Why is $R^2$ so difficult to calculate for mixed models (both for the model as a whole and the fixed effects)? I have been using a package to calculate $R^2$ values for mixed models. The documentation for the package has the following quote from Harry Singmann:

"The fact that calculating a global measure of model fit (such as $R^2$)
  is already riddled with complications and that no simple single number
  can be found, should be a hint that doing so for a subset of the model
  parameters (i.e., main-effects or interactions) is even more
  difficult. Given this, I would not recommend trying finding a measure
  of standardized effect sizes for mixed models."

Why exactly is this the case?
More over I am interested in why finding $R^2$ for each individual predictor is problematic?
Edit: I'm also curious as to whether the squared semi-partial correlation for each predictor is also problematic.
 A: Apart from the problems Doug Bates mentions in his statement about $R^2$ in mixed models, I would say that you could obtain "standardized effect sizes" for mixed models by standardizing the predictors you include in the model.
There are some discussions on the disadvantages of doing that, and especially in the manner you do the standardization. Namely, if you are going to standardize based on the mean and standard deviation the predictors you have in your sample (and you do not report these values), then you have the problem that your results will not be easily generalizable to other situations/samples. Instead, you could standardize the predictors using a mean value and a standard deviation that come from a widely accepted "normal" range for the predictor. To give an example, if one of the predictors is systolic blood pressure, then you know that the normal range is between 90 and 120 mm Hg. Hence, you could standardize the blood pressure values in your sample by subtracting 105 mm Hg and dividing with 5 mm Hg.
A: Have you read Douglas Bates' explanation of the "mixed model R2" conundrum? It's posted here, and gives a good overview of the issues. 
