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

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.

A comparison of latent class, K-means, and K-median methods for clustering dichotomous data.

“…Subsequent to the original study by Verweij et al (1996), the transitive reasoning data have been analyzed by Van der Ark (2012) and Brusco, Köhn, and Steinley (2015). In each of these studies, however, the focus was principally on Mokken (1971) scaling of the 12 dichotomous items rather than the clustering of the 425 schoolchildren.…”

“…Van der Ark et al (2008) have identified two components of Mokken scaling: (1) evaluating a collection of ordered items to detect the presence of one or more scales, and (2) follow-up analyses to discover relevant psychometric characteristics of those scales. Thus, the first part of this process can be viewed as a problem of partitioning the scale items, and that was the problem tackled by Van der Ark (2012) and Brusco et al (2015). Contrastingly, we do not seek to partition the 12 scale items here.…”

An evaluation of exact methods for the multiple subset maximum cardinality selection problem

“…A second important application of the MSMCSP occurs in the area of non‐parametric item response theory (IRT) and, in particular, in the context of exploratory Mokken scale analysis (Brusco, Köhn, & Steinley, ; Mokken, ; Straat, Van der Ark, & Sijtsma, ; Van Abswoude, Vermunt, Hemker, & Van der Ark, ; Van der Ark, Croon, & Sijtsma, ). For exploratory Mokken scaling of binary items, the goal is to obtain the largest possible subset of items, subject to constraints that guarantee that all pairwise scalability coefficients among selected items equal or exceed some threshold.…”

“…Brusco et al . () recently developed an integer programming formulation for the MSMCSP in the context of non‐parametric IRT. Their model incorporated constraints on the pairwise item correlations, but also constraints on the partial correlations among triples of items in accordance with necessary conditions developed by Ellis () for the monotone homogeneity model.…”

“…Brusco et al . () also proposed a standalone branch‐and‐bound algorithm for obtaining an exact solution to their MSMCSP formulation…”