Non-metric Scaling Research Articles

Variation In Plant Volatiles and Attraction Of The ParasitoidDiadegma semiclausum(Hellén)

Differences in allelochemistry of plants may influence their ability to attract parasitoids. We studied responses of Diadegma semiclausum (Hellén), a parasitoid of the diamondback moth (Plutella xylostella L.), to inter- and intraspecific variation in odor blends of crucifers and a non-crucifer species. Uninfested Brussels sprout (Brassica oleracea L. gemmifera), white mustard (Sinapis alba L.), a feral Brassica oleracea, and malting barley (Hordeum vulgare L.) were compared for their attractivity to D. semiclausum in a Y-tube bioassay. Odors from all plants were more attractive to the parasitoid than clean air. However, tested against each other, parasitoids preferred the volatile blend from the three cruciferous species over that of malting barley. Wasps also discriminated between uninfested crucifers: mustard was as attractive as feral B. oleracea, and both were more attractive than Brussels sprout. Attractivity of uninfested plants was compared with that of plants infested by larvae of the host P. xylostella. Host-infested mustard and Brussels sprout were more attractive than uninfested conspecifics. Interestingly, the volatile blends of uninfested white mustard and infested Brussels sprout were equally attractive. We also compared the volatile composition of different plant sources by collecting headspace samples and analysing them with GC-MS. Similarities of volatile profiles were determined by hierarchic clustering and non-metric scaling based on the Horn-index. Due to the absence of several compounds in its blend, the volatile profile of barley showed dissimilarities from blends of crucifers. The odor profile of white mustard was distinctly different from the two Brassicaceae. Feral Brassica oleracea odor profile was different from infested Brussels sprout, but showed overlap with uninfested Brussels sprout. Odor blends from infested and uninfested Brussels sprout were similar, and mainly quantitative differences were found. D. semiclausum appears to discriminate based on subtle differences in volatile composition of odor blends from infested and uninfested plants.

Adequateness and interpretability of objective functions in ordinal data analysis

Objective functions that are applied in ordinal data analysis must be adequate, i.e. carefully adapted to the structure of the observed data. In addition, any analysis of data that is based upon objective functions must lead to interpretable results. After a general characterization of adequate objective functions in ordinal data analysis, therefore, the particular problems of constructing adequate and interpretable dissimilarity coefficients and correlation coefficients in ordinal data analysis, stress measures (stress functions) in non-metric scaling and generalized stress measures or correlation coefficients in any theory of rank estimation will be discussed.

Multidimensional scaling used in multivariate statistical process control

This paper considers the use of multidimensional scaling techniques in multivariate statistical process control. Principal components analysis, multiple principal components analysis, partial least squares and PARAFAC models have already been established as useful methods for such, but it should be possible to widen the portfolio of techniques to include others that come under the multidimensional scaling class. Some of these are briefly described-namely classical scaling, non-metric scaling, biplots, Procrustes analysis-and are then used on some gas transportation data provided by Transco.

Uncertainty of measurement results in various scales

The relationship between the concepts of “error” and “uncertainty” of measurement results is examined. The concepts of “standard uncertainty,” “combined standard uncertainty,” and “expanded uncertainty” are shown to be inapplicable in nonmetric scales of quantities and properties, in which the general concept of “uncertainty” in the broad sense is recommended.

Comparison of chemometric methods for classification of fungal extracts based on rapid fluorescence spectroscopy

Rapid fluorescence landscapes were recorded on 137 extracts of five species of Penicillium and seven extracts of pure growth medium. The ability of the following methods to show six classes was investigated: principal component analysis; parallel factor analysis; discriminant partial least squares regression and N-PLS with dummy Y matrix; correspondence analysis; soft independent modelling of class analogy; principal co-ordinate analysis; non-metric scaling; and cluster analysis on similarity matrices with different similarity measures. Cluster analysis was also performed on sets of score vectors. The best measure of similarity was the correlation coefficient. None of the methods was entirely successful, because two species had some highly fluorescent compounds in common. All the tested chemometric methods obtained a level of correct classification, which normally requires mycological experts, for three species. All methods were able to capture most of the same information, but cluster analysis gave the clearest result for both classes and outliers and was the easiest method to perform and evaluate. Copyright © 2000 John Wiley & Sons, Ltd.

