Cross-Level-Interaction without varying slope in Multilevel [duplicate]

Question

Is it useful to include cross-level-interaction in multilevel (hierarchical) models without varying slopes (only varying intercept)?

A short example in R:

m1 <- lme(y~x_1+x_2+z_1+z_2+x_1:z_1, random = ~1|county, data = data)

Where:

y = Response variable

x = covariate from individual-Level

z = covariate from group-Level

Technically these models can be fitted in R without warnings. But is it useful from a theoretical point of view? I once read that cross-level-interactions are only possible with individual-Level variables that possess a varying slope. Is that correct?

Answer 1

I have the same question. The “similar question” suggested by dmartin provides some insights, but I am missing some references. To summarize: (1) Jake Westfall clearly suggests that it is not required to allow the slope of the Level-1 variable to vary in a cross-level interaction (Level-1 variable x Level-2 variable). However, he does not provide any citation for this claim. I was wondering if anyone knows about research that employs cross-level interactions without specifying a random slope for the Level-1 variable? (2) In the “similar question”, Bento suggests that the paper by LaHuis & Ferguson (2009) “The accuracy of significance tests for slope variance components in multilevel random coefficient models” [2009, Organizational Research Methods, 12(3), 418-435] provides some insights. I just read this paper, and in this simulation study, the authors demonstrate that it is possible to have a significant cross-level interaction even when the random-slope parameter for the Level-1 variable is not significant. Although, this could be used as a justification, there is clearly a difference between a insignificant random slope and not including a random slope term at all in a cross-level interaction model.

Answer 2

See a similar question here. While the model can be estimated, it doesn't make sense to have a cross-level interaction without a random slope, as the slope variability is what cross-level interaction is trying to predict.