Component analysis with different sets of constraints on different dimensions

Many of the classical multivariate data analysis and multidimensional scaling techniques call for approximations by lower dimensional configurations. A model is proposed, in which different sets of linear constraints are imposed on different dimensions in component analysis and classical multidimensional scaling frameworks. A simple, efficient, and monotonically convergent algorithm is presented for fitting the model to the data by least squares. The basic algorithm is extended to cover across-dimension constraints imposed in addition to the dimensionwise constraints, and to the case of a symmetric data matrix. Examples are given to demonstrate the use of the method.The work reported in this paper has been supported by the Natural Sciences and Engineering Research Council of Canada, grant number A6394, and by the McGill-IBM Cooperative Grant, both granted to the first author. The research of H. A. L. Kiers has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. We thank Michael Hunter for his helpful comments on earlier drafts of this paper.     

Generating Multivariate Ordinal Data via Entropy Principles

When conducting robustness research where the focus of attention is on the impact of non-normality, the marginal skewness and kurtosis are often used to set the degree of non-normality. Monte Carlo methods are commonly applied to conduct this type of research by simulating data from distributions with skewness and kurtosis constrained to pre-specified values. Although several procedures have been proposed to simulate data from distributions with these constraints, no corresponding procedures have been applied for discrete distributions. In this paper, we present two procedures based on the principles of maximum entropy and minimum cross-entropy to estimate the multivariate observed ordinal distributions with constraints on skewness and kurtosis. For these procedures, the correlation matrix of the observed variables is not specified but depends on the relationships between the latent response variables. With the estimated distributions, researchers can study robustness not only focusing on the levels of non-normality but also on the variations in the distribution shapes. A simulation study demonstrates that these procedures yield excellent agreement between specified parameters and those of estimated distributions. A robustness study concerning the effect of distribution shape in the context of confirmatory factor analysis shows that shape can affect the robust \(\chi ^2\) and robust fit indices, especially when the sample size is small, the data are severely non-normal, and the fitted model is complex.     

Magnitude estimation of short electrocutaneous pulses

Summary A comparison was made of electrocutaneous magnitude estimation data across two experiments with contextual differences not involving stimulus parameters, such as number and range of stimuli and relative position of the standard in the stimulus range. The data were fitted by 2-parameter linear, log-linear and power functions. When the data are fitted by either linear or log-linear equations, both intercept and slope parameters are significantly affected by the different contextual factors. When the data are fitted by a power function, however, only the intercept is altered; the slope remains invariant despite contextual changes introduced in the second experiment.Although the empirically derived psychophysical power law has been applied to magnitude estimation data for all other sensory modalities, its application to electrocutaneous stimuli has been less successful.     

A method of score conversion through item statistics

A method is presented for converting the scores on one form of a test to those on another form of the same test. The method is particularly applicable to the case where each form has been administered to a different group and the only link between the two forms is a subset of items common to both. The proposed method, called theitem method of conversion, has been applied to several tests for which other methods of conversion are available for comparison. The necessary data are limited to tests for which the total score is the criterion for item analyses. The method gives highly satisfactory results for all the tests to which it has been applied, particularly when the two groups are rather different, in which case the delta method (a different item method) is inappropriate.The authors are only two of a group, including W. H. Angoff, F. M. Lord, and M. K. Schultz, all of whom have made important contributions to this paper.     

Some clarifications of the TUCKALS2 algorithm applied to the IDIOSCAL problem

Kroonenberg and de Leeuw have suggested fitting the IDIOSCAL model by the TUCKALS2 algorithm for three-way components analysis. In theory, this is problematic because TUCKALS2 produces two possibly different coordinate matrices, that are useless for IDIOSCAL unless they are equal. Kroonenberg has claimed that, when IDIOSCAL is fitted by TUCKALS2, the resulting coordinate matrices will be identical. In the present paper, this claim is proven valid when the data matrices are semidefinite. However, counterexamples for indefinite matrices are also constructed, by examining the global minimum in the case where the data matrices have the same eigenvectors. Similar counterexamples have been considered by ten Berge and Kiers in the related context of CANDECOMP/PARAFAC to fit the INDSCAL model.     

A monotonically convergent algorithm for factals

Takane, Young, and de Leeuw proposed a procedure called FACTALS for the analysis of variables of mixed measurement levels (numerical, ordinal, or nominal). Mooijaart pointed out that their algorithm does not necessarily converge, and Nevels proposed a new algorithm for the case of nominal variables. In the present paper it is shown that Nevels' procedure is incorrect, and a new procedure for handling nominal variables is proposed. In addition, a procedure for handling ordinal variables is proposed. Using these results, a monotonically convergent algorithm is constructed for FACTALS of any mixture of variables.The authors are obliged to Jos ten Berge for stimulating comments on an earlier version of this paper. The research of H. A. L. Kiers has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. The research of Y. Takane has been supported by the Natural Sciences and Engineering Research Council of Canada, grant number A6394, and by the McGill-IBM Cooperative Grant.     

