preview

The Differences Between The And The Method Of The Covariance Matrices Of Random Effects, Nobre And Singer

Better Essays

4.5 EBLUP
In our model (1.1), Z_i b ̂_i reflects the difference between the predicted responses in the i-th subjects and the population average. Thus, b ̂ 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…

θ_j ) V^(-1) (y-Xβ)
+(y-Xβ)^T V^(-1) (∂^2 V)/(∂θ_k ∂θ_j ) V^(-1) (y-Xβ)
-(y-Xβ)^T V^(-1) ∂V/(∂θ_j ) V^(-1) ∂V/(∂θ_k ) V^(-1) (y-Xβ)
+tr(V^(-1) ∂V/(∂θ_k ) V^(-1) ∂V/(∂θ_j ))-tr(V^(-1) (∂^2 V)/(∂θ_k ∂θ_j ))} = 1/2{-(y-Xβ)^T V^(-1) ∂V/(∂θ_k ) V^(-1) ∂V/(∂θ_j ) V^(-1) (y-Xβ)
-(y-Xβ)^T V^(-1) ∂V/(∂θ_j ) V^(-1) ∂V/(∂θ_k ) V^(-1) (y-Xβ)
+tr(V^(-1) ∂V/(∂θ_k ) V^(-1) ∂V/(∂θ_j ))} as (∂^2 V)/(∂θ_k ∂θ_j ) = 0 k, j = 1, …, q ∂V/(∂θ_i ) = [■(ZWGZ^T&θ_i=〖σ^2〗_subject@I&θ_i=σ^2 )]
The next step is to find the second derivative of l(θ|W) with respect to w and θ evaluated at evaluated at θ = θ ̂ and w = w0: (∂^2 l(θ|w))/(∂w_j ∂θ_i ) = 1/2{-(y-Xβ)^T 〖V_w〗^(-1) (∂V_w)/(∂w_j ) 〖V_w〗^(-1) (∂V_w)/(∂θ_i ) V^(-1) (y-Xβ)
+(y-Xβ)^T 〖V_w〗^(-1) (∂V_w)/(∂w_j ∂θ_i ) 〖V_w〗^(-1) (y-Xβ)
-(y-Xβ)^T 〖V_w〗^(-1) (∂V_w)/(∂θ_i ) 〖V_w〗^(-1) (∂V_w)/(∂w_j ) 〖V_w〗^(-1) (y-Xβ)
+tr(〖V_w〗^(-1) (∂V_w)/(∂w_j ) 〖V_w〗^(-1) (∂V_w)/(∂θ_i ))-tr(〖V_w〗^(-1) (∂V_w)/(∂w_j ∂θ_i ))} evaluated at θ = θ ̂ and w = w0 = I, i = 1, …, q, j = 1, …, q (∂V_w)/(∂θ_i ) = [■(ZWGZ^T&θ_i=〖σ^2〗_subject@I&θ_i=σ^2 )] (∂V_w)/(∂w_j ∂θ_i ) = ∂/(∂w_j ) [■(ZWGZ^T&〖σ^2〗_subject@I&σ^2 )] = [■(Z ∂W/∂w GZ^T&0@0&0)] (∂V_w)/(∂w_j ) = [■(0&…&0@⋮&1_(j,j)&⋮@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 ∂θ_i )*(∂^2 l(θ))/(∂θ_k ∂θ_j )*(∂V_w)/(∂w_j ∂θ_i ) |dmax| = The largest absolute eigenvalue of F ̈_i for i-th subject
Plotting |dmax| verse

Get Access