Scaling methods (44)

Chair: Willem Heiser, Thursday 23rd July, 9.40 - 11.00, Castlereagh Room, Fisher Building. 

Richard A. Faldowski, Yudan Chen Wang and Gui-Young Hong, Department of Human Development and Family Studies, University of North Carolina at Greensboro, USA.  Solution dimensionality of principal components with optimal scaling: Adaptations and performance of simple methods. (240)

Willem Heiser and Matthijs J. Warrens, Department of Psychology, Leiden University, The Netherlands. Detecting positive regression dependence using the dominant axis of correspondence analysis. (215)

Kensuke Okada and Kazuo Shigemasu, Japan Society for the Promotion of Science, Tokyo Institute of Technology, Japan. Bayesian multidimensional scaling for Minkowski distances. (130)

Takashi Murakami, Department of Sociology, Chukyo University, Toyota, Japan. Multiple correspondence analysis of Likert items and explanations of nonlinear relationships between person scores. (171)

ABSTRACTS

Solution dimensionality of principal components with optimal scaling: Adaptations and performance of simple methods. (240)
Richard A. Faldowski, Yudan Chen Wang and Gui-Young Hong
Principal components analysis with optimal scaling (PCA/OS) is promising method for exploratory psychometric analyses, yet determination of solution dimensionality, or the so-called "number of components" problem, remains a challenging issue. The adequacy of principal component representations as well as optimal scaling transformations, are well-known to be conditional on selection of correct solution dimensionality. At the same time, most PCA/OS software such as SAS/Prinqual or SPSS/CATPCA algorithms fit PCA/OS models by conditioning on a priori selection of solution dimensionality. This yields PCA/OS solutions that are nested in neither the components nor optimal scaling transformations. Standard simple criteria for determining adequate principal components analysis solution dimensionality; such as Scree Criteria, Randomization-based Parallel Analyses, and Permutation-based Parallel Analyses; no longer directly apply. In this presentation, we describe adaptations of these simple criteria for covariance matrices that can be appropriately applied to PCA/OS solutions for determining solution dimensionality. Rather than working with PCA/OS solutions, all of the adapted methods operate on PCA/OS residuals and provide decisions about the need for additional solution dimensions. We note limitations that occur in practice and suggest directions for future work that may yield more robust results.

Detecting positive regression dependence using the dominant axis of correspondence analysis. (215)
Willem Heiser and Matthijs J. Warrens
Contingency tables and zero-one data arrays with possibly meaningful ordinal structure in the rows and columns may be analyzed by correspondence analyis. We discuss the old question, when does a one-dimensional solution suffice? Positive regression dependence is a well-known condition on the cell frequencies of the table for a given ordering of the rows and the columns, and is one possible definition of perfect association between two ordered categorical variables. Schriever (1983) showed that the dominant axis of a correspondence analysis on the unordered table detects the correct orderings when positive regression dependence holds. Fit measures are formulated that may be used to indicate how strongly a re-ordered contingency table or a zero-one data array satisfies positive regression dependence, in either the rows, or the columns, or in both. Several empirical examples are discussed to illustrate the diagnostic value of these fit measures. In a number of cases, we conclude that a one-dimensional solution suffices, even though not all variance is accounted for.

Bayesian multidimensional scaling for Minkowski distances. (130)
Kensuke Okada and Kazuo Shigemasu
Multidimensional scaling (MDS)-type modeling for non-Euclidean distance involves both psychological and statistical interests. MDS has frequently been used for modeling psychological phenomena, where the MDS geometry served as a model of psychological space, and the distance function as a model of mental arithmetic. In many former studies MDS solutions were first computed for several different Minkowski exponents and then the one that results in the lowest stress was chosen. However there have been few attempts to directly infer Minkowski exponent, which should be of primary interest. In this presentation, a new Bayesian multidimensional scaling method which treats Minkowski exponent as a random variable is introduced. Contrary to Lee’s (2008) method, our procedure models dissimilarity (not similarity) data. The idea is simply to extend the Euclidean model of Oh & Raftery’s (2001) Bayesian MDS to general Minkowski distance model. The performance of the proposed procedure is examined using a simulation study and real data analysis. The simulation study shows that our method is suitable for recovering the true Minkowski exponent. 

Multiple correspondence analysis of Likert items and explanations of nonlinear relationships between person scores. (171)
Takashi Murakami
It is well known that the "horse-shoe" and other nonlinear relationships often appear in a scatter diagram of object scores obtained by multiple correspondence analysis (MCA) of ordered categorical variables. Explanations and interpretations of the phenomena were looked for from theoretical and empirical viewpoints. Visual representation of scores as the two-dimensional projection of the multidimensional polygon formed by quantified ternary variables was used for investigating the shapes of distributions. Concepts of radex structures were generalized to ternary variables, and similar results to parabolic and circular distributions of binary data were found in the analysis of artificial data. Wedge-shaped distributions were also observed in erroneous data, and the first dimension corresponding to the largest characteristic root was almost linearly related to the scale value defined as the simple sum of item scores whereas the second dimension was related to the number of neutral responses. The decomposition of the matrix of MCA weights into the product of the matrix of quantifications of categories and that of loadings of variables as in nonmetric principal components analysis could derive several sets of two types of dimensions from real data consisting of many Likert items.