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An Exact Method for Partitioning Dichotomous Items Within the Framework of the Monotone Homogeneity Model

Published online by Cambridge University Press:  01 January 2025

Michael J. Brusco ,
Hans-Friedrich Köhn  and
Douglas Steinley
Michael J. Brusco*
Affiliation:
Florida State University
Hans-Friedrich Köhn
Affiliation:
University of Illinois at Urbana-Champaign
Douglas Steinley
Affiliation:
University of Missouri
*
Correspondence should be made to Michael J. Brusco, Florida State University, 821 Academic Way, Tallahassee, FL 32306-1110 USA. Email: mbrusco@fsu.edu
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Abstract

The monotone homogeneity model (MHM—also known as the unidimensional monotone latent variable model) is a nonparametric IRT formulation that provides the underpinning for partitioning a collection of dichotomous items to form scales. Ellis (Psychometrika 79:303–316, 2014, doi:10.1007/s11336-013-9341-5) has recently derived inequalities that are implied by the MHM, yet require only the bivariate (inter-item) correlations. In this paper, we incorporate these inequalities within a mathematical programming formulation for partitioning a set of dichotomous scale items. The objective criterion of the partitioning model is to produce clusters of maximum cardinality. The formulation is a binary integer linear program that can be solved exactly using commercial mathematical programming software. However, we have also developed a standalone branch-and-bound algorithm that produces globally optimal solutions. Simulation results and a numerical example are provided to demonstrate the proposed method.

Keywords

nonparametric IRTpartial correlationmokken scale analysisitem selectionexact algorithm
Type
Original Paper
Information
Psychometrika , Volume 80 , Issue 4 , December 2015 , pp. 949 - 967
Copyright
Copyright © 2015 The Psychometric Society

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