In this paper, we reconsider the merits of unfolding solutions based on loss functions involving a normalization on the variance per subject. In the literature, solutions based on Stress-2 are often diagnosed to be degenerate in the majority of cases. Here, the focus lies on two frequently occurring types of degeneracies. The first type typically locates some subject points far away from a compact cluster of the other points. In the second type of solution, the object points lie on a circle. In this paper, we argue that these degenerate solutions are well fitting and informative. To reveal the information, we introduce mixtures of plots based on the ideal point model of unfolding, the vector model, and on the signed distance model. In addition to a different representation, we provide a new iterative majorization algorithm to optimize the average squared correlation between the distances in the configuration and the transformed data per individual. It is shown that this approach is equivalent to minimizing Kruskal’s Stress-2.

Multidimensional probabilistic models of behavior following similarity and choice judgements have proven to be useful in representing multidimensional percepts in Euclidean and non-Euclidean spaces. With few exceptions, these models are generally computationally intense because they often require numerical work with multiple integrals. This paper focuses attention on a particularly general triad and preferential choice model previously requiring the numerical evaluation of a 2n-fold integral, where n is the number of elements in the vectors representing the psychological magnitudes. Transforming this model to an indefinite quadratic form leads to a single integral. The significance of this form to multidimensional scaling and computational efficiency is discussed.

The PARELLA model is a probabilistic parallelogram model that can be used for the measurement of latent attitudes or latent preferences. The data analyzed are the dichotomous responses of persons to items, with a one (zero) indicating agreement (disagreement) with the content of the item. The model provides a unidimensional representation of persons and items. The response probabilities are a function of the distance between person and item: the smaller the distance, the larger the probability that a person will agree with the content of the item. This paper discusses how the approach to differential item functioning presented by Thissen, Steinberg, and Wainer can be implemented for the PARELLA model.

Multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by approximately equal distances between points from different sets. A definition of an absolutely degenerate solution is proposed, which makes clear that these solutions only occur when an intercept is present in the transformation function. Many solutions for the degeneracy problem have been proposed and tested, but with little success so far. In this paper, we offer a substantial modification of an approach initiated bythat introduced a normalization factor based on thevariance in the usual least squares loss function. Heiser unpublishedthesis, (1981) and showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies. The factor proposed in the present paper, based on the coefficient of variation, discourages or penalizes nonmetric transformations of the proximities with small variation, so that the procedure steers away from solutions with small variation in the interpoint distances. An algorithm is described for minimizing the re-adjusted loss function, based on iterative majorization. The results of a simulation study are discussed, in which the optimal range of the penalty parameters is determined. Two empirical data sets are analyzed by our method, clearly showing the benefits of the proposed loss function.

The slide-vector scaling model attempts to account for the asymmetry of a proximity matrix by a uniform shift in a fixed direction imposed on a symmetric Euclidean representation of the scaled objects. Although no method for fitting the slide-vector model seems available in the literature, the model can be viewed as a constrained version of the unfolding model, which does suggest one possible algorithm. The slide-vector model is generalized to handle three-way data, and two examples from market structure analysis are presented.

Composite links and exploded likelihoods are powerful yet simple tools for specifying a wide range of latent variable models. Applications considered include survival or duration models, models for rankings, small area estimation with census information, models for ordinal responses, item response models with guessing, randomized response models, unfolding models, latent class models with random effects, multilevel latent class models, models with log-normal latent variables, and zero-inflated Poisson models with random effects. Some of the ideas are illustrated by estimating an unfolding model for attitudes to female work participation.

A model for preferential and triadic choice is derived in terms of weighted sums of central F distribution functions. This model is a probabilistic generalization of Coombs' (1964) unfolding model and special cases, such as the model of Zinnes and Griggs (1974), can be derived easily from it. This new form extends previous work by Mullen and Ennis (1991) and provides more insight into the same problem that they discussed.

Three-way unfolding was developed by DeSarbo (1978) and reported in DeSarbo and Carroll (1980, 1981) as a new model to accommodate the analysis of two-mode three-way data (e.g., nonsymmetric proximities for stimulus objects collected over time) and three-mode, three-way data (e.g., subjects rendering preference judgments for various stimuli in different usage occasions or situations). This paper presents a revised objective function and new algorithm which attempt to prevent the common type of degenerate solutions encountered in typical unfolding analysis. We begin with an introduction of the problem and a review of three-way unfolding. The three-way unfolding model, weighted objective function, and new algorithm are presented. Monte Carlo work via a fractional factorial experimental design is described investigating the effect of several data and model factors on overall algorithm performance. Finally, three applications of the methodology are reported illustrating the flexibility and robustness of the procedure.

The PARELLA model is a probabilistic parallelogram model that can be used for the measurement of latent attitudes or latent preferences. The data analyzed are the dichotomous responses of persons to stimuli, with a one (zero) indicating agreement (disagreement) with the content of the stimulus. The model provides a unidimensional representation of persons and items. The response probabilities are a function of the distance between person and stimulus: the smaller the distance, the larger the probability that a person will agree with the content of the stimulus. An estimation procedure based on expectation maximization and marginal maximum likelihood is developed and the quality of the resulting parameter estimates evaluated.