Nonmetric Linear Biplots

A methodology is developed for constructing linear biplots for a class of nonmetric multidimensional scaling methods for multivariate data. The nonlinear transformations of nonmetric scaling manifest themselves in irregularly spaced calibration markers. Two approaches are examined, one based on Procrustean embedding, the other on a modification of the popular regression method. The widespread use of an unmodified regression method in association with nonlinear transformations is questioned. An example is given. The methodology presented here could potentially be developed to give an optimal represention of a matrix in fewer geometric dimensions than its rank.

Mapping the mental space of rectangles.

The cognitive structure of a shape space--the space of rectangles--is explored by a nonmetric scaling technique. Our experiment was designed to extract the major transformational paths or 'modes' that characterize the mental shape space. Earlier studies of rectangle similarities using multidimensional scaling have provided conflicting evidence about whether the coordinate system of the mental rectangle space is based on height and width or on area and shape (i.e. aspect ratio). Our study reveals shape to be the single dominant factor. We suspected that earlier evidence for a height-width parameterization might have been due to the presentation of rectangles upright in a pseudo-gravitational coordinate system (whereas our rectangles are randomly rotated). In a control experiment with upright (vertical or horizontal) rectangles, the heavy bias towards shape preservation was still the dominant mode. In addition, however, a secondary bias towards change of height or width emerged, exactly following the pattern expected from the biasing change in context. This finding established a concrete path by which context and frame can influence the way shape is represented. The relevance of these findings to the cognitive organization of more complex shape spaces is discussed.

The stimulus range effect: evidence for top-down control of sensory intensity in audition.

The influence of intensity range on the perceived magnitude of a stimulus is well documented and usually attributed to response biases. Recent studies, however, have suggested that the range effect might be sensory in origin. To test this notion, we had one set of subjects compare loudness intervals in three conditions: a broad-range condition (15 tones, 23-95 dB SPL), a soft short-range condition (the lowest 10 tones from the broad-range condition), and a loud short-range condition (the highest 10 tones). Nonmetric scaling showed that the broad-range and loud short-range conditions had identical loudness functions. However, the second derivative of the loudness function was larger for the soft short-range condition than for the broad-range condition. This pattern of results is consistent with the notion of a nonlinear amplifier whose gain and degree of nonlinearity are adjusted under top-down control, so as to prevent distortion and increase discriminability.

Review of the Development of Multidimensional Scaling Methods

Multidimensional scaling methods are now a common statistical tool in psychophysics and sensory analysis. The development of these methods is charted, from the original research of Torgerson (metric scaling), Shepard and Kruskal (non-metric scaling) through individual differences scaling and the maximum likelihood methods proposed by Ramsay. 1 Introduction-what is multidimensional scaling? The set of procedures referred to as multidimensional scaling methods are concerned with constructing a configuration of n points, usually in Euclidean space, from information about the pairwise 'distances' among a set of n objects or individuals. The initial research into multidimensional scaling methods was carried out by scientists working in psychophysics and sensory analysis, in which the objects are stimuli, and the distances between them are assessments of either their similarity or dissimilarity as made either by an individual or by a panel of judges. Most multidimensional scaling methods have been developed to analyse dissimilarities, and so, since similarities can be expressed as dissimilarities subtracted from a constant, I shall generally refer only to dissimilarities in the remainder of this review. The dissimilarities are usually presented in an (n x n) lower triangular matrix, the entry in the ith row andjth column being the dissimilarity, dij, between object i and objectj. In some cases the assessments of dissimilarity are not made directly, but are obtained from observations on p variables for each of the n objects. Appropriate methods for calculating such measures of similarity and dissimilarity are discussed by Chatfield & Collins (1980), section 10.2. Jardine & Sibson (1971) defined one class of dissimilarity functions, which they called 'dissimilarity coefficients', which are required to satisfy the following conditions:

Two-Dimensional Configurations of Unidimensional Stimulus Sets in Nonmetric Multidimensional Scaling

When unidimensional stimulus sets are subjected to a nonmetric scaling in two dimensions, the stimuli fre quently form a C- or S-shaped configuration. In simu lated unidimensional data scaled in two dimensions, stimuli formed a C-shaped configuration when the monotone function relating distances to dissimilarity data was negatively accelerating. They formed an S- shaped configuration when the monotone function was positively accelerating. Results suggest that when uni dimensional stimulus sets are scaled in two dimensions using a rational starting configuration, the nature of the two-dimensional configuration can indicate the general form of the function mapping psychological dissimilarity, represented as distance in the scaling model, onto the observed response scale. Index terms: data transformations, multidimensional scaling, paired comparisons, proximity data, unidimensional scaling, unidimensionality.

