cognitive model. Evaluating model-data fit Prior to any further analysis, the cognitive model specified must accurately reflect the cognitive attributes used by the examinees. It is expected that there will be discrepancies, or slips, between observed response patterns generated by a large group of examinees and the expected response patterns. The fit of the cognitive model relative to the observed response patterns obtained from examinees can be evaluated using the Hierarchical Consistency Index. The HCI evaluates the degree to which the observed response patterns are consistent with the attribute hierarchy. The HCI for examinee i is given by: where J is the total number of items, Xij is examinee i ‘s score (i.e., 1 or 0) to item j, Sj includes items that require the subset of attributes of item j, and Nci is the total number of comparisons for correctly answered items by examinee i. The values of the HCI range from −1 to +1. Values closer to 1 indicate a good fit between the observed response pattern and the expected examinee response patterns generated from the hierarchy. Conversely, low HCI values indicate a large discrepancy between the observed examinee response patterns and the expected examinee response patterns generated from the hierarchy. HCI values above 0.70 indicate good model-data fit. Why is model-data fit important? Obtaining good model-data fit provides additional evidence to validate the specified attribute hierarchy, which is required before proceeding with determination of an examinee’s attribute mastery. If the data is not shown to fit the model, then various reasons may account for the large number of discrepancies including: a misspecification of the attributes, incorrect ordering of attributes within the hierarchy, items not measuring the specified attributes, and/or the model is not reflective of the cognitive processes used by a given sample of examinees. Therefore, the cognitive model should be correctly defined and closely aligned with the observed response patterns in order to provide a substantive framework for making inferلبلبences about a specific group of examinees’ knowledge and skills. لابلب Estimating attribute probabilities Once we establish that the model fits the data, the attribute probabilities can be calculated. The use of attribute probabilities is important in the psychometric analyses of the AHM because these probabilities provide examinees with specific information about their attribute-level performance as part of the diagnostic reporting process. To estimate the probability that examinees possess specific attributes, given their observed item response pattern,