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Unformatted text preview: Again returning to the GLMM with random intercept, an importance sampler can be constructed by using ( u i ; i , 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 were after, E { f ( y i  b i ) } = Z +  ( u i ; i , 2 i ) ( ( u i ) Q j f ( y ij  u i ) ( u i ; i , 2 i ) ) du i suggests drawing a sample ( u * 1 , . . . , u * R ) from ( u i ; i , 2 i ) and using the approximation E { f ( y i  b i ) } 1 R R X r ( ( u * r ) Q j f ( y ij  u * r ) ( u * r ; i , 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 i , 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 2 ) } dx = Z +  exp { f ( x ) } 2 ( x ; x, 2 ) dx = exp { f ( x ) } 2 , where ( x ; x, 2 ) is a normal density with mean x and variance 2 , x is the model of f ( x ) and hence of exp { f ( x ) } , and 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. Youll 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 ( b i ; 0 , ) Y j f ( y ij  b i ) = exp log ( b i ; 0 , ) 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 ( b i ; 0 , ) Y j f ( y ij  b i ) and the curvature (inverse negative Hessian) of the integrand, 2 i , plays the role of 2 . Thus, the Laplace approximation to the loglikelihood contribution from the i th subject becomes log f ( y i ; , ) log( 2 i ) + log { ( b i ; 0 , ) } + X j log f ( y ij  b i ) = log( i / p ) b 2 i / (2 ) + 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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This note was uploaded on 11/13/2011 for the course STAT 8630 taught by Professor Staff during the Fall '08 term at University of Georgia Athens.
 Fall '08
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