A generalized majorization method for least souares multidimensional scaling of pseudodistances that may be negative

The usual convergence proof of the SMACOF algorithm model for least squares multidimensional scaling critically depends on the assumption of nonnegativity of the quantities to be fitted, called the pseudodistances. When this assumption is violated, erratic convergence behavior is known to occur. Three types of circumstances in which some of the pseudodistances may become negative are outlined: nonmetric multidimensional scaling with normalization on the variance, metric multidimensional scaling including an additive constant, and multidimensional scaling under the city-block distance model. A generalization of the SMACOF method is proposed to resolve the difficulty that is based on the same rationale frequently involved in robust fitting with least absolute residuals.I am grateful to Patrick Groenen and Rian van Blokland-Vogelesang for their help with some of the computations, and to the anonymous referees for their very useful comments.     

An extension of the rating scale model with an application to the measurement of change

The polytomous unidimensional Rasch model with equidistant scoring, also known as the rating scale model, is extended in such a way that the item parameters are linearly decomposed into certain basic parameters. The extended model is denoted as the linear rating scale model (LRSM). A conditional maximum likelihood estimation procedure and a likelihood-ratio test of hypotheses within the framework of the LRSM are presented. Since the LRSM is a generalization of both the dichotomous Rasch model and the rating scale model, the present algorithm is suited for conditional maximum likelihood estimation in these submodels as well. The practicality of the conditional method is demonstrated by means of a dichotomous Rasch example with 100 items, of a rating scale example with 30 items and 5 categories, and in the light of an empirical application to the measurement of treatment effects in a clinical study.Work supported in part by the Fonds zur Förderung der Wissenschaftlichen Forschung under Grant No. P6414.     

Evaluating the desirability of meals: an illustrative multiattribute decision analysis procedure to assess portfolios with interdependent items

This paper presents an exploratory study designed to address complex scaling problems in applying value/utility theory to measuring preferences over alternative combinations (or portfolios) of multiattributed items. This application considers meals served to nursing home residents as portfolios of food items and seeks to maximize meal appeal. We consider food interrelation and link single food measurements to overall meal measurements in the process of constructing meal desirability scales. We test the resulting scales to capture one expert's professional judgments and discuss issues raised in this specific application in the context of general portfolio evaluation problems. © 1998 John Wiley & Sons, Ltd.     

Optimal scaling by alternating length-constrained nonnegative least squares, with application to distance-based analysis

An important feature of distance-based principal components analysis, is that the variables can be optimally transformed. For monotone spline transformation, a nonnegative least-squares problem with a length constraint has to be solved in each iteration. As an alternative algorithm to Lawson and Hanson (1974), we propose the Alternating Length-Constrained Non-Negative Least-Squares (ALC-NNLS) algorithm, which minimizes the nonnegative least-squares loss function over the parameters under a length constraint, by alternatingly minimizing over one parameter while keeping the others fixed. Several properties of the new algorithm are discussed. A Monte Carlo study is presented which shows that for most cases in distance-based principal components analysis, ALC-NNLS performs as good as the method of Lawson and Hanson or sometimes even better in terms of the quality of the solution. Supported by The Netherlands Organization for Scientific Research (NWO) by grant nr. 030-56403 for the “PIONEER” project “Subject Oriented Multivariate Analysis” to the third author. We would like to thank the anonymous referees for their valuable remarks that have improved the quality of this paper.     

On the loss of information in conditional maximum likelihood estimation of item parameters

In item response models of the Rasch type (Fischer & Molenaar, 1995), item parameters are often estimated by the conditional maximum likelihood (CML) method. This paper addresses the loss of information in CML estimation by using the information concept of F-information (Liang, 1983). This concept makes it possible to specify the conditions for no loss of information and to define a quantification of information loss. For the dichotomous Rasch model, the derivations will be given in detail to show the use of the F-information concept for making comparisons for different estimation methods. It is shown that by using CML for item parameter estimation, some information is almost always lost. But compared to JML (joint maximum likelihood) as well as to MML (marginal maximum likelihood) the loss is very small. The reported efficiency in the use of information of CML to JML and to MML in several comparisons is always larger than 93%, and in tests with a length of 20 items or more, larger than 99%.     

Exact person fit indexes for the rasch model for arbitrary alternatives

A new algorithm for obtaining exact person fit indexes for the Rasch model is introduced which realizes most powerful tests for a very general family of alternative hypotheses, including tests concerning DIF as well as model-deviating item correlations. The method is also used as a goodness-of-fit test for whole data sets where the item parameters are assumed to be known. For tests with 30 items at most, exact values are obtained, for longer tests a Monte Carlo-algorithm is proposed. Simulated examples and an empirical investigation demonstrate test power and applicability to item elimination.The author wishes to thank Elisabeth Ponocny-Seliger and the reviewers for many helpful comments. All exact goodness-of-fit tests proposed in this article are implemented in the menu-driven program T-Rasch 1.0 by Ponocny and Ponocny-Seliger (1999) which can be obtained from ProGAMMA (WWW: http://www.gamma.rug.nl) and also performs nonparametric tests.     

Stimulus affectivity of the Danish Word Association Test as measured by response heterogeneity and Rasch scaled number of prolonged reaction times

With the purpose of increasing the knowledge of the psychometric properties of the 70-item Danish Word Association Test, data from three samples of non-patients and psychiatric patients (N = 326) were used to provide two measures of affectivity of the stimulus words, response heterogeneity and reaction time prolongation. It was possible to fit an item response theory one-parameter measurement (Rasch) model to the number of reaction time prolongations (> or =3 seconds) for 54 of the stimulus words. Correlation between Rasch-model item parameters and response heterogeneity was high (r = 0.86), while no correlation was found between either of these measures and frequency of the stimulus words in the Danish language. Both measures of stimulus affectivity supported a theoretically based classification of stimulus words as emotional or neutral. Response heterogeneity measures and Rasch measurement item and person parameters for reaction time prolongations are provided.     

Random effects diagonal metric multidimensional scaling models

By assuming a distribution for the subject weights in a diagonal metric (INDSCAL) multidimensional scaling model, the subject weights become random effects. Including random effects in multidimensional scaling models offers several advantages over traditional diagonal metric models such as those fitted by the INDSCAL, ALSCAL, and other multidimensional scaling programs. Unlike traditional models, the number of parameters does not increase with the number of subjects, and, because the distribution of the subject weights is modeled, the construction of linear models of the subject weights and the testing of those models is immediate. Here we define a random effects diagonal metric multidimensional scaling model, give computational algorithms, describe our experiences with these algorithms, and provide an example illustrating the use of the model and algorithms.We would like to thank J. Douglas Carroll for early consultation of this research, and Robert I. Jennrich for commenting on an earlier draft of this paper and for help on the computational algorithms. James O. Ramsay and Forrest W. Young were instrumental in providing the example data. This work was supported in part by National Institute of Mental Health grant 1 R43 MH57559-01. We would also like to thank the anonymous referees for comments that helped to clarify our work.     

HSK主观考试评分的Rasch实验分析

主观评分中存在的不一致性导致主观评分的信度降低。多面Rasch模型基于项目反应理论,可以应用于评分员效应的识别和消除,从而提高主观评分的信度。该文介绍多面Rasch模型的理论和应用框架,设计了基于该模型的HSK主观考试评分质量控制应用框架,利用HSK作文评分数据进行了实验验证。