Optimal scaling of paired comparison and rank order data: An alternative to guttman's formulation

A formulation, which is different from Guttman's is presented. The two formulations are both called the optimal scaling approach, and are proven to provide identical scale values. The proposed formulation has at least two advantages over Guttman's. Namely, (i) the former serves to clarify close relations of the optimal scaling approach to those of Slater and the vector model of preferential choice, and (ii) in addition to the stimulus scale values, it provides scores for the subjects, which indicate the degrees of response consistency (transitivity), relative to the optimum solution. The method is assumption-free and capable of multidimensional analysis.This study was partly supported by the National Research Council Grant (No. A4581) to S. Nishisato. The author is indebted to Dr. Bert F. Green, Jr., Mr. Tomoichi Ishizuka, and anonymous reviewers for their valuable comments on an earlier draft.     

Approximate interval estimation for a certain intraclass correlation coefficient

When the raters participating in a reliability study are a random sample from a larger population of raters, inferences about the intraclass correlation coefficient must be based on the three mean squares from the analysis of variance table summarizing the results: between subjects, between raters, and error. An approximate confidence interval for the parameter is presented as a function of these three mean squares.Dr. Fleiss is also with the Biometrics Research Unit of the New York State Psychiatric Institute. This work was supported in part by grant DE 04068 from the National Institute of Dental Research.     

Contributions to factor analysis of dichotomous variables

A new method is proposed for the factor analysis of dichotomous variables. Similar to the method of Christoffersson this uses information from the first and second order proportions to fit a multiple factor model. Through a transformation into a new set of sample characteristics, the estimation is considerably simplified. A generalized least-squares estimator is proposed, which asymptotically is as efficient as the corresponding estimator of Christoffersson, but which demands less computing time.This research was supported by the Bank of Sweden Tercentenary Foundation under project Structural Equation Models in the Social Sciences, project director Karl G. Jöreskog.     

Statistical inference based on ranks

This paper develops a unified approach, based on ranks, to the statistical analysis of data arising from complex experimental designs. In this way we answer a major objection to the use of rank procedures as a major methodology in data analysis. We show that the rank procedures, including testing, estimation and multiple comparisons, are generated in a natural way from a robust measure of scale. The rank methods closely parallel the familiar methods of least squares, so that estimates and tests have natural interpretations.This research was supported in part by grant MCS76-07292 from the National Science Foundation.     

A maximum likelihood approach to the “test of inclusion”

The method of scoring is used to obtain maximum likelihood estimates of the parameters in the White and Clark learning hierarchy validation model. From the estimate of the proportion of the population possessing only the superordinate skill in a pair of hierarchical skills, and its variance, the hypothesis of inclusion is tested. An illustrative example of the procedure is given.This study is based on doctoral dissertation submitted to the University of Ottawa. The author is indebted to Martin Cooper, Marvin Boss and Jean-Paul Dionne for their assistance.     

Optimal item difficulty for the three-parameter normal ogive response model

In tailored testing, it is important to determine the optimal difficulty of the next item to present to the examinee. This paper shows that the difference that maximizes information for the three-parameter normal ogive response model is approximately 1.7 times the optimal difference –b for the three-parameter logistic model. Under the normal model, calculation of the optimal difficulty for minimizing the Bayes risk is equivalent to maximizing an associated information function.The views expressed herein, are those of the author and do not necessarily reflect those of the Department of the Navy.     

Estimating the parameters of the latent population distribution

Under consideration is a test battery of binary items. The responses ofn individuals are assumed to follow a Rasch model. It is further assumed that the latent individual parameters are distributed within a given population in accordance with a normal distribution. Methods are then considered for estimating the mean and variance of this latent population distribution. Also considered are methods for checking whether a normal population distribution fits the data. The developed methods are applied to data from an achievement test and from an attitude test.     

The bias function of the maximum likelihood estimate of ability for the dichotomous response level

Samejima has recently given an approximation for the bias function for the maximum likelihood estimate of the latent trait in the general case where item responses are discrete, generalizing Lord's bias function in the three-parameter logistic model for the dichotomous response level. In the present paper, observations are made about the behavior of this bias function for the dichotomous response level in general, and also with respect to several widely used mathematical models. Some empirical examples are given.     

A latent class approach to fitting the weighted Euclidean model,clascal

A weighted Euclidean distance model for analyzing three-way proximity data is proposed that incorporates a latent class approach. In this latent class weighted Euclidean model, the contribution to the distance function between two stimuli is per dimension weighted identically by all subjects in the same latent class. This model removes the rotational invariance of the classical multidimensional scaling model retaining psychologically meaningful dimensions, and drastically reduces the number of parameters in the traditional INDSCAL model. The probability density function for the data of a subject is posited to be a finite mixture of spherical multivariate normal densities. The maximum likelihood function is optimized by means of an EM algorithm; a modified Fisher scoring method is used to update the parameters in the M-step. A model selection strategy is proposed and illustrated on both real and artificial data.The second author is supported as Bevoegdverklaard Navorser of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek.     

A thurstonian pairwise choice model with univariate and multivariate spline transformations

A probabilistic choice model is developed for paired comparisons data about psychophysical stimuli. The model is based on Thurstone's Law of Comparative Judgment Case V and assumes that each stimulus is measured on a small number of physical variables. The utility of a stimulus is related to its values on the physical variables either by means of an additive univariate spline model or by means of multivariate spline model. In the additive univariate spline model, a separate univariate spline transformation is estimated for each physical dimension and the utility of a stimulus is assumed to be an additive combination of these transformed values. In the multivariate spline model, the utility of a stimulus is assumed to be a general multivariate spline function in the physical variables. The use of B splines for estimating the transformation functions is discussed and it is shown how B splines can be generalized to the multivariate case by using as basis functions tensor products of the univariate basis functions. A maximum likelihood estimation procedure for the Thurstone Case V model with spline transformation is described and applied for illustrative purposes to various artificial and real data sets. Finally, the model is extended using a latent class approach to the case where there are unreplicated paired comparisons data from a relatively large number of subjects drawn from a heterogeneous population. An EM algorithm for estimating the parameters in this extended model is outlined and illustrated on some real data.The first author is supported as Bevoegdverklaard Navorser of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek. The authors are indebted to Ulf Böckenholt and Yoshio Takane for useful comments on an earlier draft of this paper.