# ranef - Random Eects Model Notes Andrew Womack 1 Basic...

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Random Effects Model Notes Andrew Womack November 21, 2011 1 Basic Model We have the following hierarchical model y ij | μ j , μ, σ 2 , τ 2 ∼ N ( μ j , σ 2 ) μ j | μ, σ 2 , τ 2 ∼ N ( μ, τ 2 ) where j = 1 , . . . , J is our group index and i = 1 , . . . , n j are the individuals in group j . This can also be viewed as putting a positive correlation between indi- viduals in the same group y · j | μ, σ 2 , τ 2 ∼ N n j ( μ 1 n j , Σ j ) where Σ j = σ 2 I n j + τ 2 B n j where I n j is the identity matrix and B n j is a matrix of ones. This gives a variance which is σ 2 + τ 2 for a unit and correlation ρ = τ 2 τ 2 + σ 2 for units in the same group. Units in different groups have a different zero autocorrelation. Before discussing priors for μ, τ 2 , σ 2 (or μ, ρ, σ 2 ), we should consider the likelihood and the hierarchical structure for possible conjugate prior struc- tures. First, consider the likelihood itself. It is f ( y | μ, σ 2 , ρ ) = J Y j =1 " 1 - ρ 2 πσ 2 n j 2 (1 + n j ρ ) - 1 2 exp - 1 - ρ 2 σ 2 ( y · j - μ 1 n j ) 0 I n j - ρ ( n j - 1) ρ - 1 B n j ( y · j - μ 1 n j ) 1

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From this likelihood, it is clear that prior for μ and σ 2 are fairly easy to consider, for example it is easy enough to use a normal prior for μ and a gamma prior for σ 2 . What remains to be considered is the prior for ρ | μ, σ 2 .
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