Pitch distribution,melodic contour or both? Modeling makam schema with multidimensional scaling and self-organizing maps

Experienced listeners internalize musical tonal knowledge via statistical learning of pitch distributions as a result of exposure to musical environment. Cross-cultural studies of music cognition offer new perspectives to investigate the acquisition of tonal schema. Makam music is a rich musical system characterized by modal structures defined by micro-tonal pitch sets, and melodic progression patterns (aka seyir features). Makam schema is possibly acquired by internalizing the seyir in addition to pitch features. In the current study, we examined whether an ideal model of makam schema is built with multidimensional scaling analysis and with self-organizing maps (SOMs). We were interested in whether statistical information about seyir features, in addition to pitch distributions, would form an acceptable makam schema model. We qualitatively analyzed topographical organizations in the models to understand whether they reflect complex relations between makams. Multidimensional scaling analyses did not produce an acceptable model for makam schema. The SOM trained with pitch distributions provided an adequate model for makam schema. However, the SOM trained with both pitch distributions and seyir features was better in capturing the complex relations between makams. Further behavioral research is necessary to understand whether melodic progression patterns are intrinsic features of the tonal knowledge of the experienced listeners of makam music.     

A monte carlo study of kruskal's variance based measure on stress

Researchers in the past ten years have studied various parameters involved in nonmetric multidimensional scaling by utilizing Monte Carlo procedures. This paper develops stress distributions using Kruskal's second stress formula based upon a null hypothesis of equal likelihood in the ranking of a set of proximities. These distributions can serve to determine whether a set of data has other than random structure.This research was supported in part by a grant from the Baruch College Scholar Assistance program.     

On scaling models applied to data from several groups

Statistical methods are presented to facilitate a more complete analysis of results obtained when a scaling model is applied to data from two or more groups. These methods can be used to (a) compare the corresponding estimated latent distributions obtained using the scaling model applied to the different groups, (b) compare the corresponding estimated item reliabilities (or item response error rates) for the different groups, and (c) test whether the scaling model applied to the several groups can be replaced by a more parsimonious scaling model that includes various homogeneity constraints (i.e., constraints that describe which parameters in the model are the same for the several groups). Various kinds of scaling models are considered here in the multiple-group context.Support for this research was provided in part by the National Science Foundation, to Clogg by Grant No. SES-7823759 and to Goodman by Grant No. SES-8303838. Clogg and Goodman were Fellows at the Center for Advanced Study in the Behavioral Sciences when part of the research was done, with financial support provided in part by National Science Foundation grant BNS-8011494 to the Center. The authors are indebted to Mark P. Becker and James W. Shockey for helpful comments.     

An axiomatic formulation and generalization of successive intervals scaling

A formal set of axioms is presented for the method of successive intervals, and directly testable consequences of the scaling assumptions are derived. Then by a systematic modification of basic axioms the scaling model is generalized to non-normal stimulus distributions of both specified and unspecified form.     

A method of scalogram analysis using summary statistics

A method of Guttman scalogram analysis is presented that does not involve sorting and rearranging the entries in the item response matrix. The method requires dichotomous items. Formulas are presented for estimating the reproducibility of the scale and estimating the expected value of the chance reproducibility. An index of consistency is suggested for evaluating the reproducibility. An illustrative example is presented in detail. The logical basis of the method is discussed. Finally, several methods are suggested for dealing with non-dichotomous items.Lois K. Anderson assisted the author materially in the many computations required for this paper. The research reported in this paper was supported in part by the Department of Economics and Social Sciences at M.I.T. and in part, jointly, by the Army, Navy and Air Force under contract with the Massachusetts Institute of Technology.     

Examining the generality of children's preference for contingent reinforcement via extension to different responses, reinforcers, and schedules

Studies that have assessed whether children prefer contingent reinforcement (CR) or noncontingent reinforcement (NCR) have shown that they prefer CR. Preference for CR has, however, been evaluated only under continuous reinforcement (CRF) schedules. The prevalence of intermittent reinforcement (INT) warrants an evaluation of whether preference for CR persists as the schedule of reinforcement is thinned. In the current study, we evaluated 2 children's preference for contingent versus noncontingent delivery of highly preferred edible items for academic task completion under CRF and INT schedules. Children (a) preferred CR to NCR under the CRF schedule, (b) continued to prefer CR as the schedule of reinforcement became intermittent, and (c) exhibited a shift in preference from CR to NCR as the schedule became increasingly thin. These findings extend the generality of and provide one set of limits to the preference for CR. Applied implications, variables controlling preferences, and future research are discussed.     

Interdependence of Saccadic and Fixational Fluctuations

Recent research suggests that the different components of eye movements (fixations, saccades) are not strictly separate but are interdependent processes. This argument rests on observations that gaze-step sizes yield unimodal distributions and exhibit power-law scaling, indicative of interdependent processes coordinated across timescales. The studies that produced these findings, however, employed complex tasks (visual search, scene perception). Thus, the question is whether the observed interdependence is a fundamental property of eye movements or emerges in the interplay between cognitive processes and complex visual stimuli. In this study, we used a simple eye movement task where participants moved their eyes in a prescribed sequence at several different paces. We outlined diverging predictions for this task for independence versus interdependence of fixational and saccadic fluctuations and tested these predictions by assessing the spectral properties of eye movements. We found no clear peak in the power spectrum attributable exclusively to saccadic fluctuations. Furthermore, changing the pace of the eye movement sequence yielded a global shift in scaling relations evident in the power spectrum, not just a localized shift for saccadic fluctuations. These results support the conclusion that fixations and saccades are interdependent processes.     

