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Gaussian mixture models (GMMs) are a popular and versatile tool for exploring heterogeneity in multivariate continuous data. Arguably the most popular way to estimate GMMs is via the expectation–maximization (EM) algorithm combined with model selection using the Bayesian information criterion (BIC). If the GMM is correctly specified, this estimation procedure has been demonstrated to have high recovery performance. However, in many situations, the data are not continuous but ordinal, for example when assessing symptom severity in medical data or modeling the responses in a survey. For such situations, it is unknown how well the EM algorithm and the BIC perform in GMM recovery. In the present paper, we investigate this question by simulating data from various GMMs, thresholding them in ordinal categories and evaluating recovery performance. We show that the number of components can be estimated reliably if the number of ordinal categories and the number of variables is high enough. However, the estimates of the parameters of the component models are biased independent of sample size. Finally, we discuss alternative modeling approaches which might be adopted for the situations in which estimating a GMM is not acceptable.

Background and objectives Facioscapulohumeral muscular dystrophy (FHSD) is a debilitating inherited muscle disease for which various therapeutic strategies are being investigated. Thus far, little attention has been given in FSHD to the development of scientifically sound outcome measures fulfilling regulatory authority requirements. The aim of this study was to design a patient‐reported Rasch‐built interval scale on activity and participation for FSHD. Methods A pre‐phase FSHD‐Rasch‐built overall disability scale (pre‐FSHD‐RODS; consisting of 159 activity/participation items), based on the World Health Organization international classification of disease‐related functional consequences was completed by 762 FSHD patients (Netherlands: n = 171; UK: n = 287; United States: n = 221; France: n = 52; Australia: n = 32). A proportion of the patient cohort completed it twice (n = 230; interval 2–4 weeks; reliability studies). The pre‐FSHD‐RODS was subjected to Rasch analyses to create a model fulfilling its requirements. Validity studies were performed through correlation with the motor function measure. Results The pre‐FSHD‐RODS did not meet the Rasch model expectations. Based on determinants such as misfit statistics and misfit residuals, differential item functioning, and local dependency, we systematically removed items until a final 38‐inquiry (originating from 32 items; six items split) FSHD‐RODS was constructed achieving Rasch model expectations. Adequate test‐retest reliability and (cross‐cultural and external) validity scores were obtained. Conclusions The FSHD‐RODS is a disease‐specific interval measure suitable for detecting activity and participation restrictions in patients with FSHD with good item/person reliability and validity scores. The use of this scale is recommended in the near future, to determine the functional deterioration slope in FSHD per year as a preparation for the upcoming clinical intervention trials in FSHD., This paper presents the development and validation of a patient‐reported Rasch‐built interval scale on activity and participation for facioscapulohumeral muscular dystrophy (FSHD‐RODS). The final 38‐inquiry scale met Rasch model expectations and showed adequate discriminative power, test‐retest reliability and validity. The use of this scale is recommended in the near future to determine the functional deterioration slope in facioscapulohumeral muscular dystrophy (FSHD) per year as a preparation for the upcoming clinical intervention trials in FSHD.

Governments, multinational companies, and researchers today collect unprecedented amounts of data on human feelings. These data provide information on citizens' happiness, levels of customer satisfaction, employees' satisfaction, mental stress, societal trust, and other important variables. Yet a key scientific difficulty tends to be downplayed, or even ignored, by many users of such information. Human feelings are not measured in objective cardinal units. This paper aims to address some of the ensuing empirical challenges. It suggests an analytical way to approach the scientific complications of ordinal data. The paper describes a dichotomous-around-the-median (DAM) test, which, crucially, uses information only on direction within an ordering and deliberately discards the potentially unreliable statistical information in ordered data. Applying the proposed DAM approach, the paper demonstrates that it is possible to check and replicate some of the key conclusions of previous research—including earlier work on the effects upon human well-being of higher income.