Program transformation is an appealing technique which allows to improve run-time efficiency, space-consumption, and more generally to optimize a given program. Essentially, it consists of a sequence of syntactic program manipulations which preserves some kind of semantic equivalence. Unfolding is one of the basic operations used by most program transformation systems and consists of the replacement of a procedure call by its definition. While there is a large body of literature on the transformation and unfolding of sequential programs, very few papers have addressed this issue for concurrent languages. This paper defines an unfolding system for Constraint Handling Rules programs. We define an unfolding rule, show its correctness and discuss some conditions that can be used to delete an unfolded rule while preserving the program meaning. We also prove that, under some suitable conditions, confluence and termination are preserved by the above transformation.

Effects of guanidine hydrochloride (GdnHCl) on the structure and dynamics of wild-type Humicola lanuginosa lipase (HLL) and its two mutants were studied. The latter were S146A (with the active site Ser replaced by Ala) and the single Trp mutant W89m, with substitutions W117F, W221H, and W260H. Steady-state, stopped-flow, and time-resolved laser-induced fluorescence spectroscopy were carried out as a function of [GdnHCl]. The maximum emission wavelength and fluorescence lifetimes revealed the microenvironment of the tryptophan(s) in these lipases to become more polar upon increasing [GdnHCl]. However, significant extent of tertiary structure in GdnHCl is suggested by the observation that both wild-type HLL and W89m remain catalytically active at rather high GdnHCl concentrations of >6 and 4.0 M, respectively. Changes in steady-state emission anisotropy, as well as variation in rotational correlation times and residual anisotropy values, demonstrate that upon increasing [GdnHCl] the structure of the lipases became more loose, with increasing amplitude of structural fluctuations. Finally, intermediate states in the course of exposure of the proteins to GdnHCl were revealed by stopped-flow fluorescence measurements.

Denaturation with guanidine hydrochloride (GdnHCl) or urea is one of the primary ways of measuring the conformational stability of proteins and comparing the stability of mutant proteins. Despite the widespread use of these two denaturants to provide quantitative data for the free energies of unfolding, the mode of action of these agents is not well understood. We are not even certain whether the action of these agents on proteins is direct and can be regarded as ligand binding, or indirect and involves a change in the properties of solvent (water) in the presence of GdnHCl and urea. In this paper, an extensive kinetic study of the inhibition of ribonuclease A and papain by urea has been performed. The results suggest that the effect of urea on activities of these enzymes can be well described by the denaturant binding model. The binding constants of urea determined by the present method are nearly identical to that determined from a variety of different studies on model compounds and proteins.

Insulin-like growth factor (IGF-1) contains three disulfide bonds. In the presence of denaturant and thiol catalyst, IGF-1 shuffles its native disulfide bonds and denatures to form a mixture of scrambled isomers. The composition of scrambled IGF varies under different denaturing conditions. Among the 14 possible scrambled IGF isomers, the yield of the beads-form isomer is shown to be directly proportional to the strength of the denaturing condition. This paper demonstrates a new approach to quantify the extent of unfolding of the denatured protein.

The pressure-induced unfolding of lysozyme was investigated in an aqueous guanidinium chloride solution by means of ultraviolet spectroscopy. Assuming a two-state transition model, volume changes were calculated from the slope of free energy vs. pressure plots over a temperature range of 10 to 60 °C. Between 25 and 60 °C, almost constant volume changes were observed in the transition region, which was reflected in almost identical slopes of the free energy change vs. pressure plots. On the other hand, the different slopes were observed in the pressure dependence of free energy change at temperatures lower than 25 °C. These data were interpreted as suggesting that a two-state model is not appropriate at low temperature, but instead one or more intermediates are present under these conditions. The volume changes for unfolding became less negative at temperatures higher than 25 °C.

The consequences of amino acid substitutions at the dimer interface for the strength of the interactions between the monomers and for the catalytic function of the dimeric enzyme alkaline phosphatase from Escherichia coli have been investigated. The altered enzymes R10A, R10K, R24A, R24K, T59A, and R10A/R24A, which have amino acid substitutions at the dimer interface, were characterized using kinetic assays, ultracentrifugation, and transverse urea gradient gel electrophoresis. The kinetic data for the wild-type and altered alkaline phosphatases show comparable catalytic behavior with kcat values between 51.3 and 69.5 s−1 and Km values between 14.8 and 26.3 μM. The ultracentrifugation profiles indicate that the wild-type enzyme is more stable than all the interface-modified enzymes. The wild-type enzyme is dimeric in the pH range of pH 4.0 and above, and disassembled at pH 3.5 and below. All the interface-modified enzymes, however, are apparently monomeric at pH 4.0, begin assembly at pH 5.0, and are not fully assembled into the dimeric form until pH 6.0. The results from transverse urea gradient gel electrophoresis show clear and reproducible differences both in the position and the shape of the unfolding patterns; all these modified enzymes are more sensitive to the denaturant and begin to unfold at urea concentrations between 1.0 and 1.5 M; the wild-type enzyme remains in the folded high mobility form beyond 2.5 M urea. Alkaline phosphatase H370A, modified at the active site and not at the dimer interface, resembles the wild-type enzyme both in ultracentrifugation and electrophoresis studies. The results obtained suggest that substitution of a single amino acid at the interface sacrifices not only the integrity of the assembled dimer, but also the stability of the monomer fold, even though the activity of the enzyme at optimal pH remains unaffected and does not appear to depend on interface stability.