Causation and Multilevel Modeling
Message/Author
 Wesley Anderson posted on Tuesday, March 17, 2015 - 10:30 am
Say you have two variables X and Xbar, the former measured on persons and the latter measured on groups of persons. And another response variable measured on persons Y. When you use a random-intercept model you assume Xbar causes Y through Ybar. When you use a random-slope model you assume Xbar causes Y through the partial regression coefficient of Y on X. Is there no way that Xbar can be modeled to directly cause Y? I appreciate it.
Best,
 Bengt O. Muthen posted on Tuesday, March 17, 2015 - 11:08 am
No, because the variables on the 2 levels are taken to be uncorrelated, just like in regular random effects anova. But this doesn't seem to me a limitation.
 Wesley Anderson posted on Tuesday, March 17, 2015 - 11:18 am
But of course X and Xbar are correlated if Xbar is the mean of X in a group. I am not clear how the correlation or lack thereof (perhaps you mean after statistically control) between these two variables makes it the case that Xbar can or cannot be modeled to cause Y directly. Could you elaborate please? Also, I am not trying to discover a limitation in the methods used in Mplus. I am merely making sure I understand what's going on correctly.
 Bengt O. Muthen posted on Tuesday, March 17, 2015 - 12:11 pm

Is there no way that Xbar can be modeled to directly cause Y?

I would say that when Xbar influences the between-level latent part of Y, where the observed Y is decomposed into 2 uncorrelated parts,

Y = Y_w + Y_B,

this is directly influencing the observed Y.

Or, do you see a need for something different?
 Wesley Anderson posted on Tuesday, March 17, 2015 - 12:28 pm
I wouldn't say this. I would say Xbar-->Ybar-->Y. Consider a case in evolutionary biology known as soft selection. Ybar (group fitness) is fixed. There is frequency dependent selection so Xbar must influence Y (individual fitness) not through Ybar (by definition fixed) but through interaction with Xbar. Here you might say Xbar directly influences Y. For example, Y = f(X,Xbar) where the function includes an additive term aX, an interactive term bXXbar but no additive term cXbar. The latter cannot be included otherwise it looks like E[Ybar] = cXbar and Ybar by hypothesis is fixed. But you might also say Xbar indirectly influences Y through influencing partial regression a of Y on X (as in Multilevel models in Mplus).
 Bengt O. Muthen posted on Tuesday, March 17, 2015 - 3:08 pm
From your last sentence I conclude that one could translate this as Xbar influencing the random slope of Y on X.
 Wesley Anderson posted on Tuesday, March 17, 2015 - 3:11 pm
Correct.