In square contingency tables, analysis of agreement between the row and column classifications is of interest. In such tables, the kappa-like statistics are used as a measure of reliability. In addition to the kappa coefficients, several authors discussed agreement in terms of log-linear models. Log-linear agreement models are suggested for use to summarize the degree of agreement between nominal variables. To analyze the agreement between ordinal categories, the association models with agreement parameter can be used. In the recent studies, researchers pay more attention to the assessment of agreement among more than two raters’ decisions, especially in areas of medical and behavioral sciences. This article focuses on the approaches to study of uniform and non-uniform association with inter-rater agreement for multi-raters with ordered categories. In this article, we proposed different modifications of association plus agreement models and illustrate use of the approaches over two numerical examples., Statistica, Vol 77, No 4 (2017)

It seems common to believe that measures reported by participants in the behavioral sciences attain ordinal but not interval status. We consider three different measurement situations: Where one value is obtained from each of a number of respondents; where the measure is formed from combining other measures; and where the measure is obtained as a result of an observer perceiving stimuli over a period of time. In each case the presumption that the scale is ordinal produces serious theoretical and practical problems. The issue of where a measure is presumed to be located and the possibility that different measures are useable for the same construct are important considerations. The overall conclusion is that scales which are ordinal but not interval are only rarely available or useful in psychology.

To our knowledge, all applications of the quantum framework in social sciences are used to model measurements done on a discrete nominal scale. However, especially in cognition, experiments often produce data on an ordinal scale, which implies some internal structure between the possible outcomes. Since there are no ordinal scales in physics, orthodox projection-valued measurement (PVM) lacks the tools and methods to deal with these ordinal scales. Here, we sketch out an attempt to incorporate the ordinal structure of outcomes into the subspaces representing these outcomes. This will also allow us to reduce the dimensionality of the resulting Hilbert spaces, as these often become too high in more complex quantum-like models. To do so, we loosen restrictions placed upon the PVM (and even POVM) framework. We discuss the two major consequences of this generalization: scaling and the loss of repeatability. We also present two applications of this approach, one in game theory and one concerning Likert scales.

The Happiness Scale Interval Study deals with survey questions on happiness, using verbal response options, such as 'very happy' and 'pretty happy'. The aim is to estimate what degrees of happiness are denoted by such terms in different questions and languages. These degrees are expressed in numerical values on a continuous [0 ,10] scale, which are then used to compute 'transformed' means and standard deviations. Transforming scores on different questions to the same scale allows to broadening the World Database of Happiness considerably. The central purpose of the Happiness Scale Interval Study is to identify the happiness values at which respondents change their judgment from e.g. ‘very happy’ to ‘pretty happy’ or the reverse. This paper deals with the methodological/statistical aspects of this approach. The central question is always how to convert the frequencies at which the different possible responses to the same question given by a sample into information on the happiness distribution in the relevant population. The primary (cl)aim of this approach is to achieve this in a (more) valid way. To this end, a model is introduced that allows for dealing with happiness as a latent continuous random variable, in spite of the fact that it is measured as a discrete one. The [0,10] scale is partitioned in as many contiguous parts as the number of possible ratings in the primary scale sums up to. Any subject with a (self-perceived) happiness in the same subinterval is assumed to select the same response. 2 For the probability density function of this happiness random variable, two options are discussed. The first one postulates a uniform distribution within each of the different subintervals of the [0,10] scale. On the basis of these results, the mean value and variance of the complete distribution can be estimated. The method is described, including the precision of the estimates obtained in this way. The second option assumes the happiness distribution to be described as a beta distribution on the interval [0,10] with two shape parameters (? and ?). From their estimates on the basis of the primary information, the mean value and the variance of the happiness distribution in the population can be estimated. An illustration is given in which the method is applied to existing measurement results of 20 surveys in The Netherlands in the period 1990 – 2008. The results clarify our recommendation to apply the model with a uniform distribution within each of the category intervals, in spite of a better validity of the alternative on the basis of a beta distribution. The reason is that the recommended model allows to construct a confidence interval for the true but unknown population happiness distribution. The paper ends with a listing of actual and potential merits of this approach, which has been described here for verbal happiness questions, but which is also applicable to phenomena which are measured along similar lines.