预测视角下双因子模型与高阶因子模型的一般性模拟比较

高阶因子模型本质上是一种特殊的双因子模型, 应用中却常被当做双因子模型的竞争模型。已有研究以满足比例约束的双因子模型(此时等价于一个高阶因子模型)为真实测量模型产生模拟数据, 比较了用双因子模型和高阶因子模型作为测量模型的预测效果。本文使用不满足比例约束的双因子模型(此时不与任何高阶因子模型等价)为真实测量模型产生模拟数据进行比较, 所得结果与满足比例约束的双因子模型的结果有很大差别, 双因子模型结构系数的相对偏差较小、检验力较高, 但第Ⅰ类错误率略高。结论是, 在比例约束条件成立时可以使用高阶因子模型, 否则, 从统计角度看, 一般情况下使用双因子模型进行预测比较好。     

Weighted least squares fitting using ordinary least squares algorithms

A general approach for fitting a model to a data matrix by weighted least squares (WLS) is studied. This approach consists of iteratively performing (steps of) existing algorithms for ordinary least squares (OLS) fitting of the same model. The approach is based on minimizing a function that majorizes the WLS loss function. The generality of the approach implies that, for every model for which an OLS fitting algorithm is available, the present approach yields a WLS fitting algorithm. In the special case where the WLS weight matrix is binary, the approach reduces to missing data imputation.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author.     

Similarity and multidimensional ratio estimation with simultaneous qualitative and quantitative variation

Abstract.— Previous work on content models of multidimensional similarity have mostly been concerned with the case of only qualitative variation. In this study a set of emotion words that varied both in content and intensity was used as stimuli. Subjects gave multidimensional ratio estimations, similarity estimations and also unidimensional ratings of intensity. Both individual data and group data were collected. Five different previously suggested content models of similarity failed to fit these data that showed large individual idiosyncratic variability. It is pointed out that previous work, where some of the models fitted rather well, was concerned with cases that were largely insensitive to deviations fram the models.     

Constraints on Constraints: Surveying the Epigenetic Landscape

It is now relatively commonplace to advocate the need for some sorts of constraints on learning and knowledge acquisition. The critical issues to cognitive science concern the sorts of constraints that are able to best model various phenomena of learning and development. Four types of constraints on learning are proposed to be used as an interpretative framework within which to: 1. Better understand the nature of current research: 2. Allow the exploration of alternative models of learning related phenomena, and 3. See more clearly needs for further research. Superficially similar learning phenomena can be modeled by very different configurations of underlying constraints with strong implications for the sorts of representational states that are involved. Each of the five papers in this issue (Brown, Gelman, Markman, Newport, and Spelke) is considered in terms of the configuration of constraints after which each author intends to model their phenomena and in terms of alternate configurations. The papers are construed as illustrating a diverse set of models of how constraints might guide learning, and while the evidence generally favors the configurations suggested by the authors, in each case alternative models are possible and motivate quite specific future research questions. More broadly, it is suggested that asking detailed questions about the sorts of Constraints types that could potentially model complex cases of natural knowledge acquisition helps motivate fundamental questions about learning and the nature of knowledge and that the five papers in this issue are superb examples of how adopting this kind of perspective has been fruitful research orientation.     

Analysis of contingency tables by ideal point discriminant analysis

Cross-classified data are frequently encountered in behavioral and social science research. The loglinear model and dual scaling (correspondence analysis) are two representative methods of analyzing such data. An alternative method, based on ideal point discriminant analysis (DA), is proposed for analysis of contingency tables, which in a certain sense encompasses the two existing methods. A variety of interesting structures can be imposed on rows and columns of the tables through manipulations of predictor variables and/or as direct constraints on model parameters. This, along with maximum likelihood estimation of the model parameters, allows interesting model comparisons. This is illustrated by the analysis of several data sets.Presented as the Presidential Address to the Psychometric Society's Annual and European Meetings, June, 1987. Preparation of this paper was supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada. Thanks are due to Chikio Hayashi of University of the Air in Japan for providing the ISM data, and to Jim Ramsay and Ivo Molenaar for their helpful comments on an earlier draft of this paper.     

Multivariate thurstonian models

The recent development of probabilistic Thurstonian choice models for analyzing preference data has been motivated by the need to describe both inter- and intra-individual difference, the multidimensional nature of choice objects, and the effects of similarity and comparability among choice objects. A common feature of these models is that they focus on asingle preference judgment. It is customary, however, to ask subjects not only for an overall preference judgment but also for additional paired comparison responses regarding specific attributes. This paper proposes a generalization of Thurstonian probabilistic choice models for analyzing both multiple preference responses and their relationships. The approach is illustrated by modeling data from two multivariate preference experiments.I am grateful to James, Austin, Ingo Böckenholt, and anonymous referees for helpful comments on this research. This paper is based on research presented at the Meeting of the Society for the Multivariate Analysis in the Behavioral Sciences, Groningen, The Netherlands, December 1988.     

