Trend in correlated proportions

A random effects probit model is developed for the case in which the same units are sampled repeatedly at each level of an independent variable. Because the observed proportions may be correlated under these conditions, estimating their trend with respect to the independent variable is no longer a standard problem for probit, logit or loglinear analysis. Using a qualitative analogue of a random regressions model, we employ instead marginal maximum likelihood to estimate the average latent trend line. Likelihood ratio tests of the hypothesis of no trend in the average line, and the hypothesis of no differences in average trend lines between experimental treatments, are proposed. We illustrate the model both with simulated data and with observed data from a clinical experiment in which psychiatric patients on two drug therapies are rated on five occasions for the presence or absence of symptoms.Supported by a grant from the MacArthur Foundation and National Science Foundation Grant BNS85-11774.The authors are indebted to James Heckman for calling our attention to the Clark algorithm.     

The rank of reduced dispersion matrices

Psychometricians working in factor analysis and econometricians working in regression with measurement error in all variables are both interested in the rank of dispersion matrices under variation of the diagonal elements. Psychometricians concentrate on cases in which low rank can be attained, preferably rank one, the Spearman case. Econometricians cocentrate on cases in which the rank cannot be reduced below the number of variables minus one, the Frisch case. In this paper we give an extensive historial discussion of both fields, we prove the two key results in a more satisfactory and uniform way, we point out various small errors and misunderstandings, and we present a methodological comparison of factor analysis and regression on the basis of our results.Financial support by the Netherlands Organization for the Advancement of Pure Research (ZWO) is gratefully acknowledged.     

Application of model-selection criteria to some problems in multivariate analysis

A review of model-selection criteria is presented, with a view toward showing their similarities. It is suggested that some problems treated by sequences of hypothesis tests may be more expeditiously treated by the application of model-selection criteria. Consideration is given to application of model-selection criteria to some problems of multivariate analysis, especially the clustering of variables, factor analysis and, more generally, describing a complex of variables.     

Some properties of the correlation matrix of dichotomous guttman items

Both the elements and the eigenvalues of the Pearson correlation matrix of dichotomous Guttman-scalable items can be expressed as simple functions of the number of items if the score distribution is uniform and there is an equal number of items at each difficulty level. Even when these special conditions do not hold, the correlations can often be expressed in a simple form by assuming a particular score distribution.The author thanks Neil Dorans, Bob Mislevy, Ledyard R Tucker, and Howard Wainer for their comments.     

The fallacy of fallacies

Several of the so-called fallacies in Aristotle are not in fact mistaken inference-types, but mistakes or breaches of rules in the questioning games which were practiced in the Academy and in the Lyceum. Hence the entire Aristotelian theory of fallacies ought to be studied by reference to the author's interrogative model of inquiry, based on his theory of questions and answers, rather than as a part of the theory of inference. Most of the fallacies mentioned by Aristotle can in fact be diagnosed by means of the interrogative model, including petitio principii, multiple questions, babbling, etc., and so can Aristotle's alleged anticipation of the fallacy of argumentum ad hominem. The entire Aristotelian conception of inquiry is an interrogative one. Deductive conclusions caught Aristotle's attention in the form of answers that every rational interlocutor must give, assuming only his own earlier answers. Several features of Aristotle's methodology can be understood by means of the interrogative model, including the role of endoxa in it. Theoretically, there is also considerable leeway as to whether fallacies are conceived of as mistakes in questioning or as breaches of the rules that govern questioning games.     

Parent-perceived child behavior problems: Comparison of normative and clinical factors of the Adelaide-Conners Parent Rating Scale

The Adelaide-Conners Parent Rating Scale (APRS), an instrument developed by studying a large, representative group of schoolchildren, was used with a group of psychiatry attenders. Multimethod factor analysis found satisfactory agreement between the factor structures of the clinical and the normative groups. The patterns of scores on the 12 APRS scales were also compared. Two higher-order factors (Conflict with the Environment and Conflict within the Self) were identified in the clinical sample as previously found in the normative group. Comparison of the factor solutions with previous empirical efforts to identify parent-perceived patterns of child behavior disorder showed that the APRS compares well with other instruments and supports the strategy of proceeding from the study of normative populations to the study of clinically defined groups.This work was supported in part by a grant from the Department of Health, Canberra.     

A stochastic growth model applied to repeated tests of academic knowledge

In the course of the medical program at the University of Limburg, students complete a total of 24 progress tests, consisting of items drawn from a constant itembank. A model is presented for the growth of knowledge reflected by these results. The Rasch model is used as a starting point, but both ability and difficulty parameters are taken to be random, and moreover the logistic distribution is replaced by the normal. Both individual and group abilities are estimated and explained through simple linear regression. Application to real data shows that the model fits very well.     

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.     

A multidimensional scaling model for the size-weight illusion

The kinds of individual differences in perceptions permitted by the weighted euclidean model for multidimensional scaling (e.g., INDSCAL) are much more restricted than those allowed by Tucker's Three-mode Multidimensional Scaling (TMMDS) model or Carroll's Idiosyncratic Scaling (IDIOSCAL) model. Although, in some situations the more general models would seem desirable, investigators have been reluctant to use them because they are subject to transformational indeterminacies which complicate interpretation. In this article, we show how these indeterminacies can be removed by constructing specific models of the phenomenon under investigation. As an example of this approach, a model of the size-weight illusion is developed and applied to data from two experiments, with highly meaningful results. The same data are also analyzed using INDSCAL. Of the two solutions, only the one obtained by using the size-weight model allows examination of individual differences in the strength of the illusion; INDSCAL can not represent such differences. In this sample, however, individual differences in illusion strength turn out to be minor. Hence the INDSCAL solution, while less informative than the size-weight solution, is nonetheless easily interpretable.This paper is based on the first author's doctoral dissertation at the Department of Psychology, University of Illinois at Urbana-Champaign. The aid of Professor Ledyard R Tucker is gratefully acknowledged.     

Factor analysis by instrumental variables methods

Three alternative estimation procedures for factor analysis based on the instrumental variables method are presented. These procedures are justified by the method of least squares. Formulas for asymptotic standard errors of factor loadings are derived. The procedures are empirically compared to the method of maximum likelihood. The conclusion, based on the data used in this study, is that two of the procedures seem to work well.