The Differences Between The And The Method Of The...

4.5 EBLUP In our model (1.1), Z_i b Ì‚_i reflects the difference between the predicted responses in the i-th subjects and the population average. Thus, b Ì‚ verse subject indices can be used for identifying the outlying subjects. To assess the sensitiveness of subjects to the homogeneity of the covariance matrices of the random effects, Nobre and Singer develop the method of influence methods from Cook (1986). The idea is to put some weights to the var(b), i.e. var(b) = WG and then calculate |dmax|, which is the normalized eigenvector associated with the direction of largest normal curvature of the influence graph under a perturbation of the covariance matrix of the random effects (for detail, see appendix or Cook (1986)). First, we need†¦show more content†¦ÃŽ ¸_j ) V^(-1) (y-XÃŽ ²) +(y-XÃŽ ²)^T V^(-1) (∂^2 V)/(∂Î ¸_k ∂Î ¸_j ) V^(-1) (y-XÃŽ ²) -(y-XÃŽ ²)^T V^(-1) ∂V/(∂Î ¸_j ) V^(-1) ∂V/(∂Î ¸_k ) V^(-1) (y-XÃŽ ²) +tr(V^(-1) ∂V/(∂Î ¸_k ) V^(-1) ∂V/(∂Î ¸_j ))-tr(V^(-1) (∂^2 V)/(∂Î ¸_k ∂Î ¸_j ))} = 1/2{-(y-XÃŽ ²)^T V^(-1) ∂V/(∂Î ¸_k ) V^(-1) ∂V/(∂Î ¸_j ) V^(-1) (y-XÃŽ ²) -(y-XÃŽ ²)^T V^(-1) ∂V/(∂Î ¸_j ) V^(-1) ∂V/(∂Î ¸_k ) V^(-1) (y-XÃŽ ²) +tr(V^(-1) ∂V/(∂Î ¸_k ) V^(-1) ∂V/(∂Î ¸_j ))} as (∂^2 V)/(∂Î ¸_k ∂Î ¸_j ) = 0 k, j = 1, †¦, q ∂V/(∂Î ¸_i ) = [ââ€"  (ZWGZ^TÃŽ ¸_i=ã€â€"ÏÆ'^2ã€â€"_subject@IÃŽ ¸_i=ÏÆ'^2 )] The next step is to find the second derivative of l(ÃŽ ¸|W) with respect to w and ÃŽ ¸ evaluated at evaluated at ÃŽ ¸ = ÃŽ ¸ Ì‚ and w = w0: (∂^2 l(ÃŽ ¸|w))/(∂w_j ∂Î ¸_i ) = 1/2{-(y-XÃŽ ²)^T ã€â€"V_wã€â€"^(-1) (∂V_w)/(∂w_j ) ã€â€"V_wã€â€"^(-1) (∂V_w)/(∂Î ¸_i ) V^(-1) (y-XÃŽ ²) +(y-XÃŽ ²)^T ã€â€"V_wã€â€"^(-1) (∂V_w)/(∂w_j ∂Î ¸_i ) ã€â€"V_wã€â€"^(-1) (y-XÃŽ ²) -(y-XÃŽ ²)^T ã€â€"V_wã€â€"^(-1) (∂V_w)/(∂Î ¸_i ) ã€â€"V_wã€â€"^(-1) (∂V_w)/(∂w_j ) ã€â€"V_wã€â€"^(-1) (y-XÃŽ ²) +tr(ã€â€"V_wã€â€"^(-1) (∂V_w)/(∂w_j ) ã€â€"V_wã€â€"^(-1) (∂V_w)/(∂Î ¸_i ))-tr(ã€â€"V_wã€â€"^(-1) (∂V_w)/(∂w_j ∂Î ¸_i ))} evaluated at ÃŽ ¸ = ÃŽ ¸ Ì‚ and w = w0 = I, i = 1, †¦, q, j = 1, †¦, q (∂V_w)/(∂Î ¸_i ) = [ââ€"  (ZWGZ^TÃŽ ¸_i=ã€â€"ÏÆ'^2ã€â€"_subject@IÃŽ ¸_i=ÏÆ'^2 )] (∂V_w)/(∂w_j ∂Î ¸_i ) = ∂/(∂w_j ) [ââ€"  (ZWGZ^Tã€â€"ÏÆ'^2ã€â€"_subject@IÏÆ'^2 )] = [ââ€"  (Z ∂W/∂w GZ^T0@00)] (∂V_w)/(∂w_j ) = [ââ€"  (0†¦0@â‹ ®1_(j,j)â‹ ®@0†¦0)] Finally, we can calculate F ̈_i and get the largest absolute eigenvalue, |dmax| for every subject i: F ̈_i = (∂V_w)/(∂w_j ∂Î ¸_i )*(∂^2 l(ÃŽ ¸))/(∂Î ¸_k ∂Î ¸_j )*(∂V_w)/(∂w_j ∂Î ¸_i ) |dmax| = The largest absolute eigenvalue of F ̈_i for i-th subject Plotting |dmax| verse