Correlation between plaque size and genetic variation of type 3 poliovirus from a vaccinate.

Vaccine strain derivatives could be serially isolated from the feces of a healthy infant for about 30 days after the second administration of a trivalent poliovirus vaccine. These were all identified as poliovirus type 3, and no other types were detected. The titer in feces increased gradually until day 19, when it reached a maximum, after which it decreased. The plaque size altered roughly in parallel with the changes in the titer. On day 11 it began to increase, becoming the highest abruptly on day 15, and then fell again. Oligonucleotide mapping analysis of RNAs of isolated viruses showed that the samples from days 1 and 11 exhibited the same patterns as those shown by the vaccine strain RNA. However, on day 15 many of the preexisting spots had decreased, and new ones appeared. On day 27, the pattern was again similar to that of the original virus, rather than that of the day-15 virus. A nonmetric distance scaling and cluster analysis suggested that all the strains were derived from the same origin and that the polioviruses on days 1, 11, 21, and 27 had closer relationships, but the poliovirus on day 15 was different.

Multidimensional scaling: Some possibilities for counseling psychology.

Although counseling psychologists conduct a great deal of research that attempts to reveal the structure of a given data set, rarely if ever do they utilize scaling procedures, preferring instead to rely on factor analytic strategies. In this article, we give a short introduction to the use of multidimensional scaling (MDS), with specific emphasis on applications in counseling and vocational psychology, including an example of one standard nonmetric scaling method. We conclude with a discussion of some conceptual and practical considerations associated with the use of MDS, along with a description of its possible applications to a variety of substantive issues in counseling and vocational psychology.

A two-stage algorithm for assessing violations of additivity via axiomatic and numerical conjoint analysis

An algorithm for assessing additivity conjunctively via both axiomatic conjoint analysis and numerical conjoint scaling is described. The algorithm first assesses the degree of individual differences among sets of rankings of stimuli, and subsequently examines either individual or averaged data for violations of axioms necessary for an additive model. The axioms are examined at a more detailed level than has been previously done. Violations of the axioms are broken down into different types. Finally, a nonmetric scaling of the data can be done based on either or both of two different badness-of-fit scaling measures. The advantages of combining all of these features into one algorithm for improving the diagnostic value of axiomatic conjoint measurement in evaluating additivity are discussed.

Applying Metric and Nonmetric Multidimensional Scaling to Ecological Studies: Some New Results

Metric (eigenanalysis) and nonmetric multidimensional scaling strategies for ecological ordination were compared. The result, based on simulated coenoplane data showing varying degrees of species turnover on two independent environmental axes, suggested some strong differences between metric and nonmetric scaling methods in their ability to recover underlying nonlinear data structures. Prior data standardization had important effects on the results of both metric and nonmetric scaling, though the effect varied with the ordination method used. Nonmetric multidimensional scaling based on Euclidean distance following stand norm standardization proved to be the best strategy for recovering simulated coenoplane data. Of the metric strategies compared, correspondence analysis and the detrended form were the most successful. While detrending improved ordination configurations in some cases, in others it led to a distortion of results. It is suggested that none of the currently available ordination strategies is appropriate under all circumstances, and that future research in ordination methodology should emphasize a statistical rather than empirical approach.

Genetic differentiation in populations of Canada geese (Branta canadensis)

Electrophoretic data for 35 structural gene loci were obtained from 249 individuals in 11 populations of Branta canadensis across North America. Twenty-four loci were polymorphic in at least one population sample, average observed heterozygosity ranged from 0.031 to 0.083, Rogers' genetic distance ranged from 0.014 to 0.063, and Wright's measure of genetic structuring among populations was 0.065. Samples of B. c. minima from Willows, California, and B. c. interior from Akimiski Island, Northwest Territories, exhibited both the highest levels of genetic variability within populations and the greatest divergence from other populations. Principal coordinates analysis on the genetic distances, followed by nonmetric scaling in three dimensions and supplemented by construction of a minimum spanning tree, gave a more informative and less distorted picture of relationships among populations than cluster analysis. Differentiation in structural genes was neither equivalent to morphometric differentiation (in magnitude and pattern) nor consistent with conventional subspecies designations. This suggests that structural genes are selectively neutral in Branta canadensis and are differentiating through random drift, in contrast to morphometric characters which have diverged markedly under different selective pressures.

