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Receiver operating characteristic (ROC) curves have been widely used in evaluation of the goodness of the diagnostic method in many study fields, such as disease diagnosis in medicine. The area under the ROC curve (AUC) naturally became one of the most used variables in gauging the goodness of the diagnosis (Mossman, Somoza 1991). Since medical diagnosis often is not dichotomous, the ROC curve and AUC need to be generalized to a multi-dimensional case. The generalization of AUC to multi-class case has been studied by many researchers in the past decade. Most recently, Nakas & Yiannoutsos (2004) considered the ordered d classes ROC analysis by only considering the sensitivities of each class. Hence, their dimension is only d. Cha (2005) considered more types of mis-classification in the ordered multiple-class case, but reduced the dimension of Ferri, at.el. from d(d-1) to 2(d-1). In this dissertation we are trying to adjust and calculate the VUS for an ordered multipleclass with Cha’s 2(d-1)-dimension method. Our methodology of finding the VUS is introduced. We present the method of adjusting and calculating VUS and their statistical inferences for the 2(d-1)-dimension. Some simulation results are included and a real example will be presented. Intellectual outcomes in pediatric brain-tumor patients were investigated in a prospective longitudinal study. The Standard-Binet Intelligence Scale-Fourth Edition (SB-IV) Standard Age Score (SAS) and Composite intelligence quotient (IQ) score are examined as cognitive outcomes in pediatric brain-tumor patients. Treatment factors, patient factors and time since diagnosis are taken into account as the risk factors. Hierarchical linear/quadratic models and Gompertz based hierarchical nonlinear growth models were applied to build linear and nonlinear longitudinal curves. We use PRESS and Volume Under the Surface (VUS) as the criterions to compare these two methods. Some model interpretations are presented in this dissertation.