An incomplete data approach to the analysis of covariance structures

In this paper, linear structural equation models with latent variables are considered. It is shown how many common models arise from incomplete observation of a relatively simple system. Subclasses of models with conditional independence interpretations are also discussed. Using an incomplete data point of view, the relationships between the incomplete and complete data likelihoods, assuming normality, are highlighted. For computing maximum likelihood estimates, the EM algorithm and alternatives are surveyed. For the alternative algorithms, simplified expressions for computing function values and derivatives are given. Likelihood ratio tests based on complete and incomplete data are related, and an example on using their relationship to improve the fit of a model is given.This research forms part of the author's doctoral thesis and was supported by a Commonwealth Postgraduate Research Award. The author also wishes to acknowledge the support of CSIRO during the preparation of this paper and the referees' comments which led to substantial improvements.     

The structures of the WISC-R subtests: a comparison of the IQ-profiles of reading impaired and autistic subjects

The present paper is an analysis of the WISC-R test profiles of reading impaired subjects and autistic subjects. It is argued that well-known classification systems such as Bannatyne's categories (1974) and Kaufman's factors (1975) cannot explain differences in the peaks and troughs across the two populations. A new classification system is then developed. The 11 different WISC-R subtests are characterised in terms of a combination of three modes of cognitive functioning: a knowledge mode (declarative-procedural), a processing mode (transformation-preservation of information), and a verbal-nonverbal mode (verbal-nonverbal-processing). When reanalysed within this reference frame, the IQ profiles of reading impaired subjects (N = 114) and autistic subjects (data from 12 studies reviewed by Happé, 1994) were similar regarding differences in the verbal-nonverbal dimension. However, the IQ profiles showed an opposite pattern in the declarative-procedural dimension, and an opposite pattern in the preservation-transformation dimension. Thus, the new classification system allows one to interpret the IQ-profiles of both groups within a single and coherent reference frame.     

Circumplex models for correlation matrices

Structural models that yield circumplex inequality patterns for the elements of correlation matrices are reviewed. Particular attention is given to a stochastic process defined on the circle proposed by T. W. Anderson. It is shown that the Anderson circumplex contains the Markov Process model for a simplex as a limiting case when a parameter tends to infinity.Anderson's model is intended for correlation matrices with positive elements. A replacement for Anderson's correlation function that permits negative correlations is suggested. It is shown that the resulting model may be reparametrzed as a factor analysis model with nonlinear constraints on the factor loadings. An unrestricted factor analysis, followed by an appropriate rotation, is employed to obtain parameter estimates. These estimates may be used as initial approximations in an iterative procedure to obtain minimum discrepancy estimates.Practical applications are reported.Presented as the 1992 Psychometric Society Presidential Address. I am greatly indebted to Stephen Du Toit for help in the development of the computer program employed here. Part of this research was carried out at the University of South Africa and at the Institute for Statistical Research of the South African Human Sciences Research Council.     

A simple Gauss-Newton procedure for covariance structure analysis with high-level computer languages

An implementation of the Gauss-Newton algorithm for the analysis of covariance structures that is specifically adapted for high-level computer languages is reviewed. With this procedure one need only describe the structural form of the population covariance matrix, and provide a sample covariance matrix and initial values for the parameters. The gradient and approximate Hessian, which vary from model to model, are computed numerically. Using this approach, the entire method can be operationalized in a comparatively small program. A large class of models can be estimated, including many that utilize functional relationships among the parameters that are not possible in most available computer programs. Some examples are provided to illustrate how the algorithm can be used.We are grateful to M. W. Browne and S. H. C. du Toit for many invaluable discussions about these computing ideas. Thanks also to Scott Chaiken for providing the data in the first example. They were collected as part of the U.S. Air Force's Learning Ability Measurement Project (LAMP), sponsored by the Air Force Office of Scientific Research (AFOSR) and the Human Resource Directorate of the Armstrong Laboratory (AL/HRM).     

The effects of strong belief structures on information-processing evaluations and choice

This study examined the influence of the strength of belief structures on selected aspects of the decision-making process. To examine these issues, a business-acquisition decision scenario was studied in an experiment. Subjects played the role of a CEO of an electronics firm and were asked to evaluate the attractiveness of six potential acquisition candidates and to rate various aspects of the associated decision process. We presented half the subjects with information that the belief structure of their organization was extreme, agreed upon, and clear. The other half was presented with information that there was disagreement about the belief structure and that it was more ambiguous and less extreme. The results clearly showed that the decision process is different for people who were presented with an agreed-upon, extreme, and tightly constructed belief structure when compared to those who received information reflecting a weak and loosely constructed belief structure. A strong belief structure resulted in less positive evaluations, information requested, and money allocated to explore incompatible acquisition candidates (and vice versa for a highly compatible candidate) when compared to subjects using a weak belief structure. In addition, subjects in the strong-belief condition reported that their decision process would be characterized by less doubt, less time, less difficulty, and less conflict compared to subjects in the weak-belief structure condition. Implications for both decision theory and practical decision processes are discussed.     

A three-stage estimation procedure for structural equation models with polytomous variables

This paper is concerned with the analysis of structural equation models with polytomous variables. A computationally efficient three-stage estimator of the thresholds and the covariance structure parameters, based on partition maximum likelihood and generalized least squares estimation, is proposed. An example is presented to illustrate the method.This research was supported in part by a research grant DA01070 from the U.S. Public Health Service. The production assistance of Julie Speckart is gratefully acknowledged.     

Spatial,non-spatial and hybrid models for scaling

In this paper, hierarchical and non-hierarchical tree structures are proposed as models of similarity data. Trees are viewed as intermediate between multidimensional scaling and simple clustering. Procedures are discussed for fitting both types of trees to data. The concept of multiple tree structures shows great promise for analyzing more complex data. Hybrid models in which multiple trees and other discrete structures are combined with continuous dimensions are discussed. Examples of the use of multiple tree structures and hybrid models are given. Extensions to the analysis of individual differences are suggested.1976 Psychometric Society Presidential Address.While people too numerous to list here have contributed ideas, inspiration, and other help, I particularly wish to acknowledge the contributions of Sandra Pruzansky, without whom this paper could not have been written. I would also like to acknowledge the past contributions of my long-time colleague Jih-Jie Chang, without whose help I probably would not have beenasked to write it.     

A least squares algorithm for fitting additive trees to proximity data

A least squares algorithm for fitting additive trees to proximity data is described. The algorithm uses a penalty function to enforce the four point condition on the estimated path length distances. The algorithm is evaluated in a small Monte Carlo study. Finally, an illustrative application is presented.The author is Aspirant of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek. The author is indebted to Professor J. Hoste for providing computer facilities at the Institute of Nuclear Sciences at Ghent.     

The Logic of Pragmatic Truth

The mathematical concept of pragmatic truth, first introduced in Mikenberg, da Costa and Chuaqui (1986), has received in the last few years several applications in logic and the philosophy of science. In this paper, we study the logic of pragmatic truth, and show that there are important connections between this logic, modal logic and, in particular, Jaskowski's discussive logic. In order to do so, two systems are put forward so that the notions of pragmatic validity and pragmatic truth can be accommodated. One of the main results of this paper is that the logic of pragmatic truth is paraconsistent. The philosophical import of this result, which justifies the application of pragmatic truth to inconsistent settings, is also discussed.