Thanks JJ. I'll take up on your suggestion about writing about LMMs and goodness-of-fit measures for LMMs. For comparing goodness-of-fit of two LMMs, my current choice is Likelihood based measures such as AIC (seems like you already use it) or a simple Likelihood Ratio test. The major problem I have with AIC (or Adjusted-R^2) is that it treats the contributions of all explanatory variables equally.

There are R^2 based measures proposed (some of which are contested) to tease apart the relative contribution of the FE and the RE portions of the LMM in explaining the overall variance. I have personally not used any sum of squares based measures for LMMs. But I will use your question as motivation to investigate them and write about them :=)

'best