Interpreting cross-level interactions in mixed

[Samuli Koponen]

Hi, all!

I'm running mixed models in Stata 15 in order to find out if level-1 variable is mediating the effect of level-2 variable to life satisfaction. The models are as follows:

MODEL 1: mixed satisfaction level2variable || country:, mle
MODEL 2: mixed satisfaction level2variable level1variable || country: level1variable, mle cov(uns)
MODEL 3: mixed satisfaction c.level2variable##c.level1variable || country: level1variable, mle cov(uns)

I'm primarily interested in changes at level2variable's estimate after adding level-1 variables and cross-level interaction term. There's two problems I'm struggling with:

1) The standardized beta coefficients for level-2 variable are 0.149 (model 1), 0.077 (model 2) and 0.139 (model 3). So the effect of level-2 variable decreases about 50% when the level-1 variable is accounted for, which I'm interpreting as mediating effect. But the value returns almost to original value when the cross-level interaction is added implicating there's no mediating effect. How can this be?
2) Is it even appropriate to include id: level1variable when there's already interaction term in the model? If so, how can it be interpreted?

[Clyde Schechter]

The standardized beta coefficients for level-2 variable are 0.149 (model 1), 0.077 (model 2) and 0.139 (model 3). So the effect of level-2 variable decreases about 50% when the level-1 variable is accounted for, which I'm interpreting as mediating effect. But the value returns almost to original value when the cross-level interaction is added implicating there's no mediating effect.

It is impossible to provide specific advice here because you show only the commands, but not the output. But in general terms, when you add an interaction, the meaning of the coefficient for level2variable changes. In the model without the interaction, it is a unified, overall effect of level2variable on outcome satisfaction. When you add the interaction, there is no longer any term in the model that represents a unified overall effect of level2variable. In fact, by using an interaction, you are proclaiming that no such unified overall effect even exists. The coefficient of level2variable in the model containing an interaction is a different beast altogether: it is the marginal effect of level2variable on satisfiaction conditional on level1variable = 0. Since the coefficients have completely different meanings in the two models, there is no reason to expect them to be the same or even close, and there is no reason to expect them to be different in any particular way. And, in particular, there is no basis for interpreting the difference (or lack thereof) of these coefficients across models as saying anything one way or another about mediation.

Is it even appropriate to include id: level1variable when there's already interaction term in the model? If so, how can it be interpreted?

Yes, it's perfectly legitimate. The inclusion of -id: level1variable- serves to turn the slope of the satisfaction:level1variable relationship from a fixed constant into a random slope, with a different value for each id:. This remains true even when there is an interaction term among the fixed effects. The interaction's interpretation becomes a bit more complicated, but basically the way yours is set up, the interaction term's effect is applied equally to each id's particular (random) slope. In other words, there is no fixed slope of the satisfaction:level1variable relationship. Instead, the slope varies across id's, and it is also a linear function of the value of level2variable in that observation.

That said, whether it is appropriate for your particular goals is hard to say. As you have not described what these variables are, nor set out your expectations for how they relate to each other, it is anybody's guess whether this is a reasonable specification of a plausible data-generating process or not. It's not a question of syntax, or even of statistics. It's a substantive question in your discipline: what is a reasonable model for this process?