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
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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
are repeated across several sets of stimuli (Beeson, & Robey, 2006). Additionally in a change in criterion design delayed or withdrawal of treatment is not required, as is the case for the multiple-baseline design or withdrawal designs (McDougall, Hawkins, Brady, & Jenkins, 2006). Although Baseline collection is continued throughout the series of the ABAB designs the intention is that the targeted behavior will either be increase of decreased from the original baseline in the direction of the experiment
3. Design an algorithm in pseudocode to solve the problem. Make sure to include steps to get each input and to report each output.
5. When all subjects receive all levels of the independent variable, this is referred to as (J) Within Subject design.
Iterations of analysis eliminated data points that were listed as “unusual observations,” or any data point with a large standardized residual. After 5 iterations, the analysis showed improved residual plots. Randomness in the versus fits and versus order plots means that the linear regression model is appropriate for the data; a straight line in the normal probability plot illustrates the linearity of the data, and a bell shaped curve in the histogram illustrates the normality of the data.
by the Trapezoidal rule is Etrap = O(h2 ) and by Simpson’s rule is ESimp = O(h4 ). i) For each of these numerical integration rules, what conditions are required on the integrand f so these error estimates are valid? ii) Suppose that the error using h = 5 × 10−3 is E0 = 1.19 × 10−4 when using either the Trapezoidal rule or Simpson’s rule. For both rules, ˆ estimate the error if an interval width of h = 1 × 10−3 is used. iii) The Matlab command [z, w] = gauleg(N); calculates the N Gauss-Legendre nodes z and weights w for the interval [−1, 1]. Show how z and w can be used to numerically calculate
6. Twiddle factor is a.W_N=e^(j2π⁄N) b.W_N=e^((-2π)⁄N) c.W_N=e^(2π⁄N) d.〖W_N=e〗^((-j2π)⁄N) 7. DFT of x (n) = {1, 0, 1, 0}.
Independence model | .352 | .465 | .313 | .362 | Baseline Comparisons Model | NFI Delta1 | RFI rho1 | IFI Delta2 | TLI rho2 | CFI | Saturated model | 1.000 | | 1.000 | | 1.000 | Independence model | .000 | .000 | .000 | .000 | .000 | Parsimony-Adjusted Measures
In this equation, “Y represents the predicted value of each outcome variable for each individual (i) in the classroom (j); Female, Race, and ESL represent a series of dummy variables for the demographic control variables included in the analysis; and e represents random error” (Aseltine & DeMartino, 2004, p. 448).
The NeuroSolutions Infinity, a product of Neurodimensions Inc. of Florida (2005), the SPSS software package (SPSS 17.0), IBM SPSS Modeler (IBM SPSS 21.0) and the STATA (STATA 12.0) (http://www.stata.com), the most widely usable software were utilized to develop non–parametric SVM
equation to loop 1 of Fig. 2 and current balance equation to node 1, the state space equations (1) and (2)
u and v are found by solving the equation shown in (3.6). Ix and Iy are the hori-
similar test settings available in the literature [Daniels et al., 1995,Bano et al., 2009,Capozzoli et al., 2012].
In contrast to the other flexible NB model, QR models are more robust to outliers and scale effects since the estimation minimizes least absolute deviation. In addition, NB models, which focus on the conditional mean, may not be the best approximator for a skewed-location distribution. When scale and location effects are a concern, such as this analysis, then researchers should choose the more flexible QR
The absolute value of intercept (alpha) in the Three-Factor Model for Low Portfolio is 1.1159, 0.1347 for 5 Portfolio, 0.6354 for High Portfolio, and 1.4591 for High-Minus-Low Portfolio. They all are larger than those in the Four-Factor Model, given 0.2081 for Low Portfolio, 0.0961 for 5 Portfolio, 0.0412 for High Portfolio, and 0.0402 for High-Minus-Low Portfolio. Therefore, alpha, a measurement of deviation from the model, is much smaller in the Four-Factor Model than in the Three-Factor Model. It shows that the Four-Factor Model fits the real situation better. In addition, we could see that R-Square (percentage of data that can be explained by the model) for the
With our best subset method, we can leverage our lowest BIC metric to select the best model. We can plot out best subset method and pinpoint the number of variables to select. The plot below showcases that the lowest point or value of BIC, contains 6 variables. We leverage the BIC metric because it places a penalty on models with more or many variables. Meaning, the mode variable a model has, the bigger the penalty. We can review the coefficients, standard errors, t-value, and the p-values for the best subset method with the significant six variables. Our MSE metric for the 6-variable best subset model is 3090.483, a slight decrease from our linear regression model.