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)?
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.
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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.
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.