Interactions and Bayesian Anova

Gregor Gorjanc writes:

I am working on a old problem in the area of animal breeding and genetics. You probably remember the pioneering work of late Henderson on mixed models used for genetic evaluation of dairy bulls. In essence his model was:

y ~ Normal(mu, sigma^2_e)

mu = alpha + F1 + F2 + … + HYS + a

a ~ Normal(0, A * sigma^2_a)

where the only difference from a “pure mixed model” is the matrix A, which is constructed based on the pedigree information, while F1, F2, …, and HYS are some factors.

I would like to ask you something about the HYS effect. This is basically a triple interaction between effect of herd (H), year (Y) and season (S). This effect is very strong as you can imagine – differences between years, seasons (temparature, price of feed, …) as well as different herds (management, knowledge, …). If we fit this effect as “fixed” or in a Bayesian sense we do not estimate the variance in the prior but we assign some large value to it, i.e., HYS ~ Normal(0, “some large value”), then this equivalent to fitting all main effects and all possible interactions, i.e., H + Y + S + HY + HS + YS + HYS.

All fine up to now. In some countries (like in mine) herds are quite small and there is not much information to precisely estimate the effect of all the levels for HYS or even HY and HS. The usual approach is to fit HYS as a “random” effect and estimate the variance between levels to perform some shrinkage for levels with smaller number of records. Usually some other components of H + Y + S + HY + HS + YS are not fitted. I am now wondering if you can point me to any relevant source about such issues? I am particularly interested which “HYS” components could be largely “absorbed” in the HYS term in it is fitted as “random”.

My reply: It sounds like you should treat each batch of variables–H, Y, S, HY, HS, YS, and HYS–as having its own variance component, with partial pooling within each batch. This is the Bayesian analogue to Anova, and this idea is discussed (with examples) in the Anova chapter of my book with Jennifer and also in my Annals of Statistics article on Anova. For computational reasons it can sometimes be helpful to include all factors except for HYS and then esitmate these (Bayesianly) using a resitual analysis. We did this with some of three-way models of public opinion that I’ve been posting recently on the blog (those two-way grids of maps).