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Growth mixture modeling is generally used for two purposes: (1) to identify mixtures of normal subgroups and (2) to approximate oddly shaped distributions by a mixture of normal components. Often in applied research this methodology is applied to both of these situations indistinctly: using the same fit statistics and likelihood ratio tests. This can lead to the overextraction of latent classes and the attribution of substantive meaning to these spurious classes. The goals of this study are (1) to explore the performance of the Bayesian information criterion, sample-adjusted BIC, and bootstrap likelihood ratio test in growth mixture modeling analysis with nonnormal distributed outcome variables and (2) to examine the effects of nonnormal time invariant covariates in the estimation of the number of latent classes when outcome variables are normally distributed. For both of these goals, we will include nonnormal conditions not considered previously in the literature. Two simulation studies were conducted. Results show that spurious classes may be selected and optimal solutions obtained in the data analysis when the population departs from normality even when the nonnormality is only present in time invariant covariates.

A measure of "clusterability" serves as the basis of a new methodology designed to preserve cluster structure in a reduced dimensional space. Similar to principal component analysis, which finds the direction of maximal variance in multivariate space, principal cluster axes find the direction of maximum clusterability in multivariate space. Furthermore, the principal clustering approach falls into the class of projection pursuit techniques. Comparisons are made with existing methodologies both in a simulation study and analysis of real-world data sets. Furthermore, a demonstration of how to interpret the results of the principal cluster axes is provided on the analysis of Supreme Court voting data and similarities between the interpretation of competing procedures (e.g., factor analysis and principal component analysis) are provided. In addition to the Supreme Court analysis, we analyze several data sets often used to test cluster analysis procedures, including Fisher's Iris data, Agresti's Crab data, and a data set on glass fragments. Finally, discussion is provided to help determine when the proposed procedure will be the most beneficial to the researcher. (Contains 5 footnotes, 5 tables, and 8 figures.)

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. (Contains 4 tables.)

Steinley (2007) provided a lower bound for the sum-of-squares error criterion function used in K-means clustering. In this article, on the basis of the lower bound, the authors propose a method to distinguish between 1 cluster (i.e., a single distribution) versus more than 1 cluster. Additionally, conditional on indicating there are multiple clusters, the procedure is extended to determine the number of clusters. Through a series of simulations, the proposed methodology is shown to outperform several other commonly used procedures for determining both the presence of clusters and their number. (Contains 3 figures, 4 tables and 5 footnotes.)

McLachlan (2011) and Vermunt (2011) each provided thoughtful replies to our original article (Steinley & Brusco, 2011). This response serves to incorporate some of their comments while simultaneously clarifying our position. We argue that greater caution against overparamaterization must be taken when assuming that clusters are highly elliptical in nature. Specifically, users of mixture model clustering techniques should be wary of overreliance on fit indices, and the importance of cross-validation is highlighted. Additionally, we note that "K"-means clustering is part of a larger family of discrete partitioning algorithms, many of which are designed to solve problems identical to those for which mixture modeling approaches are often touted. (Contains 3 footnotes.)

This article provides a large-scale investigation into several of the properties of mixture-model clustering techniques (also referred to as latent class cluster analysis, latent profile analysis, model-based clustering, probabilistic clustering, Bayesian classification, unsupervised learning, and finite mixture models; see Vermunt & Magdison, 2002). Focus is given to the multivariate normal distribution, and 9 separate decompositions (i.e., class structures) of the covariance matrix are investigated. To provide a link to the current literature, comparisons are made with K-means clustering in 3 detailed Monte Carlo studies. The findings have implications for applied researchers in that mixture-model clustering techniques performed best when the covariance structure and number of clusters were known. However, as the information about the shape and number of clusters became unknown, degraded performance was observed for both K-means clustering and mixture-model clustering. (Contains 5 figures, 5 tables and 8 footnotes.)

Structural balance theory (SBT) has maintained a venerable status in the psychological literature for more than 5 decades. One important problem pertaining to SBT is the approximation of structural or generalized balance via the partitioning of the vertices of a signed graph into "K" clusters. This "K"-balance partitioning problem also has more general psychological applications associated with the analysis of similarity/dissimilarity relationships among stimuli. Accordingly, "K"-balance partitioning can be gainfully used in a wide variety of SBT applications, such as attraction and child development, evaluation of group membership, marketing and consumer issues, and other psychological contexts not necessarily related to SBT. We present a branch-and-bound algorithm for the "K"-balance partitioning problem. This new algorithm is applied to 2 synthetic numerical examples as well as to several real-world data sets from the behavioral sciences literature. (Contains 6 tables and 4 figures.)

The "p"-median clustering model represents a combinatorial approach to partition data sets into disjoint, nonhierarchical groups. Object classes are constructed around "exemplars", that is, manifest objects in the data set, with the remaining instances assigned to their closest cluster centers. Effective, state-of-the-art implementations of "p"-median clustering are virtually unavailable in the popular social and behavioral science statistical software packages. We present "p"-median clustering, including a detailed description of its mechanics and a discussion of available software programs and their capabilities. Application to a complex structured data set on the perception of food items illustrates "p"-median clustering. (Contains 2 figures and 2 tables.)

