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Multilevel models with only one or two groups

Gregor Gorjanc writes,

On page 18 of your paper on analysis of variance you talk about uniform prior on hierarchical standard deviations and note that at least two degrees of freedom are needed to obtain a proper posterior and that in cases with lower df ML is essentially employed. I do not follow this fully. What do you actually do, when you have a batch with df = 1? Do you assign uniform prior and … or you just treat this batch as “fixed”??

Another questions relates to comments about contrast in page 50 (the rejoinder). You write there finite-population contrast and state that super-population contrast is appropriatelly scaled. How is this scaling done?

My reply:

In answer to your first question, for a factor with only one or two groups, multilevel modeling typically can’t pull much out of the data beyond a classical regression model. There’s just not enough information to estimate the population variance for that factor. The two exceptions are: (a) if you have a lot of factors, each with a small number of groups, you can fit a hierarchical model to the group sd’s themselves (as in Section 6 of this paper); and (b) if you’re willing to use prior information to bound or otherwise restrict the variance parameter.

You might need to do this, even though it seems difficult from a statistical perspective, if you need to make inferences or predictions for new groups not in the data.

Regarding your question about page 50: I can’t remember now, but I think I’d rescale by dividing by 2 so the scale goes from -1 to 1.