Forgive me if this question is too general. I have used HLM to determine the siginificance of a cross-level interaction in a multilevel model. However, I would like to compare the fit of the overall path model, to a second nested model. I have decided to use SEM and Mplus to do this. Yet, it is not clear to me if I can generate a cross-level interaction with a multilevel SEM. I haven't seen this in the literature either. The models and literature by both Muthens has been very helpful, but I hadn't seen anything about this possibility. You are innovators with regard to multilevel SEMs, so I am sure you can offer some information.
I just wanted to confirm that the Verena Hahn's post on Tuesday, August 7th, 2012 is still what you would suggest. I have a multilevel model where repeated measures are taken at two time points within persons. I'm only interested in the fixed effect of the within person slope but to also allow a random intercept. I also expect that the fixed slope will be moderated by a level 2 variable. In other words, I'm only interested in examining whether the average within person effect changes as a function of a level 2 variable. As such, I'm interested in a cross-level interaction where the level 1 slope is not randomly varying. Would you still recommend implementing this in Mplus as:
MODEL: %WITHIN%; beta1 | y ON x; %BETWEEN%; y on w; beta1 ON w; beta1 @0;
That seems like a correct setup. I wouldn't say that the slope isn't varying because it does vary as a function of w. I also don't see the need for fixing the residual variance at zero for the slope on between - that says w totally determines the variation in the slope.
Thank you Dr. Muthen. I see what mean about the slope varying across w. If I omit the residual constraint (i.e., beta1 @0) won't that estimate the random effect for the level 1 slope? I was under the impression that with only two level 1 observations a model with both a random intercept and slope would be underidentified and that the residual constraint is necessary for model identification.
I have a quick question. When we are testing cross-level interactions, do we test the main effects between the two level 1 variables in the hypothesized interactive model?
For example: Within s1 | Y ON X; YW ON X;
Between s1 WITH YB; YB s1 ON Z1 Z2;
In the above model, the effect of YW ON X is negative and non-significant. Whereas in the main effects model, the effect was positive and significant. I saw some papers report similar effects but argued that lower level effects should not be interpreted in the presence of higher level effects. Is that the way to go? And what the reasoning behind such an effect? Thank you.