XWITH or random slope interaction?
Message/Author
 Eli Awtrey posted on Friday, December 29, 2017 - 9:35 pm
When doing a multilevel cross-level interaction between two latent variables (one between and one within), what is the conceptual difference between (a) using XWITH to interact the two latent variables and (b) generating a random slope at the within level and then regressing it on the between variable?

Background: I'm conducting a multilevel moderation analysis in which all observed variables are measured at the within level, but I want to interact the latent between and within variables of an IV. In other words, my model would look roughly like this:

y = xb + xw + xb*xw

where xb and xw are latent variables defined using BY at the corresponding level. Following the lead of Preacher et al 2016 (example 20 from the appendix on Preacher's website), I initially tried modeling this with a random slope (s1 | y ON xw), and then regressing that on (s1 ON xb). These models never converged well. However, if I define an interaction at the within level between the latent variables (int | xb XWITH xw), then I see the kind of results I would expect.

What is the difference between these two approaches (XWITH vs random slope)?
 Bengt O. Muthen posted on Saturday, December 30, 2017 - 4:24 pm
I don't see how you can have int | xb XWITH xw on Within if xb is defined on Between.

Typically, the difference between using interactions and random slopes is the residual of the random slope.

You may want to send relevant output to Support along with your license number.
 Eli Awtrey posted on Tuesday, January 02, 2018 - 10:42 am
I have similar questions, particularly since I do see that syntax used in other parts of the aforementioned Preacher et al. appendix (e.g. example 2). I will follow up with him for clarification.

I was more interested generally in how the two different approaches were handled internally by MPlus so I could understand the assumptions I would make with each one. To be clear, the models do "work" technically with the XWTIH approach, but I wondered if I am making a mistake with that usage.
 Bengt O. Muthen posted on Tuesday, January 02, 2018 - 1:55 pm
XWITH is used for interactions between two variables that are measured on the same level. The random slope creates an interaction between a variable on a lower level with that of a variable on a higher level.