Multilevel modeling (Controlling for a Level 2 variable)

[Oliver]

Hi everyone,

I recently got a comment from a reviewer regarding the multilevel modeling approach that was used in one of my papers recently submitted and I'd like to have your input.

Here's the issue: I've built a model designed to examine the association between a continous Level 1 variable (i.e., mood; 0-10) and a continous Level 1 outcome (i.e., fatigue; 0-10). The Level 1 independent variable (IV) was centered within cluster/person, and the association (i.e., fixed effect) between the IV and the DV was significant. After running this analysis, I wondered whether this Level 1 association would remain significant even after controlling for a Level 2 variable (i.e., gender). I thus subsequently built the same model, but included gender (Level 2; Men=0; Women=1) as a fixed effect with the other Level 1 independent variable (mood). The Level 1 association did remain significant despite minor changes in some model parameters for the Level 1 fixed effect (e.g., coefficient, Std. Error).

One of the reviewers essentially told me that it does not make sense to test the association between Level 1 variables while controlling for a Level 2 variable. The reviewer's words were as follows: "variance components of the outcome variable on levels 1 and 2 are independent by definition, such that adding covariates on level 2 does not control for any relationships on level 1 (in other words, "pure" level 2 predictors explain only level 2 variance, but they have no "impact" on level 1 variance)".

Given the minor minor changes that were observed in the beta value and Std.Error for the fixed effect, I guess the reviewer might be right. If the reviewer is right, then what would be the best modeling approach in order to completely "rule out" the influence of a Level 2 variable on a Level 1 association ? Perhaps modeling the interaction between the Level 1 and Level 2 variables, and showing that this interaction is not significant ?  

Thanks in advance for your input.
O.
 

[Andy W]

This is a fairly advanced topic, one that is not necessarily very amenable to a short forum post, but I generally agree with the reviewer. Especially since you de-meaned WITHIN person (which is basically a fixed effects design), including the level 2 covariate is not necessary - you have already controlled for them.

So if we have:

Mood_ij = B1*Fatigue_ij + L*Gender_j             [Eq A]

And then we add all of the micro level observations up to the macro level:

  Mood_1j = B1*Fatigue_1j + L*Gender_j
  Mood_2j = B1*Fatigue_1j + L*Gender_j
  Mood_3j = B1*Fatigue_1j + L*Gender_j
  .
  .
+ Mood_nj = B1*Fatigue_nj + L*Gender_j
------------------------------------------
Sum(Mood_nj) = B1*Sum(Fatigue_nj) + n*L*Gender_j  [Eq B]

Now note that if you divide Sum(Mood_nj) by n you have the mean within group values. So divide each side by n leaves:

Mean(Mood_j) = B1*Mean(Fatigue_j) + L*Gender      [Eq C]

Now if you subtract [Eq C] from [Eq A], you then have your within group de-meaned fixed effects model

Mood_ij - Mean(Mood_j) = B1*[Fatigue_ij - Mean(Fatigue_j)] + L*(Gender_j - Gender_j)

Notice the last term cancels out, so there is no need to include it in the model.

For references Snijders and Bosker's Multilevel Modelling book has a walk through of this fixed effects derivation, as does Angrist and Pischke's Mostly Harmless Econometrics.

Multi-level modelling with the within grouped demeaned variables I would characterize as controversial. See Bell and Jones (2014) "Explaining Fixed Effects: Random Effects modelling of Time-Series Cross-Sectional and Panel Data" as a proponent, but "The Effect of Different Forms of Centering in Hierarchical Linear Models" (Kreft et al., 1995) as a differing view. In general, the demeaning within groups will only make the lower level effects unbiased, but not 2nd level ones. (Also the standard errors should be adjusted.)

For additional references about identifying the second level effects in the model see these two cross validated posts

 - http://stats.stackexchange.com/a/90759/1036
 - http://stats.stackexchange.com/q/100168/1036