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.     

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 (l/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.     

Analyses of Model Fit and Robustness. A New Look at the PISA Scaling Model Underlying Ranking of Countries According to Reading Literacy

This paper addresses methodological issues that concern the scaling model used in the international comparison of student attainment in the Programme for International Student Attainment (PISA), specifically with reference to whether PISA’s ranking of countries is confounded by model misfit and differential item functioning (DIF). To determine this, we reanalyzed the publicly accessible data on reading skills from the 2006 PISA survey. We also examined whether the ranking of countries is robust in relation to the errors of the scaling model. This was done by studying invariance across subscales, and by comparing ranks based on the scaling model and ranks based on models where some of the flaws of PISA’s scaling model are taken into account. Our analyses provide strong evidence of misfit of the PISA scaling model and very strong evidence of DIF. These findings do not support the claims that the country rankings reported by PISA are robust.     

A comparison of AME and CR100 for scaling perceived exertion

In psychophysical studies of the relation between perceived magnitude and physical stimulus, interest has focused on the preciseness of growth functions. Very little interest has been devoted to natural levels of an "absolute" character and to the validity of direct measurements. In this article, the Category (C)-Ratio (R) (CR) scaling methodology, developed by Borg (1973, 1977, 1982) is presented together with a new CR scale: the CR100 scale. In an experiment on perceived exertion, CR100 was compared with Absolute Magnitude Estimation (AME). Perceived exertion has the advantage of having easily measured physiological variables that can function as validity criteria. 32 persons (16 men and 16 women) participated as subjects on two different occasions, and the presentation order was counterbalanced. Results from CR100 and AME could be described equally well with psychophysical functions. The exponents obtained with CR100 were n = 1.60 for women and n = 1.69 for men, and the corresponding exponents for AME were n = 1.60 for women and n = 1.46 for men (with a "noise" constant included in the function). ANOVA showed that CR100 could make the predicted differentiation between sexes, whereas AME could not. This drawback with AME was also found when using the methods to predict working capacity.     

Application of a variable criterion model to auditory reaction time as a function of the type of catch trial

A variable criterion model has been applied to data from an RT experiment by LaBerge in which catch trial stimuli varied in similarity to a single auditory RT signal. Estimates of a sensory recruitment function and the parameters of hypothetical criterion distributions were obtained by a scaling solution. Auditory recruitment was an exponential growth function, and the criterion distributions were normal. In combination, these estimates successfully predicted the empirical RT distributions.     

An internal consistency check for scale values determined by the method of successive intervals

The method of successive intervals is a psychological scaling procedure in which stimuli are classified into successive intervals according to the degree of some defined attribute which they are judged to possess. A psychological continuum is defined and the scale values are then taken as the medians of the distributions of judgments on the psychological continuum. It is assumed that the distributions of judgments for each stimulus are normal on the psychological continuum as defined.An internal consistency check indicates that the cumulative distributions of empirical judgments for the various stimuli can be reproduced by means of a limited number of parameters with an average error that compares favorably with that usually reported for paired comparison data. Furthermore, the scale values obtained by successive interval scaling, for the data reported, are shown to be linearly related to those obtained by the method of paired comparisons.This paper was written while the first author was a post-doctoral Research Training Fellow of the Social Science Research Council studying at the University of Chicago. It reports research undertaken in cooperation with the Quartermaster Food and Container Institute for the Armed Forces. The views or conclusions contained in this report are those of the authors. They are not to be construed as necessarily reflecting the views or endorsement of the Department of Defense.     

Chance events in career development: Influence, control and multiplicity

This article reports three studies on the nature and impact of chance events. The first study investigated chance events in terms of the dimensions of influence and control. The second and third studies investigated the effects of multiplicity of chance events on career development are in terms of respondents’ own careers and then in terms of career scenarios. Results indicated that chance events can be characterized by the interactive influence of influence and control. High influence and low control chance events had the biggest impact on career development. When there is a connection between multiple chance events, the impact is greater than when unrelated chance events occur. The finding was confirmed regardless of whether individuals rated their own experiences or those presented in scenarios. However, it was also found that negative outcome chance events had the greatest impact regardless of whether such events were single or multiple influences on individuals’ careers. The results provide further support for the integration of chance events into the Chaos Theory of Careers.     

