Extracting partially ordered clusters from ordinal polytomous data

Abstract

In practical applications of knowledge space theory, knowledge states can be conceived as partially ordered clusters of individuals. Existing extensions of the theory to polytomous data lack methods for building "polytomous" structures. To this aim, an adaptation of the k-median clustering algorithm is proposed. It is an extension of k-modes to ordinal data in which the Hamming distance is replaced by the Manhattan distance, and the central tendency measure is the median, rather than the mode. The algorithm is tested in a series of simulation studies and in an application to empirical data. Results show that there are theoretical and practical reasons for preferring the k-median to the k-modes algorithm, whenever the responses to the items are measured on an ordinal scale. This is because the Manhattan distance is sensitive to the order on the levels, while the Hamming distance is not. Overall, k-median seems to be a promising data-driven procedure for building polytomous structures.

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“…k‐means) relying on the mean for defining the centroid of each cluster. Among alternative algorithms are k‐modes and k‐medians (relying on the mode and on the median, respectively); the output of these algorithms would be the same, all the variables at hand being dichotomous. Therefore, we used k‐modes .…”