The selection of a subset of variables from a pool of candidates is an important problem in several areas of multivariate statistics. Within the context of principal component analysis (PCA), a number of authors have argued that subset selection is crucial for identifying those variables that are required for correct interpretation of the components. In this paper, we adapt the variable neighborhood search (VNS) paradigm to develop two heuristics for variable selection in PCA. The performances of these heuristics were compared to those obtained by a branch-and-bound algorithm, as well as forward stepwise, backward stepwise, and tabu search heuristics. In the first experiment, which considered candidate pools of 18 to 30 variables, the VNS heuristics matched the optimal subset obtained by the branch-and-bound algorithm more frequently than their competitors. In the second experiment, which considered candidate pools of 54 to 90 variables, the VNS heuristics provided better solutions than their competitors for a large percentage of the test problems. An application to a real-world data set is provided to demonstrate the importance of variable selection in the context of PCA.

This paper proposes an order-constrained K-means cluster analysis strategy, and implements that strategy through an auxiliary quadratic assignment optimization heuristic that identifies an initial object order. A subsequent dynamic programming recursion is applied to optimally subdivide the object set subject to the order constraint. We show that although the usual K-means sum-of-squared-error criterion is not guaranteed to be minimal, a true underlying cluster structure may be more accurately recovered. Also, substantive interpretability seems generally improved when constrained solutions are considered. We illustrate the procedure with several data sets from the literature.

Eight different variable selection techniques for model-based and non-model-based clustering are evaluated across a wide range of cluster structures. It is shown that several methods have difficulties when non-informative variables (i.e., random noise) are included in the model. Furthermore, the distribution of the random noise greatly impacts the performance of nearly all of the variable selection procedures. Overall, a variable selection technique based on a variance-to-range weighting procedure coupled with the largest decreases in within-cluster sums of squares error performed the best. On the other hand, variable selection methods used in conjunction with finite mixture models performed the worst.

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. (Contains 6 figures, 9 tables, and 3 footnotes.)

Clusterwise linear regression is a multivariate statistical procedure that attempts to cluster objects with the objective of minimizing the sum of the error sums of squares for the within-cluster regression models. In this article, we show that the minimization of this criterion makes no effort to distinguish the error explained by the within-cluster regression models from the error explained by the clustering process. In some cases, most of the variation in the response variable is explained by clustering the objects, with little additional benefit provided by the within-cluster regression models. Accordingly, there is tremendous potential for overfitting with clusterwise regression, which is demonstrated with numerical examples and simulation experiments. To guard against the misuse of clusterwise regression, we recommend a benchmarking procedure that compares the results for the observed empirical data with those obtained across a set of random permutations of the response measures. We also demonstrate the potential for overfitting via an empirical application related to the prediction of reflective judgment using high school and college performance measures. (Contains 1 figure and 3 tables.)

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.

Given that a minor condition holds (e.g., the number of variables is greater than the number of clusters), a nontrivial lower bound for the sum-of-squares error criterion in K-means clustering is derived. By calculating the lower bound for several different situations, a method is developed to determine the adequacy of cluster solution based on the observed sum-of-squares error as compared to the minimum sum-of-squares error.

Similarities between latent class models with K classes and linear factor models with K-1 factors are investigated. Specifically, the mathematical equivalence between the covariance structure of the two models is discussed, and a Monte Carlo simulation is performed using generated data that represents both latent factors and latent classes with known amounts of overlap. It is shown that, under certain conditions, the distribution of factor scores can be related to the continuity of the latent space via tests of multimodality as suggested by McDonald (1967).

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.

Using the cluster generation procedure proposed by D. Steinley and R. Henson (2005), the author investigated the performance of K-means clustering under the following scenarios: (a) different probabilities of cluster overlap; (b) different types of cluster overlap; (c) varying samples sizes, clusters, and dimensions; (d) different multivariate distributions of clusters; and (e) various multidimensional data structures. The results are evaluated in terms of the Hubert-Arabie adjusted Rand index, and several observations concerning the performance of K-means clustering are made. Finally, the article concludes with the proposal of a diagnostic technique indicating when the partitioning given by a K-means cluster analysis can be trusted. By combining the information from several observable characteristics of the data (number of clusters, number of variables, sample size, etc.) with the prevalence of unique local optima in several thousand implementations of the K-means algorithm, the author provides a method capable of guiding key data-analysis decisions.

This article provides an investigation of cluster validation indices that relates 4 of the indices to the L. Hubert and P. Arable (1985) adjusted Rand index--the cluster validation measure of choice (G. W. Milligan & M. C. Cooper, 1986). It is shown how these other indices can be "roughly" transformed into the same scale as the adjusted Rand index. Furthermore, in-depth explanations are given of why classification rates should not be used in cluster validation research. The article concludes by summarizing several properties of the adjusted Rand index across many conditions and provides a method for testing the significance of observed adjusted Rand indices.