Three notes on ALSCAL

It is reported that (1) a new coordinate estimation routine is superior to that originally proposed for ALSCAL; (2) an oversight in the interval measurement level case has been found and corrected; and (3) a new initial configuration routine is superior to the original.     

A note on item information in any direction for the multidimensional three-parameter logistic model

The purpose of this note is twofold: (a) to present the formula for the item information function (IIF) in any direction for the Multidimensional 3-Parameter Logistic (M3-PL) model and (b) to give the equation for the location of maximum item information (θ_max) in the direction of the item discrimination vector. Several corollaries are given. Implications for future research are discussed.This research was supported in part by an Educational Testing Service (ETS) Harold T. Gulliksen Psychometric Research Fellowship to the author.This revised article was published online in August 2005 with the PDF paginated correctly.     

Optimal Least-Squares Unidimensional Scaling: Improved Branch-and-Bound Procedures and Comparison to Dynamic Programming

There are two well-known methods for obtaining a guaranteed globally optimal solution to the problem of least-squares unidimensional scaling of a symmetric dissimilarity matrix: (a) dynamic programming, and (b) branch-and-bound. Dynamic programming is generally more efficient than branch-and-bound, but the former is limited to matrices with approximately 26 or fewer objects because of computer memory limitations. We present some new branch-and-bound procedures that improve computational efficiency, and enable guaranteed globally optimal solutions to be obtained for matrices with up to 35 objects. Experimental tests were conducted to compare the relative performances of the new procedures, a previously published branch-and-bound algorithm, and a dynamic programming solution strategy. These experiments, which included both synthetic and empirical dissimilarity matrices, yielded the following findings: (a) the new branch-and-bound procedures were often drastically more efficient than the previously published branch-and-bound algorithm, (b) when computationally feasible, the dynamic programming approach was more efficient than each of the branch-and-bound procedures, and (c) the new branch-and-bound procedures require minimal computer memory and can provide optimal solutions for matrices that are too large for dynamic programming implementation.The authors gratefully acknowledge the helpful comments of three anonymous reviewers and the Editor. We especially thank Larry Hubert and one of the reviewers for providing us with the MATLAB files for optimal and heuristic least-squares unidimensional scaling methods.This revised article was published online in June 2005 with all corrections incorporated.     

Stability of heartbeat interval distributions in chronic high altitude hypoxia

Recent studies of nonlinear dynamics of the long-term variability of heart rate have identified nontrivial long-range correlations and scale-invariant power-law characteristics (1/f noise) that were remarkably consistent between individuals and were unrelated to external or environmental stimuli (Meyer et al., 1998a). The present analysis of complex nonstationary heartbeat patterns is based on the sequential application of the wavelet transform for elimination of local polynomial nonstationary behavior and an analytic signal approach by use of the Hilbert transform (Cumulative Variation Amplitude Analysis). The effects of chronic high altitude hypoxia on the distributions and scaling functions of cardiac intervals over 24 hr epochs and 4 hr day/nighttime subepochs were determined from serial heartbeat interval time series of digitized 24 hr ambulatory ECGs recorded in 9 healthy subjects (mean age 34 yrs) at sea level and during a sojourn at high altitude (5,050 m) for 34 days (Ev-K2-CNR Pyramid Laboratory, Sagarmatha National Park, Nepal). The results suggest that there exists a hidden, potentially universal, common structure in the heterogeneous time series. A common scaling function with a stable Gamma distribution defines the probability density of the amplitudes of the fluctuations in the heartbeat interval time series of individual subjects. The appropriately rescaled distributions of normal subjects at sea level demonstrated stable Gamma scaling consistent with a single scaled plot (data collapse). Longitudinal assessment of the rescaled distributions of the 24 hr recordings of individual subjects showed that the stability of the distributions was unaffected by the subject’s exposure to a hypobaric (hypoxic) environment. The rescaled distributions of 4 hr subepochs showed similar scaling behavior with a stable Gamma distribution indicating that the common structure was unequivocally applicable to both day and night phases and, furthermore, did not undergo systematic changes in response to high altitude. In contrast, a single function stable over a wide range of time scales was not observed in patients with congestive heart failure or patients after cardiac transplantation. The functional form of the scaling in normal subjects would seem to be attributable to the underlying nonlinear dynamics of cardiac control. The results suggest that the observed Gamma scaling of the distributions in healthy subjects constitutes an intrinsic dynamical property of normal heart function that would not undergo early readjustment or late acclimatization to extrinsic environmental physiological stress, e.g., chronic hypoxia.     

