model: A new, more reliable approach to group-level effects in contextual studies. Psychological Methods, 13, 203-229.
Asparouhov, T., & Muthén, B. (2019). Latent variable centering of predictors and mediators in multilevel and time-series models. Structural Equation Modeling: A Multidisciplinary Journal, 26, 119-142. DOI: 10.1080/10705511.2018.1511375