Analysis of pairwise preference data using integrated B-splines

Pairwise preference data are represented as a monotone integral transformation of difference on the underlying stimulus-object or utility scale. The class of monotone transformations considered is that in which the kernel of the integral is a linear combination of B-splines. Two types of data are analyzed: binary and continuous. The parameters of the transformation and the underlying scale values or utilities are estimated by maximum likelihood with inequality constraints on the transformation parameters. Various hypothesis tests and interval estimates are developed. Examples of artificial and real data are presented.This investigation was supported in part by a research grant from the Natural Sciences and Engineering Research Council Canada.     

A second generation nonlinear factor analysis

Nonlinear common factor models with polynomial regression functions, including interaction terms, are fitted by simultaneously estimating the factor loadings and common factor scores, using maximum-likelihood-ratio and ordinary-least-squares methods. A Monte Carlo study gives support to a conjecture about the form of the distribution of the likelihood-ratio criterion.The research reported in this paper was partly supported by Natural Sciences and Engineering Research Grant No. A6346.     

The analysis of multitrait-multimethod matrices via constrained components analysis

Multitrait-Multimethod (MTMM) matrices are often analyzed by means of confirmatory factor analysis (CFA). However, fitting MTMM models often leads to improper solutions, or non-convergence. In an attempt to overcome these problems, various alternative CFA models have been proposed, but with none of these the problem of finding improper solutions was solved completely. In the present paper, an approach is proposed where improper solutions are ruled out altogether and convergence is guaranteed. The approach is based on constrained variants of components analysis (CA). Besides the fact that these methods do not give improper solutions, they have the advantage that they provide component scores which can later on be used to relate the components to external variables. The new methods are illustrated by means of simulated data, as well as empirical data sets.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the first author. The authors are obliged to three anonymous reviewers and an associate editor for constructive suggestions on the first version of this paper.     

Multidimensional successive categories scaling: A maximum likelihood method

A single-step maximum likelihood estimation procedure is developed for multidimensional scaling of dissimilarity data measured on rating scales. The procedure can fit the euclidian distance model to the data under various assumptions about category widths and under two distributional assumptions. The scoring algorithm for parameter estimation has been developed and implemented in the form of a computer program. Practical uses of the method are demonstrated with an emphasis on various advantages of the method as a statistical procedure.The research reported here was partly supported by Grant A6394 to the author by Natural Sciences and Engineering Research Council of Canada. Portions of this research were presented at the Psychometric Society meeting in Uppsala, Sweden, in June, 1978. MAXSCAL-2.1, a program to perform the computations discussed in this paper may be obtained from the author. Thanks are due to Jim Ramsay for his helpful comments.     

Minimization of a class of matrix trace functions by means of refined majorization

A procedure is described for minimizing a class of matrix trace functions. The procedure is a refinement of an earlier procedure for minimizing the class of matrix trace functions using majorization. It contains a recently proposed algorithm by Koschat and Swayne for weighted Procrustes rotation as a special case. A number of trial analyses demonstrate that the refined majorization procedure is more efficient than the earlier majorization-based procedure. The research of H. A. L. Kiers has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences.     

Practice and training in bimanual coordination tasks: strategies and constraints.

There are a number of structural constraints on the coordination of the two hands. Understanding bimanual coordination involves not only identification of these constraints but also how they can be overcome or modulated. The performance of polyrhythms has been used to examine these issues. In this article the constraints on coordination are outlined and research on the acquisition of polyrhythms is reviewed. It is suggested that the constraints on the production of multifrequency ratios are overcome by integrating the timing of the two hands. Oscillator models of coordination are then considered and a two-process model of motor timing is suggested.     

On quantifying different types of categorical data

In quantifying categorical data, constraints play an important role in characterizing the outcome. In the Guttman-type quantification of contingency tables and multiple-choice data (incidence data), the trivial solution due to the marginal constraints is typically removed before quantification; this removal, however, has the effect of distorting the shape of the total space. Awareness of this is important for the interpretation of the quantified outcome. The present study provides some relevant formulas for those cases that are affected by the trivial solution and those cases that are not. The characterization of the total space used by the Guttman-type quantification and pertinent discussion are presented.This study was supported by a grant from The Natural Sciences and Engineering Research Council of Canada to S. Nishisato.     

Restricted multidimensional scaling models for asymmetric proximities

Restricted multidimensional scaling models [Bentler & Weeks, 1978] allowing constraints on parameters, are extended to the case of asymmetric data. Separate functions are used to model the symmetric and antisymmetric parts of the data. The approach is also extended to the case in which data are presumed to be linearly related to squared distances. Examples of several models are provided, using journal citation data. Possible extensions of the models are considered. This research was supported in part by USPHS Grant 0A01070, P. M. Bentler, principal investigator, and NIMH Grant MH-24819, E. J. Anthony and J. Worland, principal investigators. The authors wish to thank E. W. Holman and several anonymous reviewers for their valuable suggestions concerning this research.