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.     

Checking the Assumptions of Rasch's Model for Speed Tests

Two new tests for a model for the response times on pure speed tests by Rasch (1960) are proposed. The model is based on the assumption that the test response times are approximately gamma distributed, with known index parameters and unknown rate parameters. The rate parameters are decomposed in a subject ability parameter and a test difficulty parameter. By treating the ability as a gamma distributed random variable, maximum marginal likelihood (MML) estimators for the test difficulty parameters and the parameters of the ability distribution are easily derived. Also the model tests proposed here pertain to the framework of MML. Two tests or modification indices are proposed. The first one is focused on the assumption of local stochastic independence, the second one on the assumption of the test characteristic functions. The tests are based on Lagrange multiplier statistics, and can therefore be computed using the parameter estimates under the null model. Therefore, model violations for all items and pairs of items can be assessed as a by-product of one single estimation run. Power studies and applications to real data are included as numerical examples.     

A person-fit index for polytomous rasch models,latent class models,and their mixture generalizations

A normally distributed person-fit index is proposed for detecting aberrant response patterns in latent class models and mixture distribution IRT models for dichotomous and polytomous data.This article extends previous work on the null distribution of person-fit indices for the dichotomous Rasch model to a number of models for categorical data. A comparison of two different approaches to handle the skewness of the person-fit index distribution is included.Major parts of this paper were written while the first author worked at the Institute for Science Education, Kiel, Germany. Any opinions expressed in this paper are those of the authors and not necessarily of Educational Testing Service. The results presented in this paper were improved by valuable comments from J. Rost, K. Yamamoto, N.D. Verhelst, E. Bedrick and two anonymous reviewers.     

Canonical analysis of two convex polyhedral cones and applications

Canonical analysis of two convex polyhedral cones consists in looking for two vectors (one in each cone) whose square cosine is a maximum. This paper presents new results about the properties of the optimal solution to this problem, and also discusses in detail the convergence of an alternating least squares algorithm. The set of scalings of an ordinal variable is a convex polyhedral cone, which thus plays an important role in optimal scaling methods for the analysis of ordinal data. Monotone analysis of variance, and correspondence analysis subject to an ordinal constraint on one of the factors are both canonical analyses of a convex polyhedral cone and a subspace. Optimal multiple regression of a dependent ordinal variable on a set of independent ordinal variables is a canonical analysis of two convex polyhedral cones as long as the signs of the regression coefficients are given. We discuss these three situations and illustrate them by examples.     

A class of multidimensional IRT models for testing unidimensionality and clustering items

We illustrate a class of multidimensional item response theory models in which the items are allowed to have different discriminating power and the latent traits are represented through a vector having a discrete distribution. We also show how the hypothesis of unidimensionality may be tested against a specific bidimensional alternative by using a likelihood ratio statistic between two nested models in this class. For this aim, we also derive an asymptotically equivalent Wald test statistic which is faster to compute. Moreover, we propose a hierarchical clustering algorithm which can be used, when the dimensionality of the latent structure is completely unknown, for dividing items into groups referred to different latent traits. The approach is illustrated through a simulation study and an application to a dataset collected within the National Assessment of Educational Progress, 1996. The author would like to thank the Editor, an Associate Editor and three anonymous referees for stimulating comments. I also thank L. Scaccia, F. Pennoni and M. Lupparelli for having done part of the simulations.     

Residualanalysis in the polytomous rasch model

Residuals for check of model fit in the polytomous Rasch model are examined. Comparisons are made between using counts for all response pattern and using item totals for score groups for the construction of the residuals. Comparisons are also, for the residuals based on score group totals, made between using as basis the item totals, or using the estimated item parameters. The developed methods are illustrated by two examples, one from a psychiatric rating scale, one from a Danish Welfare Study.     

Some neglected problems in IRT

The paper addresses three neglected questions from IRT. In section 1, the properties of the “measurement” of ability or trait parameters and item difficulty parameters in the Rasch model are discussed. It is shown that the solution to this problem is rather complex and depends both on general assumptions about properties of the item response functions and on assumptions about the available item universe. Section 2 deals with the measurement of individual change or “modifiability” based on a Rasch test. A conditional likelihood approach is presented that yields (a) an ML estimator of modifiability for given item parameters, (b) allows one to test hypotheses about change by means of a Clopper-Pearson confidence interval for the modifiability parameter, or (c) to estimate modifiability jointly with the item parameters. Uniqueness results for all three methods are also presented. In section 3, the Mantel-Haenszel method for detecting DIF is discussed under a novel perspective: What is the most general framework within which the Mantel-Haenszel method correctly detects DIF of a studied item? The answer is that this is a 2PL model where, however, all discrimination parameters are known and the studied item has the same discrimination in both populations. Since these requirements would hardly be satisfied in practical applications, the case of constant discrimination parameters, that is, the Rasch model, is the only realistic framework. A simple Pearsonx ² test for DIF of one studied item is proposed as an alternative to the Mantel-Haenszel test; moreover, this test is generalized to the case of two items simultaneously studied for DIF.     

Adaptation in Visuomanual Tracking Depends on Intact Proprioception

The role of arm proprioception in motor learning was investigated in experiments in which, by moving the arm, subjects followed the motion of a target displayed on a monitor screen. Adaptive capabilities were tested in visuomanual tracking tasks following alterations in the relationship between the observer's actual arm movement and visual feedback of the arm movement given by a cursor motion on the screen. Tracking performance and adaptive changes, measured in terms of spatiotemporal error, tracking trajectory curvature, and spatial gain, were compared in 7 control subjects (CSs) and in 1 deafferented subject (DS). CSs adapted appropriately to altered visuomanual relationships; those changes were present in trials immediately after restoration of normal scaling. In contrast, although the DS modified his tracking strategy from trial to trial according to the altered conditions, he did not show plastic changes in internal visuomanual scaling. Like the results of prismatic adaptation experiments, the present results suggest that arm proprioception contributes to the plastic changes that follow alterations in the scaling of visuomanual gain.     

Reducing Measurement Error in Student Achievement Estimation

The achievement level is a variable measured with error, that can be estimated by means of the Rasch model. Teacher grades also measure the achievement level but they are expressed on a different scale. This paper proposes a method for combining these two scores to obtain a synthetic measure of the achievement level based on the theory developed for regression with covariate measurement error. In particular, the focus is on ordinal scaled grades, using the SIMEX method for measurement error correction. The result is a measure comparable across subjects with smaller measurement error variance. An empirical application illustrates the method.