Statistical classification of item-response patterns into misconception groups in rule space

In an effort to further explore a specific aspect of rule-space theory, identification of students with respect to their error rules, three major questions were raised: (a) How do the discriminant-function approach (DA) and the k-nearest-neighbor approach (KNN) perform for the classification of each bivariate-normally- and non-bivariate-normally-distributed item-response patterns in rule space? (b) Is the use of rule-space model better than not using it for classifications? (c) What is the minimum pairwise distance for a 90% correct classification?A Monte Carlo study was executed. Two independent and identical sample sets of item-response patterns were simulated. For each sample, two hundred item-response patterns in each rule group, for a total of fifteen groups, were generated. However, only the item-response patterns of nine rule groups located near the 0 axis were used for classifications. The excluded item-response patterns were of the groups that regressed steeply from extreme ZETA toward ZETA = 0. An optimum k = 9 was chosen for KNN. Three kinds of error rates--leave-one-out error rate (Ecv), apparent error rate (App), and error rate using separate sample groups (Ess)--were used for the hypothesis tests.Major findings were: (a) DA was robust against non-bivariate-normality over the ability continuum for classification in rule space. (b) The use of rule-space variables as classification variables was proved to be more reliable than the use of item scores. (c) The minimum pairwise distance for 90% correct classifications was given by rather a large interval between 3.57 and 14.06.Discussions were given in relation with the regression of sample distributions, an optimum k for KNN, and the three major findings. Based on the discussions, correction of the regression, generalization studies for the robustness of DA, unsupervised classification, classification of the observations into merged groups were suggested for further studies.U of I OnlyETDs are only available to UIUC Users without author permissio