A New Measurement Instrument for Music-Related Argumentative Competence: The MARKO Competency Test and Competency Model

In this paper, we introduce the MARKO competency test and competency model, a new measurement instrument for music-related argumentative competence (MARKO: Musikbezogene ARgumentationsKOmpetenz; German for music-related argumentative competence). This competence, which plays an essential role in school curricula, refers to the ability to justify, and defend judgments about music. The two main goals of this study were 1) to design an assessment test for music-related argumentation that fulfills psychometric criteria and 2) to derive competency levels based on empirical data to describe the cognitive dispositions that are necessary when engaging in argumentation about music. Based on a theoretical framework, we developed a competency test to assess music-related argumentative competence. After two pretests (n = 391), we collected data from 440 students from grade nine to the university level. The final test consisted exclusively of open-ended items, which were rated with coding schemes that had been designed for each item. After ensuring inter-rater reliability, we composed an item pool that met psychometric criteria (e.g., local stochastic independence and item homogeneity) and represented content-related aspects in a meaningful way. Based on this item pool, we estimated a one-dimensional partial credit model. Following a standard-setting approach, four competency levels were derived from the empirical data. While individuals on the lowest competency level expressed their own opinions about the music by referring to salient musical attributes, participants on the highest level discussed different opinions on the music, and considered the social and cultural context of the music. The proficiency scores significantly varied between grades. Our findings empirically support some theoretical assumptions about music-related argumentation and challenge others.

New results on perturbation-based copulas

A prominent example of a perturbation of the bivariate product copula (which characterizes stochastic independence) is the parametric family of Eyraud-Farlie-Gumbel-Morgenstern copulas which allows small dependencies to be modeled. We introduce and discuss several perturbations, some of them perturbing the product copula, while others perturb general copulas. A particularly interesting case is the perturbation of the product based on two functions in one variable where we highlight several special phenomena, e.g., extremal perturbed copulas. The constructions of the perturbations in this paper include three different types of ordinal sums as well as flippings and the survival copula. Some particular relationships to the Markov product and several dependence parameters for the perturbed copulas considered here are also given.

Non-commutative stochastic independence and cumulants

In a fundamental lemma we characterize “generating functions” of certain functors on the category of algebraic non-commutative probability spaces. Special families of such generating functions correspond to “unital, associative universal products” on this category, which again define a notion of non-commutative stochastic independence. Using the fundamental lemma, we prove the existence of cumulants and of “cumulant Lie algebras” for all independences coming from a unital, associative universal product. These include the five independences (tensor, free, Boolean, monotone, anti-monotone) appearing in Muraki’s classification, c-free independence of Bożejko and Speicher, the indented product of Hasebe and the bi-free independence of Voiculescu. We show how the non-commutative independence can be reconstructed from its cumulants and cumulant Lie algebras.