Clustering, seriation, and subset extraction of confusion data

The study of confusion data is a well established practice in psychology. Although many types of analytical approaches for confusion data are available, among the most common methods are the extraction of 1 or more subsets of stimuli, the partitioning of the complete stimulus set into distinct groups, and the ordering of the stimulus set. Although standard commercial software packages can sometimes facilitate these types of analyses, they are not guaranteed to produce optimal solutions. The authors present a MATLAB *.m file for preprocessing confusion matrices, which includes fitting of the similarity-choice model. Two additional MATLAB programs are available for optimally clustering stimuli on the basis of confusion data. The authors also developed programs for optimally ordering stimuli and extracting subsets of stimuli using information from confusion matrices. Together, these programs provide several pragmatic alternatives for the applied researcher when analyzing confusion data. Although the programs are described within the context of confusion data, they are also amenable to other types of proximity data.     

Compact integer-programming models for extracting subsets of stimuli from confusion matrices

This paper presents an integer linear programming formulation for the problem of extracting a subset of stimuli from a confusion matrix. The objective is to select stimuli such that total confusion among the stimuli is minimized for a particular subset size. This formulation provides a drastic reduction in the number of variables and constraints relative to a previously proposed formulation for the same problem. An extension of the formulation is provided for a biobjective problem that considers both confusion and recognition in the objective function. Demonstrations using an empirical interletter confusion matrix from the psychological literature revealed that a commercial branch-and-bound integer programming code was always able to identify optimal solutions for both the single-objective and biobjective formulations within a matter of seconds. A further extension and demonstration of the model is provided for the extraction of multiple subsets of stimuli, wherein the objectives are to maximize similarity within subsets and minimize similarity between subsets.     

A Comparison of Heuristic Procedures for Minimum Within-Cluster Sums of Squares Partitioning

Perhaps the most common criterion for partitioning a data set is the minimization of the within-cluster sums of squared deviation from cluster centroids. Although optimal solution procedures for within-cluster sums of squares (WCSS) partitioning are computationally feasible for small data sets, heuristic procedures are required for most practical applications in the behavioral sciences. We compared the performances of nine prominent heuristic procedures for WCSS partitioning across 324 simulated data sets representative of a broad spectrum of test conditions. Performance comparisons focused on both percentage deviation from the “best-found” WCSS values, as well as recovery of true cluster structure. A real-coded genetic algorithm and variable neighborhood search heuristic were the most effective methods; however, a straightforward two-stage heuristic algorithm, HK-means, also yielded exceptional performance. A follow-up experiment using 13 empirical data sets from the clustering literature generally supported the results of the experiment using simulated data. Our findings have important implications for behavioral science researchers, whose theoretical conclusions could be adversely affected by poor algorithmic performances.     

Heuristic cognitive diagnosis when the Q‐matrix is unknown

Cognitive diagnosis models of educational test performance rely on a binary Q‐matrix that specifies the associations between individual test items and the cognitive attributes (skills) required to answer those items correctly. Current methods for fitting cognitive diagnosis models to educational test data and assigning examinees to proficiency classes are based on parametric estimation methods such as expectation maximization (EM) and Markov chain Monte Carlo (MCMC) that frequently encounter difficulties in practical applications. In response to these difficulties, non‐parametric classification techniques (cluster analysis) have been proposed as heuristic alternatives to parametric procedures. These non‐parametric classification techniques first aggregate each examinee's test item scores into a profile of attribute sum scores, which then serve as the basis for clustering examinees into proficiency classes. Like the parametric procedures, the non‐parametric classification techniques require that the Q‐matrix underlying a given test be known. Unfortunately, in practice, the Q‐matrix for most tests is not known and must be estimated to specify the associations between items and attributes, risking a misspecified Q‐matrix that may then result in the incorrect classification of examinees. This paper demonstrates that clustering examinees into proficiency classes based on their item scores rather than on their attribute sum‐score profiles does not require knowledge of the Q‐matrix, and results in a more accurate classification of examinees.     

Examining the effect of initialization strategies on the performance of Gaussian mixture modeling

Mixture modeling is a popular technique for identifying unobserved subpopulations (e.g., components) within a data set, with Gaussian (normal) mixture modeling being the form most widely used. Generally, the parameters of these Gaussian mixtures cannot be estimated in closed form, so estimates are typically obtained via an iterative process. The most common estimation procedure is maximum likelihood via the expectation-maximization (EM) algorithm. Like many approaches for identifying subpopulations, finite mixture modeling can suffer from locally optimal solutions, and the final parameter estimates are dependent on the initial starting values of the EM algorithm. Initial values have been shown to significantly impact the quality of the solution, and researchers have proposed several approaches for selecting the set of starting values. Five techniques for obtaining starting values that are implemented in popular software packages are compared. Their performances are assessed in terms of the following four measures: (1)?the ability to find the best observed solution, (2)?settling on a solution that classifies observations correctly, (3)?the number of local solutions found by each technique, and (4)?the speed at which the start values are obtained. On the basis of these results, a set of recommendations is provided to the user.     

