an artificial neural network approach is used. Brief description of a neural network The neural network is a type of parallel-processing architecture that transforms any stimulus received by the input unit (i.e., stimulus units) to a signal for the output unit (i.e., response units) through a series of mid-level hidden units. Each unit in the input layer is connected to each unit in the hidden layer and, in turn, to each unit in the output layer. Generally speaking, a neural network requires the following steps. To begin, each cell of the input layer receives a value (0 or 1) corresponding to the response values in the exemplar vector. Each input cell then passes the value it receives to every hidden cell. Each hidden cell forms a linearly weighted sum of its input and transforms the sum using the logistic function and passes the result to every output cell. Each output cell, in turn, forms a linearly weighted sum of its inputs from the hidden cells and transforms it using the logistic function, and outputs the result. Because the result is scaled using the logistic transformation, the output values range from 0 to 1. The result can be interpreted as the probability the correct or target value for each output will have a value of 1. The output targets in the response units (i.e., the examinee attributes) are compared to the pattern associated with each stimulus input or exemplar (i.e., the expected response patterns). The solution produced initially with the stimulus and association connection weights is likely to be discrepant resulting in a relatively large error. However, this discrepant result can be used to modify the connection weights thereby leading to a more accurate solution and a smaller error term. One popular approach for approximating the weights so the error term is minimized is with a learning algorithm called the generalized delta rule that is incorporated in a training procedure called back propagation of error. Specification of the neural network Calculation of attribute probabilities begins by presenting the neural network with both the generated expected examinee response patterns from Stage 1, with their associated attribute patterns which is derived from the cognitive model (i.e., the transpose of the Qr matrix), until the network learns each association. The result is a set of weight matrices that will be used to calculate the probability that an examinee has mastered a particular cognitive attribute