lec3 - Again returning to the GLMM with random intercept an...

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Unformatted text preview: Again returning to the GLMM with random intercept, an importance sampler can be constructed by using φ ( u i ; μ i , τ 2 i ), the normal density approximation to the posterior density f ( u i | y i ). That is, the representation from p.196 of the integral we’re after, E { f ( y i | b i ) } = Z + ∞-∞ φ ( u i ; μ i , τ 2 i ) ( φ ( u i ) Q j f ( y ij | √ ψu i ) φ ( u i ; μ i , τ 2 i ) ) du i suggests drawing a sample ( u * 1 , . . . , u * R ) from φ ( u i ; μ i , τ 2 i ) and using the approximation E { f ( y i | b i ) } ≈ 1 R R X r ( φ ( u * r ) Q j f ( y ij | √ ψu * r ) φ ( u * r ; μ i , τ 2 i ) ) • This method is akin to AGQ, which can be viewed as a determinis- tic version of importance sampling. Both methods require an extra computation of μ i , τ 2 i , the mean and variance of the normal approx- imation to the posterior distribution. • Importance sampling is implemented in PROC NLMIXED. 201 3. Analytic Approximation of the Likelihood: • Most of the methods that fall into this category are based upon taking a Laplace approximation to the integral involved in the GLMM loglikelihood. For a unidimensional integral, the Laplace approximation can be written as Z + ∞-∞ exp { f ( x ) } dx ≈ Z + ∞-∞ exp { f (˜ x )- ( x- ˜ x ) 2 / (2 σ 2 ) } dx = Z + ∞-∞ exp { f (˜ x ) } √ 2 πσφ ( x ; ˜ x, σ 2 ) dx = exp { f (˜ x ) } √ 2 πσ, where φ ( x ; ˜ x, σ 2 ) is a normal density with mean ˜ x and variance σ 2 , ˜ x is the model of f ( x ) and hence of exp { f ( x ) } , and σ 2 =- ∂ 2 f ( x ) ∂x 2 fl fl fl fl x =˜ x ¶- 1 . • The approximation in the first line here is obtained by approximating f ( x ) by a second order Taylor expansion around its mode. You’ll notice that the first derivative term f (˜ x ) drops out here because it is evaluated at the mode, at which point f ( x ) = 0. 202 For the GLMM with random intercept, the integrand in the likeli- hood contribution for the i th cluster (which plays the role of exp { f ( x ) } ) is φ ( b i ; 0 , ψ ) Y j f ( y ij | b i ) = exp log φ ( b i ; 0 , ψ ) Y j f ( y ij | b i ) • As mentioned before, this quantity is proportional to the posterior distribution f ( b i | y i ). Therefore, in the Laplace approximation, the quantity playing the role of ˜ x is the posterior mode ˜ b i = arg max b i φ ( b i ; 0 , ψ ) Y j f ( y ij | b i ) and the curvature (inverse negative Hessian) of the integrand, σ 2 i , plays the role of σ 2 . Thus, the Laplace approximation to the loglikelihood contribution from the i th subject becomes log f ( y i ; β , ψ ) ≈ log( √ 2 πσ i ) + log { φ ( ˜ b i ; 0 , ψ ) } + X j log f ( y ij | ˜ b i ) = log( σ i / p ψ )- ˜ b 2 i / (2 ψ ) + X j log f ( y ij | ˜ b i ) . ( ** ) • This approximation is good whenever the posterior density of b i is approximately normal. This occurs for large cluster sizes....
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lec3 - Again returning to the GLMM with random intercept an...

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