Attribute Hierarchy Method - Cognitive Model Representation

Cognitive Model Representation

The Ratio and Algebra attribute hierarchy can also be expressed in matrix form. To begin, the direct relationship among the attributes is specified by a binary adjacency matrix (A) of order (k,k), where k is the number of attributes, such that each element in the A matrix represents the absence (i.e., 0) or presence (i.e., 1) of a direct connection between two attributes. The A matrix for the Ratio and Algebra hierarchy presented is shown below.

$\begin{bmatrix} 0&1&0&1&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&1&0&1\\ 0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0 \end{bmatrix}$

Each row and column the A matrix represents one attribute; the first row and column represents attribute A1 and the last row and column represents attribute A9. The presence of a 1 in a particular row denotes a direct connection between that attribute and the attribute corresponding to the column position. For example, attribute A1 is directly connected to attribute A2 because of the presence of a 1 in the first row (i.e. attribute A1) and the second column (i.e., attribute A2). The positions of 0 in row 1 indicate that A1 is neither directly connected to itself nor to attributes A3 and A5 to A9.

The direct and indirect relationships among attributes are specified by the binary reachability matrix (R) of order (k,k), where k is the number of attributes. To obtain the R matrix from the A matrix, Boolean addition and multiplication operations are performed on the adjacency matrix, meaning where n is the integer required to reach invariance, and I is the identity matrix. The R matrix for the Ratio and Algebra hierarchy is shown next.

$\begin{bmatrix} 1&1&1&1&1&1&1&1&1\\ 0&1&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0\\ 0&0&0&1&1&1&1&1&1\\ 0&0&0&0&1&1&0&0&0\\ 0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1&1&0\\ 0&0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&0&1 \end{bmatrix}$

Similar to the A matrix, each row and column in the matrix represents one attribute; the first row and column represents attribute A1 and the last row and column represents attribute A9. The first attribute is either directly or indirectly connected to all attributes A1 to A9. This is represented by the presence of 1’s in all columns of row 1 (i.e., representing attribute A1). In the R matrix, an attribute is considered related to itself resulting in 1’s along the main diagonal. Referring back to the hierarchy, it is shown that attribute A1 is directly connected to attribute A2 and indirectly to A3 through its connection with A2. Attribute A1 is indirectly connected to attributes A5 to A9 through its connection with A4.

The potential pool of items is represented by the incidence matrix (Q) matrix of order (k, p), where k is the number of attributes and p is number of potential items. This pool of items represents all combinations of the attributes when the attributes are independent of each other. However, this pool of items can be reduced to form the reduced incidence matrix (Q_r), by imposing the constraints of the attribute hierarchy as defined by the R matrix. The Q_r matrix represents items that capture the dependencies among the attributes defined in the attribute hierarchy. The Q_r matrix is formed using Boolean inclusion by determining which columns of the R matrix are logically included in each column of the Q matrix. The Q_r matrix is of order (k,) where k is the number of attributes and i is the reduced number of items resulting from the constraints in the hierarchy. For the Ratio and Algebra hierarchy, the Q_r matrix is shown next.

The Q_r matrix serves an important test item development blueprint where items can be created to measure each specific combination of attributes. In this way, each component of the cognitive model can be evaluated systematically. In this example, a minimum of 9 items are required to measure all the attribute combinations specified in the Q_r matrix.

The expected examinee response patterns can now be generated using the Q_r matrix. An expected examinee is conceptualized as a hypothetical examinee who correctly answers items that require cognitive attributes that the examinee has mastered. The expected response matrix (E) is created, using Boolean inclusion, by comparing each row of the attribute pattern matrix (which is the transpose of the Q_r matrix) to the columns of the Q_r matrix. The expected response matrix is of order (j,i), where j is the number of examinees and i is the reduced number of items resulting from the constraints imposed by the hierarchy. The E matrix for the Ratio and Algebra hierarchy is shown below.

$\begin{bmatrix} 0&0&0&0&0&0&0&0&0\\ 1&0&0&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0&0\\ 1&1&1&0&0&0&0&0&0\\ 1&0&0&1&0&0&0&0&0\\ 1&1&0&1&0&0&0&0&0\\ 1&1&1&1&0&0&0&0&0\\ 1&0&0&1&1&0&0&0&0\\ 1&1&0&1&1&0&0&0&0\\ 1&1&1&1&1&0&0&0&0\\ 1&0&0&1&1&1&0&0&0\\ 1&1&0&1&1&1&0&0&0\\ 1&1&0&1&1&1&0&0&0\\ 1&0&0&1&0&0&1&0&0\\ 1&1&0&1&0&0&1&0&0\\ 1&1&1&1&0&0&1&0&0\\ 1&0&0&1&1&0&1&0&0\\ 1&1&0&1&1&0&1&0&0\\ 1&1&1&1&1&0&1&0&0\\ 1&0&0&1&1&1&1&0&0\\ 1&1&0&1&1&1&1&0&0\\ 1&1&1&1&1&1&1&0&0\\ 1&0&0&1&0&0&1&1&0\\ 1&1&0&1&0&0&1&1&0\\ 1&1&1&1&1&0&1&0&0\\ 1&0&0&1&1&0&1&1&0\\ 1&1&0&1&1&0&1&1&0\\ 1&1&1&1&1&0&1&1&0\\ 1&0&0&1&1&1&1&1&0\\ 1&1&0&1&1&1&1&1&0\\ 1&1&1&1&1&1&1&1&0\\ 1&0&0&1&0&0&0&0&1\\ 1&1&0&1&0&0&0&0&1\\ 1&1&1&1&0&0&0&0&1\\ 1&0&0&1&1&0&0&0&1\\ 1&1&0&1&1&0&0&0&1\\ 1&1&1&1&1&0&0&0&1\\ 1&0&0&1&1&1&0&0&1\\ 1&1&0&1&1&1&0&0&1\\ 1&1&1&1&1&1&0&0&1\\ 1&0&0&1&0&0&1&0&1\\ 1&1&0&1&0&0&1&0&1\\ 1&1&1&1&0&0&1&0&1\\ 1&0&0&1&1&0&1&0&1\\ 1&1&0&1&1&0&1&0&1\\ 1&1&1&1&1&0&1&0&1\\ 1&0&0&1&1&1&1&0&1\\ 1&1&0&1&1&1&1&0&1\\ 1&1&1&1&1&1&1&0&1\\ 1&0&0&1&0&0&1&1&1\\ 1&1&0&1&0&0&1&1&1\\ 1&1&1&1&0&0&1&1&1\\ 1&0&0&1&1&0&1&1&1\\ 1&1&0&1&1&0&1&1&1\\ 1&1&1&1&1&0&1&1&1\\ 1&0&0&1&1&1&1&1&1\\ 1&1&0&1&1&1&1&1&1\\ 1&1&1&1&1&1&1&1&1 \end{bmatrix}$

If the cognitive model is true, then 58 unique item response patterns should be produced by examinees who write these cognitively-based items. A row of 0s is usually added to the E matrix which represents an examinee who has not mastered any attributes. To summarize, if the attribute pattern of the examinee contains the attributes required by the item, then the examinee is expected to answer the item correctly. However, if the examinee’s attribute pattern is missing one or more of the cognitive attributes required by the item, the examinee is not expected to answer the item correctly.

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