The nonconvergence of factals: A nonmetric common factor analysis

FACTALS is a nonmetric common factor analysis model for multivariate data whose variables may be nominal, ordinal or interval. In FACTALS an Alternating Least Squares algorithm is utilized which is claimed to be monotonically convergent.In this paper it is shown that this algorithm is based upon an erroneous assumption, namely that the least squares loss function (which is in this case a nonscale free loss function) can be transformed into a scalefree loss function. A consequence of this is that monotonical convergence of the algorithm can not be guaranteed.     

Constrained DEDICOM

The DEDICOM method for the analysis of asymmetric data tables gives representations that are identified only up to a nonsingular transformation. To identify solutions it is proposed to impose subspace constraints on the stimulus coefficients. Most attention is paid to the case where different subspace constraints are imposed on different dimensions. The procedures are discussed both for the case where the complete table is fitted, and for cases where only offdiagonal elements are fitted. The case where the data table is skew-symmetric is treated separately as well.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. The authors are obliged to Richard A. Harshman for helpful comments on an earlier version.     

Quantitative analysis of qualitative data

This paper presents an overview of an approach to the quantitative analysis of qualitative data with theoretical and methodological explanations of the two cornerstones of the approach, Alternating Least Squares and Optimal Scaling. Using these two principles, my colleagues and I have extended a variety of analysis procedures originally proposed for quantitative (interval or ratio) data to qualitative (nominal or ordinal) data, including additivity analysis and analysis of variance; multiple and canonical regression; principal components; common factor and three mode factor analysis; and multidimensional scaling. The approach has two advantages: (a) If a least squares procedure is known for analyzing quantitative data, it can be extended to qualitative data; and (b) the resulting algorithm will be convergent. Three completely worked through examples of the additivity analysis procedure and the steps involved in the regression procedures are presented.Presented as the Presidential Address to the Psychometric Society's Annual meeting, May, 1981. I wish to express my deep appreciation to Jan de Leeuw and Yoshio Takane. Our team effort was essential for the developments reported in this paper. Without this effort the present paper would not exist. Portions of this paper appear in Lantermann, E. D. & Feger, H. (Eds.)Similarity and Choice, Hans Huber, Vienna, 1980. The present paper benefits greatly from a set of detailed comments made by Joseph Kruskal on the earlier paper.     

Constrained latent class analysis: Simultaneous classification and scaling of discrete choice data

A reparameterization of a latent class model is presented to simultaneously classify and scale nominal and ordered categorical choice data. Latent class-specific probabilities are constrained to be equal to the preference probabilities from a probabilistic ideal-point or vector model that yields a graphical, multidimensional representation of the classification results. In addition, background variables can be incorporated as an aid to interpreting the latent class-specific response probabilities. The analyses of synthetic and real data sets illustrate the proposed method.The authors thank Yosiho Takane, the editor and referees for their valuable suggestions. Authors are listed in reverse alphabetical order.     

An improved solution for factals: A nonmetric common factor analysis

In FACTALS an alternating least squares algorithm is utilized. Mooijaart (1984) has shown that this algorithm is based upon an erroneous assumption. This paper gives a proper solution for the loss function used in FACTALS.     

Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features

A new procedure is discussed which fits either the weighted or simple Euclidian model to data that may (a) be defined at either the nominal, ordinal, interval or ratio levels of measurement; (b) have missing observations; (c) be symmetric or asymmetric; (d) be conditional or unconditional; (e) be replicated or unreplicated; and (f) be continuous or discrete. Various special cases of the procedure include the most commonly used individual differences multidimensional scaling models, the familiar nonmetric multidimensional scaling model, and several other previously undiscussed variants.The procedure optimizes the fit of the model directly to the data (not to scalar products determined from the data) by an alternating least squares procedure which is convergent, very quick, and relatively free from local minimum problems.The procedure is evaluated via both Monte Carlo and empirical data. It is found to be robust in the face of measurement error, capable of recovering the true underlying configuration in the Monte Carlo situation, and capable of obtaining structures equivalent to those obtained by other less general procedures in the empirical situation.This project was supported in part by Research Grant No. MH10006 and Research Grant No. MH26504, awarded by the National Institute of Mental Health, DHEW. We wish to thank Robert F. Baker, J. Douglas Carroll, Joseph Kruskal, and Amnon Rapoport for comments on an earlier draft of this paper. Portions of the research reported here were presented to the spring meeting of the Psychometric Society, 1975. ALSCAL, a program to perform the computations discussed in this paper, may be obtained from any of the authors.Jan de Leeuw is currently at Datatheorie, Central Rekeninstituut, Wassenaarseweg 80, Leiden, The Netherlands. Yoshio Takane can be reached at the Department of Psychology, University of Tokyo, Tokyo, Japan.     

