Rick DeShon writes,

As I read through your discussion paper on the analysis of variance published in the Annals of Statistics in 2005, I became a bit confused about the connections between your notion of parameter batches and prior work on the topic of fixed and random effects. Specifically, I wonder how your approach connects to Nelder’s “great mixed model muddle?”

As I’m sure you are aware, their are two distinct parameterizations used to estimate variance components (e.g., Hartley & Searle, 1969) that result in different variance component estimates when the model contains fixed effects. In Section 4 of your paper, you connect your notion of superpopulations and finite-population variances with the EMS estimates commonly associated with the constrained parameterization for mixed models. However, if I’m not mistaken, you emphasize that the models are best analyzed using Bayesian methods. To the best of my knowledge, all current Bayesian approaches are based on the unconstrained parameterization of the mixed model and would yield different results when compared to the balanced, EMS estimates.

So, I’m wondering if your conceptualization provides a resolution to the “muddle” or at least a justification for preferring one parameterization to the other. Any insight you could provide on the connection between the approach you suggested and the connection to the mixed model muddle would be greatly appreciated.

My reply: I’m not quite sure of the details here, but my thinking in this area has been much influenced by Nelder’s work (and, before that, the writings of Yates). My take on it is that the model can be formulated without reference to what the model will be used for. But different uses lead to different summaries (for example, finite-population or superpopulation variances). I really hate what Searle et al. and others have written about when to use so-called fixed or random effects. So my intuition here is that at least one of the estimates you refer to above does not make sense. I’d have to see a particular example to say more than that.

I've never got anything out of anything Searle has written (except confusion, of course). Some people condition their random effects to always sum to zero. You can do that , if you want to, but why is beyond me. But if you want to do that, then Searle, maybe, can tell you how. maybe?. But why bother? If anybody can give a practical example where it makes practical, applied , sense to condition the random effects to sum to zero, then I'm delighted.

i have been disturbed by the magnitude of differences in variance component estimates yielded by anova v. mixed model. I have tried everything trying to get the estimates to match up (e.g., suppressing the fixed constant in the mixed model), but i often get incongruous results regardless. i find this to be the case even in very simple variance decompositions, such as partitioning a single variable's variance into residual and cluster components (for instance, to calculate ICC). why is this? and which method is better?