Improving the efficiency and effectiveness of interactively selected MDS data designs

Two revisions of interactive MDS data selection procedures are presented. One revision improves the estimates of the MDS parameters by adding an analysis of the volume of the spatial coordinates of stimuli. Frames of stimuli augmented by an analysis of volume should more nearly surround the swarm of stimulus points. The second revision, based on randomly ordering the list of stimuli, permits more efficient data designs to be selected by reducing the number of judgments collected but never analyzed.This investigation was supported in part by a Research Scientist Development Award (KO2-DA00017), research grants (DA01070 and MH24149) from the U.S. Public Health Service, and a research and instruction award (IUC 2-47622-1990) from the University of California.     

On the precision of a euclidean structure

This paper is concerned with the development of a measure of the precision of a multidimensional euclidean structure. The measure is a precision index for each point in the structure, assuming that all the other points are precisely located. The measure is defined and two numerical methods are presented for its calculation. A small Monte Carlo study of the measure's behavior is performed and findings discussed.The authors are indebted to Bert F. Green, Jr., Ronald Helms, Andrea Sedlak, and three anonymous reviewers for their valuable comments on earlier drafts of this paper.     

The approximation of a group stimulus space by averaging responses to selected subsets of the stimuli

This paper presents a simple random procedure for selecting subsets of stimulus pairs for presentation to subjects. The resulting set of ratings from the group of subjects allows the construction of a group (average) space through the use of the computer program TORSCA-9 or equivalent programs. The procedure is applied to both real and simulated data. It is found that subjects need only make between 20% and 50% of the usual judgements to reconstruct a reasonably accurate group space.     

Additive similarity trees

Similarity data can be represented by additive trees. In this model, objects are represented by the external nodes of a tree, and the dissimilarity between objects is the length of the path joining them. The additive tree is less restrictive than the ultrametric tree, commonly known as the hierarchical clustering scheme. The two representations are characterized and compared. A computer program, ADDTREE, for the construction of additive trees is described and applied to several sets of data. A comparison of these results to the results of multidimensional scaling illustrates some empirical and theoretical advantages of tree representations over spatial representations of proximity data.We thank Nancy Henley and Vered Kraus for providing us with data, and Jan deLeeuw for calling our attention to relevant literature. The work of the first author was supported in part by the Psychology Unit of the Israel Defense Forces.     

Effects of conditionality on INDSCAL and ALSCAL weights

The role of conditionality in the INDSCAL and ALSCAL procedures is explained. The effects of conditionality on subject weights produced by these procedures is illustrated via a single set of simulated data. Results emphasize the need for caution in interpreting subject weights provided by these techniques.     

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.     

Goal attainment scaling: An idiosyncratic method to assess treatment effectiveness in agoraphobia

Goal attainment scaling (GAS) is an individually tailored way to measure treatment gains, using a highly standardized procedure. An advantage of the method is that it takes into account individual characteristics of the patients, and at the same time the data are suitable for quantitative analysis and comparable across patients. Despite the wide acceptance and use of the method in the evaluation of psychotherapy, data on its psychometric properties are rather scarce. In the current study, GAS was used as one of several outcome measures in a research project on the effectiveness of various treatments for panic disorder with agoraphobia. Guidelines for GAS are presented as well as data on the reliability and validity of the procedure. Results indicate that the procedure is reliable, valid, and sensitive to the improvement of patients during treatment. Comparison of GAS with standardized measures revealed considerable concordance, although the clinical end status of patients diverged somewhat dependent on the measure considered.     

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.     

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.