A Unified Framework for the Comparison of Treatments with Ordinal Responses

Different latent variable models have been used to analyze ordinal categorical data which can be conceptualized as manifestations of an unobserved continuous variable. In this paper, we propose a unified framework based on a general latent variable model for the comparison of treatments with ordinal responses. The latent variable model is built upon the location-scale family and is rich enough to include many important existing models for analyzing ordinal categorical variables, including the proportional odds model, the ordered probit-type model, and the proportional hazards model. A flexible estimation procedure is proposed for the identification and estimation of the general latent variable model, which allows for the location and scale parameters to be freely estimated. The framework advances the existing methods by enabling many other popular models for analyzing continuous variables to be used to analyze ordinal categorical data, thus allowing for important statistical inferences such as location and/or dispersion comparisons among treatments to be conveniently drawn. Analysis on real data sets is used to illustrate the proposed methods.     

Additive structure in qualitative data: An alternating least squares method with optimal scaling features

A method is developed to investigate the additive structure of data that (a) may be measured at the nominal, ordinal or cardinal levels, (b) may be obtained from either a discrete or continuous source, (c) may have known degrees of imprecision, or (d) may be obtained in unbalanced designs. The method also permits experimental variables to be measured at the ordinal level. It is shown that the method is convergent, and includes several previously proposed methods as special cases. Both Monte Carlo and empirical evaluations indicate that the method is robust.This research was supported in part by grant MH-10006 from the National Institute of Mental Health to the Psychometric Laboratory of the University of North Carolina. We wish to thank Thomas S. Wallsten for comments on an earlier draft of this paper. Copies of the paper and of ADDALS, a program to perform the analyses discussed herein, may be obtained from the second author.     

Homogeneity analysis withk sets of variables: An alternating least squares method with optimal scaling features

Homogeneity analysis, or multiple correspondence analysis, is usually applied tok separate variables. In this paper we apply it to sets of variables by using sums within sets. The resulting technique is called OVERALS. It uses the notion of optimal scaling, with transformations that can be multiple or single. The single transformations consist of three types: nominal, ordinal, and numerical. The corresponding OVERALS computer program minimizes a least squares loss function by using an alternating least squares algorithm. Many existing linear and nonlinear multivariate analysis techniques are shown to be special cases of OVERALS. An application to data from an epidemiological survey is presented.This research was partly supported by SWOV (Institute for Road Safety Research) in Leidschendam, The Netherlands.     

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.     

Simulating Ordinal Data

The increasing use of ordinal variables in different fields has led to the introduction of new statistical methods for their analysis. The performance of these methods needs to be investigated under a number of experimental conditions. Procedures to simulate from ordinal variables are then required. In this article, we deal with simulation from multivariate ordinal random variables. We propose a new procedure for generating samples from ordinal random variables with a prespecified correlation matrix and marginal distributions. Its features are examined and compared with those of its main competitors. A software implementation in R is also provided along with examples of its application.     

Estimating the polychoric correlation from misclassified data

Many variables that are used in social and behavioural science research are ordinal categorical or polytomous variables. When more than one polytomous variable is involved in an analysis, observations are classified in a contingency table, and a commonly used statistic for describing the association between two variables is the polychoric correlation. This paper investigates the estimation of the polychoric correlation when the data set consists of misclassified observations. Two approaches for estimating the polychoric correlation have been developed. One assumes that the probabilities in relation to misclassification are known, and the other uses a double sampling scheme to obtain information on misclassification. A parameter estimation procedure is developed, and statistical properties for the estimates are discussed. The practicability and applicability of the proposed approaches are illustrated by analysing data sets that are based on real and generated data. Excel programmes with visual basic for application (VBA) have been developed to compute the estimate of the polychoric correlation and its standard error. The use of the structural equation modelling programme Mx to find parameter estimates in the double sampling scheme is discussed.     

Maximum likelihood additivity analysis

A maximum likelihood estimation procedure was developed to fit unweighted and weighted additive models to conjoint data obtained by the categorical rating, the pair comparison or the directional ranking method. The scoring algorithm used to fit the models was found to be both reliable and efficient, and the program MAXADD is capable of handling up to 300 parameters to be estimated. Practical uses of the procedure are reported to demonstrate various advantages of the procedure as a statistical method.The research reported here was supported by Grant A6394 to the author from the Natural Sciences and Engineering Research Council of Canada. Portions of this research were presented at the Psychometric Society meeting in Iowa City, Iowa, in May, 1980.Thanks are due to Jim Ramsay, Justine Sergent and anonymous reviewers for their helpful comments.Two MAXADD programs which perform the computations discussed in this paper may be obtained from the author.     

