## 3 Proposed Hierarchy-Based Competency Structure and Its Application in E-Evaluation
### Competency Tree Application in E-Evaluation
The lower the node in the competency tree (the bigger the level number), the smaller score should be assigned for achieving the competency as the sum of lower-level competency scores should not be greater than the score of parent competency. Therefore, we added an aggregation complexity coefficient \(k\), see in Equation (1), which will define the additional complexity (in percents) to integrate all child competencies into one competency, comparing the efforts of demonstrating all child competencies separately. As stated by [PERSON] et al. [35]: \"The brain's capacity rapidly to integrate information from  
Figure 4: Example structure of course/task competency tree. many different sources lies at the root of our cognitive abilities\". Therefore, if no clear rules exist for defining the complexity to integrate child competencies into one parent competency, we proposed to use a proportion between the number of sibling competencies \(n(L)\) in level \(L\) and the expected cognitive abilities \(C\) for certain grade students, see in Equation (1). Therefore, if we have a task with four child competencies \(n(L)\) and it has to be solved by first-grade students, whose cognitive ability \(C=3\), the complexity coefficient of the task \(k(L)\) will be equal to 1. Meanwhile, the same task for the third-grade student, whose cognitive ability is \(C=5\), the task complexity coefficient \(k(L)\) will be equal to 0.6. \[k(L)=\left\{ \begin{array}{cc}\frac{n(L)-1}{C},&n(L)\leq C\\ 1,&n(L)>C\end{array} \right. \tag{1}\]  
where \(k(L)\) is the complexity coefficient to integrate child competencies to one in level \(L\), \(n(L)\) is the number of sibling competencies in level \(L\), and \(C\) is the expected cognitive ability for certain level students. As the competency tree was assumed to be balanced during the design, all child competencies will have the same importance for the parent competency achievement. Therefore, if competency node \(X\) with score \(s\) has \(n\) child competencies, the score of the child competency node \(Y\) will be equal to \(1/n\) of the parent score \(s\) divided by the integrated task complexity coefficient \((k+1)\), see in Equation (2). For example, if the root element's score is 1 and it has four child nodes, the score for the child node will be equal to 0.125 for complexity coefficient \(k(Y)=1\) and 0.156 for complexity coefficient \(k(Y)=0.6\). If the student will solve all four tasks, the sum of these four scores will not be equal the score, where all four tasks are integrated into one because the ability to adopt multiple sub-competencies is more valuable than just knowing the separate competencies without being able to adapt it. \[s(Y)=\frac{s(X)}{n!(1+k)} \tag{2}\]  
where \(Y\) is one of n child competency of parent \(X\), \(k\) is the aggregation complexity coefficient, \(s(X)\) and \(s(Y)\) are the parent and child competency scores, respectively. The coefficients for each node of the competency tree can be calculated from top to down, and the root competency score \(s(root)\) will be equal to 1, as shown in Equation (3). \[s(root)=1 \tag{3}\]  
At the same time, the score of each competency node \(N\) can be calculated from down to top by using the formula:  
\[s(N)=\prod_{i=0}^{L-1}\frac{1}{n(L-i)!(1+k)} \tag{4}\]  
where \(s(N)\) is the score for competency node \(N\) in competency tree level \(L\), while \(n(L)\) is the number of sibling competencies, and \(k\) is the aggregation complexity coefficient. To reduce the influence of evaluator's strictness or personal preferences, each competency should be evaluated in a binary system--achieved or not. This simplifies the mark calculation in e-evaluation systems as computer-based evaluation (especially in programming area) are mostly based on the estimation of the correct or incorrect result too.