The assessment of personality factors across 25 countries

The means for each scale score from the Eysenck Personality Questionnaire were compared individually and conjointly across 25 countries. Comparisons were carried out using coded difference scores, Pearson correlations, Euclidean distances, cluster analyses and non-metric multidimensional scaling. It was possible to compare the personality (as defined by the scale scores on E, N, P and L) of each country with all other countries, culminating in a one-dimensional non-metric scale of distances for both the male and female data from each country. The assumptions underlying the rationale of the measures used and the comparisons undertaken are briefly discussed.

Fitting Unidimensional Choice Models with Nonmetric Multidimensional Scaling

A class of unidimensional choice models is de scribed. Thurstone's paired comparisons model, Case 5, and the Bradley-Terry-Luce model both fall into the class. A simple nonmetric method is presented for es timating scale values from choice data which satisfy any model in the class. In two examples, nonmetric scale values are compared to Thurstone estimates. The scaling method is extended to permit estimation of scale values in a class of unidimensional ordered-cate gory models, a class which includes the law of cate gorical judgment.

Tourist space and Touristic attraction: An analysis of the destination choices of European travelers

Abstract This paper addresses the problem of inferring and measuring touristic attraction for countries as wholes, based on the observed distribution of tourists among a set of destination countries. Embodied in the research is the understanding that even if it is the distinctiveness of countries that allows them to become touristically important, countries are not unique to the extent that a generalized measure of attraction cannot be derived. The phenomenon of tourism is expressed as constituting a system with a peculiar mode of organization in which attraction varies spatially. Interpreting the data on tourist flows in the Western European/U.S.A. system as resulting from individual expressions of choice/preference for the destination visited, a nonmetric scaling algorithm is used to derive an attraction scale. The significance of the scale is then tested against the hypothesis that the directional bias of tourist flows and the spatial variation in attraction are both functions and expressions of socioeconomic center/ periphery differences.

Multidimensional scaling in Riemann space

Although the first multidimensional (MDS) scaling algorithms developed were metric algorithms (Torgerson, 1958), the development of nonmetric methods (Shepard, 1966) led to a rapia and nearly complete abandonment of the metric procedures in favor of these newer algorithms. Recently, however, there has been a resurgence of interest in metric algorithms, particularly within the field of human communication research, where the use of metric procedures predominates. In order to understand this reversion to what many psychometricians believe to be an outdated technique it is necessary to understand the difficulties and philosophy which led to the increased interest in metric scaling. While there is little doubt that the development of nonmetric multidimensional scaling algorithms represents a great advance in the methods available to the contemporary social scientist, there exist some areas of inquiry in which the metric scaling routines may offer certain advantages. This article discusses one such case, familiar to communication researchers in particular. The reader should understand at the outset that this article is not an argument against the use of nonmetric scaling, but rather a suggestion that, for a specific theory dealing with a specific set of research questions, metric multidimensional scaling, modified in certain important ways, provides useful results obtainable either only with great difficulty or not at all by nonmetric means.

Loudness functions of short-duration tone glides from nonmetric scaling

Individual loudness functions of short-duration (20 ms) tone glides and pure tones were obtained for three listeners. Upsweep and downsweep tone glides varied linarly between endpoint frequencies of 500 and 2500 Hz; binary judgments were made by subjects at ten intensities between 30 and 90 dB SPL. Subjects judged loudness differences between tone pairs, and loudness projections were derived using a multidimensional unfolding program developed by Schneider [J. Acoust. Soc. Am. 69, 1208–1209 (1981)]. Loudness differences between the tone glides and pure tones occurred over most of the range of intensities; some differences occurred between upsweep and downsweep tone glides at the highest levels tested. These results indicate that for short-duration signals with rapid changes in frequency, loudness growth is a steeper function that for pure tones of similar duration.