Model comparison of nonlinear structural equation models with fixed covariates

Recently, it has been recognized that the commonly used linear structural equation model is inadequate to deal with some complicated substantive theory. A new nonlinear structural equation model with fixed covariates is proposed in this article. A procedure, which utilizes the powerful path sampling for computing the Bayes factor, is developed for model comparison. In the implementation, the required random observations are simulated via a hybrid algorithm that combines the Gibbs sampler and the Metropolis-Hastings algorithm. It is shown that the proposed procedure is efficient and flexible; and it produces Bayesian estimates of the parameters, latent variables, and their highest posterior density intervals as by-products. Empirical performances of the proposed procedure such as sensitivity to prior inputs are illustrated by a simulation study and a real example.This research is fully supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 4346/01H). The authors are thankful to the Editor, the Associate Editor, and anonymous reviewers for valuable comments which improve the paper significantly, and grateful to ICPSR and the relevant funding agency for allowing use of the data in the example. The assistance of Michael K.H. Leung and Esther L.S. Tam is gratefully acknowledged.     

The principal components of mixed measurement level multivariate data: An alternating least squares method with optimal scaling features

A method is discussed which extends principal components analysis to the situation where the variables may be measured at a variety of scale levels (nominal, ordinal or interval), and where they may be either continuous or discrete. There are no restrictions on the mix of measurement characteristics and there may be any pattern of missing observations. The method scales the observations on each variable within the restrictions imposed by the variable's measurement characteristics, so that the deviation from the principal components model for a specified number of components is minimized in the least squares sense. An alternating least squares algorithm is discussed. An illustrative example is given.Copies of this paper and of the associated PRINCIPALS program may be obtained by writing to Forrest W. Young, Psychometric Laboratory, Davie Hall 013-A, Chapel Hill, NC 27514.     

Generalized structured component analysis

We propose an alternative method to partial least squares for path analysis with components, called generalized structured component analysis. The proposed method replaces factors by exact linear combinations of observed variables. It employs a well-defined least squares criterion to estimate model parameters. As a result, the proposed method avoids the principal limitation of partial least squares (i.e., the lack of a global optimization procedure) while fully retaining all the advantages of partial least squares (e.g., less restricted distributional assumptions and no improper solutions). The method is also versatile enough to capture complex relationships among variables, including higher-order components and multi-group comparisons. A straightforward estimation algorithm is developed to minimize the criterion.The work reported in this paper was supported by Grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the second author. We wish to thank Richard Bagozzi for permitting us to use his organizational identification data and Wynne Chin for providing PLS-Graph 3.0.     

Towards better computational models of the balance scale task: a reply to Shultz and Takane

In contrast to Shultz and Takane [Shultz, T.R., & Takane, Y. (2007). Rule following and rule use in the balance-scale task. Cognition, in press, doi:10.1016/j.cognition.2006.12.004.] we do not accept that the traditional Rule Assessment Method (RAM) of scoring responses on the balance scale task has advantages over latent class analysis (LCA): RAM is similar to a very restricted form of LCA. The apparent shortcomings of LCA are also less severe than they suggest. Via new simulations we show that LCA detects small classes reliably. We also counter their concerns regarding the torque difference effect and we underline the problems connectionist models have with correctly responding to balance items. Despite these differences in opinion we agree with Shultz and Takane on the possible avenues for future research.     

Bayesian model comparison of nonlinear structural equation models with missing continuous and ordinal categorical data

Missing data are very common in behavioural and psychological research. In this paper, we develop a Bayesian approach in the context of a general nonlinear structural equation model with missing continuous and ordinal categorical data. In the development, the missing data are treated as latent quantities, and provision for the incompleteness of the data is made by a hybrid algorithm that combines the Gibbs sampler and the Metropolis‐Hastings algorithm. We show by means of a simulation study that the Bayesian estimates are accurate. A Bayesian model comparison procedure based on the Bayes factor and path sampling is proposed. The required observations from the posterior distribution for computing the Bayes factor are simulated by the hybrid algorithm in Bayesian estimation. Our simulation results indicate that the correct model is selected more frequently when the incomplete records are used in the analysis than when they are ignored. The methodology is further illustrated with a real data set from a study concerned with an AIDS preventative intervention for Filipina sex workers.