A Tabu-Search Heuristic for Deterministic Two-Mode Blockmodeling of Binary Network Matrices

Two-mode binary data matrices arise in a variety of social network contexts, such as the attendance or non-attendance of individuals at events, the participation or lack of participation of groups in projects, and the votes of judges on cases. A popular method for analyzing such data is two-mode blockmodeling based on structural equivalence, where the goal is to identify partitions for the row and column objects such that the clusters of the row and column objects form blocks that are either complete (all 1s) or null (all 0s) to the greatest extent possible. Multiple restarts of an object relocation heuristic that seeks to minimize the number of inconsistencies (i.e., 1s in null blocks and 0s in complete blocks) with ideal block structure is the predominant approach for tackling this problem. As an alternative, we propose a fast and effective implementation of tabu search. Computational comparisons across a set of 48 large network matrices revealed that the new tabu-search heuristic always provided objective function values that were better than those of the relocation heuristic when the two methods were constrained to the same amount of computation time.     

Heuristic Implementation of Dynamic Programming for Matrix Permutation Problems in Combinatorial Data Analysis

Dynamic programming methods for matrix permutation problems in combinatorial data analysis can produce globally-optimal solutions for matrices up to size 30×30, but are computationally infeasible for larger matrices because of enormous computer memory requirements. Branch-and-bound methods also guarantee globally-optimal solutions, but computation time considerations generally limit their applicability to matrix sizes no greater than 35×35. Accordingly, a variety of heuristic methods have been proposed for larger matrices, including iterative quadratic assignment, tabu search, simulated annealing, and variable neighborhood search. Although these heuristics can produce exceptional results, they are prone to converge to local optima where the permutation is difficult to dislodge via traditional neighborhood moves (e.g., pairwise interchanges, object-block relocations, object-block reversals, etc.). We show that a heuristic implementation of dynamic programming yields an efficient procedure for escaping local optima. Specifically, we propose applying dynamic programming to reasonably-sized subsequences of consecutive objects in the locally-optimal permutation, identified by simulated annealing, to further improve the value of the objective function. Experimental results are provided for three classic matrix permutation problems in the combinatorial data analysis literature: (a) maximizing a dominance index for an asymmetric proximity matrix; (b) least-squares unidimensional scaling of a symmetric dissimilarity matrix; and (c) approximating an anti-Robinson structure for a symmetric dissimilarity matrix. We are extremely grateful to the Associate Editor and two anonymous reviewers for helpful suggestions and corrections.     

Intercultural Study of Personal Space: A Case Study

The study of intercultural differences in personal space attempts to identify factors that could be helpful in guiding people from different cultures to live and work together in international ventures. The objective of this research was to measure interpersonal distance in a cross-cultural environment. Twenty-three participants classified in 4 cultural groups answered an 11-questions interview where the personal distance and orientation between seated interactants were measured. Results showed that Anglo Saxons used the largest zone of personal space, followed by Asians, then Caucasians, and, as expected, Mediterraneans and Latinos used the shortest distance. Where Latinos adopt a face-to-face position, Caucasians tend to orient themselves to the side of the interviewer, which has an impact on interpersonal communication.     

Identifying a Reordering of Rows and Columns for Multiple Proximity Matrices Using Multiobjective Programming

This paper is concerned with a problem where K (n×n) proximity matrices are available for a set of n objects. The goal is to identify a single permutation of the n objects that provides an adequate structural fit, as measured by an appropriate index, for each of the K matrices. A multiobjective programming approach for this problem, which seeks to optimize a weighted function of the K indices, is proposed, and illustrative examples are provided using a set of proximity matrices from the psychological literature. These examples show that, by solving the multiobjective programming model under different weighting schemes, the quantitative analyst can uncover information about the relationships among the matrices and often identify one or more permutations that provide good to excellent index values for all matrices.     

A New Variable Weighting and Selection Procedure for K-means Cluster Analysis

A variance-to-range ratio variable weighting procedure is proposed. We show how this weighting method is theoretically grounded in the inherent variability found in data exhibiting cluster structure. In addition, a variable selection procedure is proposed to operate in conjunction with the variable weighting technique. The performances of these procedures are demonstrated in a simulation study, showing favorable results when compared with existing standardization methods. A detailed demonstration of the weighting and selection procedure is provided for the well-known Fisher Iris data and several synthetic data sets.