Automatic attention to stimuli signalling chances and dangers: Moderating effects of positive and negative goal and action contexts

Current research on automatic attention allocation focuses on the questions whether there is an asymmetry in attentional biases towards negative and positive stimuli and whether these attentional biases are influenced by situational variables. In an experiment with N = 48 participants, automatic allocation of attention to chance and danger stimuli was investigated. Attentional capture was generally larger for chance stimuli than for danger stimuli. Additionally, attentional bias was influenced by the outcome focus of the actual goal orientation. Results revealed an incongruence effect of goal orientation on attentional biases: Attentional capture for the chance and danger stimuli was comparatively stronger when an outcome focus of opposite valence had been induced.     

Motor conditioning of the albino rabbit: Apparatus and procedures for acquisition of the CR

Apparatus and procedures for shock-avoidance conditioning of the albino rabbit in the rotator are described. Differences in speed of conditioning, in number of CR’s, and in the form of the acquisition curve are examined for data obtained in a study of two groups of 100 rabbits each, which were given different numbers of conditioning trials in the daily test-periods. Training proceeded to the same criterion of conditioning. Although there were no significant differences in the mean numbers of trials needed to attain the criterion, there were significant differences in the variances of the two distributions and both the numbers of CR’s and the forms of the acquisition curves were significantly different for the two groups.     

A revised law of comparative judgment

In contrast to Thurstone's Law of Comparative Judgment, a model in which a comparison pair and its complement are assumed to give rise to two different distributions of differences is considered. The consequences of this revised model on scaling problems is developed.The research in this article was supported jointly by the Army, Navy, and Air Force under contract with the Massachusetts Institute of Technology.     

Social Support in Cardiac Rehabilitation Exercise Maintenance: Associations with Self‐Efficacy and Health‐Related Quality of Life1

This study examined whether cardiac rehabilitation (CR) maintenance exercise participants who were high and moderate in social support (SS) differed in their types of self‐efficacy and health‐related quality of life, and whether SS was related to self‐efficacy. Measures of SS, self‐efficacy (i.e., walking, scheduling, in‐class efficacy), and health‐related quality of life (HRQL) were administered to 64 CR maintenance exercise participants. A one‐way, between‐groups MANOVA was significant (p < .01), indicating that CR participants higher in SS reported significantly greater self‐efficacy and physical HRQL than did their moderate counterparts. In addition, social support predicted significant variance in task self‐efficacy. The findings suggest that differential perceptions of SS are related to differences in walking, in‐class and scheduling self‐efficacy, and the physical component of HRQL among CR exercise maintainers.     

An analysis of generative representation systems

Three ideas are basic to generative theory: (a) Subjects are assumed to attend to the relations among stimuli, extracting the transformations relating pairs of stimuli; (b) the set of abstracted transformations is decomposed or reduced to an elementary set of generators; (c) subjects use the elementary generators as the basis for judging similarity. The purpose of this paper is to illustrate these ideas with an experiment in which subjects were asked to rate the similarity between stimulus pairs. The stimulus materials consisted of the permutations of a 4-item pattern with the properties of a dihedral group which insured the existence of sets of elementary transformations. Three analytic techniques were used to determine the generator set of transformations abstracted by subjects. The first analysis consisted of a monotonic regression between dissimilarity ratings and the number of elementary generators of a given permutation. The residual variance of this monotone regression, suitably normalized, was used as a quantitative goodness-of-fit measure. For the stochastic analysis, cumulative distributions of dissimilarity ratings were obtained for permutations requiring one, two, or three generators. The idea was that permutations requiring fewer generators should be associated with distributions of lower dissimilarity values (higher similarity scores) as compared to permutations predicted to be transformationally more complex. The final analysis, a multidimensional scaling of dissimilarity ratings, converted subjects' ratings into spatial structures to determine whether individual subjects' ratings exhibited the predicted spatial arrangement. The monotone regression and stochastic analyses abstracted similar generator sets for individual subjects, some of which provided perfect fits to the data. Although the scaling analysis yielded similar estimates of generators, for some subjects, transformations with the same number of generators yielded unequal “cognitive” distances resulting in some-what deformed spatial structures for these subjects. It was concluded that the results generally supported a generative model as an approximation to subjects' representations of interstimulus relationships.     

A theoretical framework for testing the additive difference model for dissimilarities data: Representing gambles as multidimensional stimuli

Implicit within the acceptance of most multidimensional scaling models as accurate representations of an individual's cognitive structure for a set of complex stimuli, is the acceptance of the more general Additive Difference Model (ADM). A theoretical framework for testing the ordinal properties of the ADM for dissimilarities data is presented and is illustrated for a set of three-outcome gambles. Paired comparison dissimilarity judgments were obtained for two sets of gambles. Judgments from one set were first analyzed using the ALSCAL individual differences scaling model. Based on four highly interpretable dimensions derived from this analysis, a predicted set of dimensions were obtained for each subject for the second set of gambles. The ordinal properties of the ADM necessary for interdimensional additivity and intradimensional subtractivity were then tested for each subject's second set of data via a new computer-based algorithm, ADDIMOD. The tests indicated that the ADM was rejected. Although violations of the axioms were significantly less than what would be expected by chance, for only one subject was the model clearly supported. It is argued that while multidimensional scaling models may be useful as data reduction techniques, they do not reflect the perceptual processes used by individuals to form judgments of similarity. Implications for further study of multidimensional scaling models are offered and discussed.