Multidimensional adaptive testing

Maximum likelihood and Bayesian procedures for item selection and scoring of multidimensional adaptive tests are presented. A demonstration using simulated response data illustrates that multidimensional adaptive testing (MAT) can provide equal or higher reliabilities with about one-third fewer items than are required by one-dimensional adaptive testing (OAT). Furthermore, holding test-length constant across the MAT and OAT approaches, substantial improvements in reliability can be obtained from multidimensional assessment. A number of issues relating to the operational use of multidimensional adaptive testing are discussed.     

Gleaning in the field of dual scaling

Some historical background and preliminary technical information are first presented, and then a number of hidden, but important, methodological aspects of dual scaling are illustrated and discussed: normed versus projected weights, the amount of information accounted for by each solution, a perfect solution to the problem of multidimensional unfolding, multidimensional quantification space, graphical display, number-of-option problems, option standardization versus item standardization, and asymmetry of symmetric (dual) scaling. Contrary to the common perception that dual scaling and similar quantification methods are now mathematically transparent, the present study demonstrates how much more needs to be clarified for routine use of the method to arrive at valid conclusions. Data analysis must be carried out in such a way that common sense, intuition and sound logic will prevail.Presidential Address delivered at the Annual Meeting of the Psychometric Society, Banff Centre for Conferences, Banff, Alberta, Canada, June 27–30, 1996. The work has been supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. I am grateful to Ira Nishisato for his comments, Ingram Olkin and Yoshio Takane for important references, and Liqun Xu for computational help.     

The tunneling method for global optimization in multidimensional scaling

This paper focuses on the problem of local minima of the STRESS function. It turns out that unidimensional scaling is particularly prone to local minima, whereas full dimensional scaling with Euclidean distances has a local minimum that is global. For intermediate dimensionality with Euclidean distances it depends on the dissimilarities how severe the local minimum problem is. For city-block distances in any dimensionality many different local minima are found. A simulation experiment is presented that indicates under what conditions local minima can be expected. We introduce the tunneling method for global minimization, and adjust it for multidimensional scaling with general Minkowski distances. The tunneling method alternates a local search step, in which a local minimum is sought, with a tunneling step in which a different configuration is sought with the same STRESS as the previous local minimum. In this manner successively better local minima are obtained, and experimentation so far shows that the last one is often a global minimum.This paper is based on the 1994 Psychometric Society's outstanding thesis award of the first author. The authros would like to thank Robert Tijssen of the CWTS Leiden for kindly making available the co-citation data of the Psychometric literature. This paper is an extended version of the paper presented at the Annual Meeting of the Psychometric Society at Champaign-Urbana, Illin., June 1994.     

Loglinear Rasch models for the analysis of stability and change

Loglinear unidimensional and multidimensional Rasch models are considered for the analysis of repeated observations of polytomous indicators with ordered response categories. Reparameterizations and parameter restrictions are provided which facilitate specification of a variety of hypotheses about latent processes of change. Models of purely quantitative change in latent traits are proposed as well as models including structural change. A conditional likelihood ratio test is presented for the comparison of unidimensional and multiple scales Rasch models. In the context of longitudinal research, this renders possible the statistical test of homogeneity of change against subject-specific change in latent traits. Applications to two empirical data sets illustrate the use of the models.The author is greatly indebted to Ulf Böckenholt, Rolf Langeheine, and several anonymous reviewers for many helpful suggestions.     

Robust multidimensional scaling

A method for multidimensional scaling that is highly resistant to the effects of outliers is described. To illustrate the efficacy of the procedure, some Monte Carlo simulation results are presented. The method is shown to perform well when outliers are present, even in relatively large numbers, and also to perform comparably to other approaches when no outliers are present.This research was supported by Grant A8351 from the Natural Sciences and Engineering Research Council of Canada to Ian Spence.     

Netscal: A network scaling algorithm for nonsymmetric proximity data

A simple property of networks is used as the basis for a scaling algorithm that represents nonsymmetric proximities as network distances. The algorithm determines which vertices are directly connected by an arc and estimates the length of each arc. Network distance, defined as the minimum pathlength between vertices, is assumed to be a generalized power function of the data. The derived network structure, however, is invariant across monotonic transformations of the data. A Monte Carlo simulation and applications to eight sets of proximity data support the practical utility of the algorithm.I am grateful to Roger Shepard and Amos Tversky for their helpful comments and guidance throughout this project. The work was supported by National Science Foundation Grant BNS-75-02806 to Roger Shepard and a National Science Foundation Graduate Fellowship to the author. Parts of this paper were drawn from a doctoral dissertation submitted to Stanford University (Hutchinson, 1981).