Random vs. fixed effects - LMM

[G David Garson]

In a message today Carol expressed confusion in giving an effect size interpretation to variables in a LMM model.

As my Statnotes website was quoted, I would like to add that the bottom line is that effect size interpretations in LMM are in debate, though most follow Raudenbush and Bryk.

Here is the most salient paragraph of mine on this confusion (drawn from G. David Garson, ed., "Hierarchical Linear Models: Guide and Applications," Sage, April 2012):     
"The concepts of "effect size" and "percent of variance explained" are problematic in linear mixed modeling, so much so that some authors avoid reporting effect size measures.  In multilevel modeling, there is unexplained variance at multiple levels, not just level 1. Variance explained at lower levels is dependent on higher levels. In fact, residual variance could be attributed to any level, including any that were omitted from the model.  In random coefficients models, where lower level slopes are modeled by higher level random effects, the random slope reflects one regression per group and "explained variance" is not uniquely defined but rather is a mean, subject to the problems of means as measures of central tendency.  Adding certain variables may affect the variance of the level 1 intercept, and therefore the percent of variance accounted for  when slopes and intercepts are correlated  (group mean centering may eliminate the correlation, but this also changes the meaning of the variable in question as noted elsewhere in this volume) . Add to this the fact that in some linear mixed models, random effects may be allowed to exhibit covariance and groups may overlap as well as be nested,  the problem of assessing effect size can become even more complex.  Nonetheless, some of the most common approaches are summarized below."