BackgroundNo consensus exists which quality of life (QoL) measure should be used in patients with inflammatory neuropathies. Moreover, most QoL measures are ordinal-based scales with their known deficiencies.ObjectivesTo establish a new disease-specific interval-based QoL questionnaire in inflammatory neuropathies (IN-QoL) using the Rasch model and evaluate its scientific properties (validity, reliability and responsiveness).Methods264 patients with inflammatory neuropathies completed six commonly used QoL questionnaires. The obtained data were stacked and subjected to Rasch analysis. Responsiveness was determined by using the concept of minimum clinically important differences related to varying individually obtained SEs (responsiveness definition: MCID-SE≥1.96 after 1-year follow-up compared with baseline).ResultsThe IN-QoL fulfilled all Rasch’s model requirements with high internal reliability values (patient separation index of 0.94), except being multidimensional. Additional factor analysis resulted in two (functional and mental) subsets that were unidimensional on their own. The IN-QoL showed good correlation with the EuroQol-health quality visual analogue scale (EQ-VAS) (Spearman’s rho 0.72). It demonstrated acceptable responsiveness in patients with Guillain-Barré syndrome (GBS) and chronic inflammatory demyelinating polyradiculoneuropathy (CIDP), as did the EQ-VAS. In patients with monoclonal gammopathy-related neuropathy and multifocal motor neuropathy, hardly any changes were seen over time.ConclusionThe IN-QoL questionnaire fulfils modern clinimetric requirements and correlates strongly with a patient’s self-assessment of their own quality of health, while also showing responsiveness in patients with GBS and CIDP. We propose using the IN-QoL and the EQ-VAS for assessing the QoL of patients with inflammatory neuropathies in future studies.

Many practical problems of quality control involve the use of ordinal scales. Questionnaires planned to collect judgments on qualitative or linguistic scales, whose levels are terms such as ‘‘good,’’ ‘‘bad,’’ ‘‘medium,’’ etc., are extensively used both in evaluating service quality and in visual controls for manufacturing industry. In an ordinal environment, the concept of distance between two generic levels of the same scale is not defined. Therefore, a population (universe) of judgments cannot be described using ‘‘traditional’’ statistical distributions since they are based on the notion of distance. The concept of ‘‘distribution shape’’ cannot be defined as well. In this article, we introduce a new statistical entity, the so-called ordinal distribution, to describe a population of judgments expressed on an ordinal scale. We also discuss which of the traditional location and dispersion measures can be used in this context and we briefly analyze some of their properties. A new dispersion measure, the ordinal range, as an extension of the cardinal range to ordinal scales, is then proposed. A practical application in the field of quality is developed throughout the article.

Multifunctionality of typical gamma-spectrometric complexes, which should solve various tasks of applied nuclear gamma-ray spectrometry is considered. These tasks include:- defining nuclide and element composition of matter;- developing the system for determining metrological registration factors of gamma-radiation of bulk samples;- developing the background stabilization and reduction system;- developing the internal security system.The description of the developed and existing research stands that solve these tasks at the gamma-spectrometric complex of the Institute of Electron Physics of the NAS of Ukraine is given. It is shown that these systems act as autonomous research stands that typically exist as independent units.The fact of efficiency of multifunctionality of gamma-spectrometric complex allows to conclude on the possibility to implement functional completeness in the gamma-spectrometric complex as a system.Operation of research stands allows to make the process of improving gamma-spectrometric complex indicators in whole and in specific cases constant.Results are related to typical gamma-spectrometric complexes, which are in the majority. Therefore, our results can be used in practice in many analytical laboratories., Рассмотрена многофункциональность типовых гамма-спектрометрических комплексов, которые должны решать различные задачи прикладной ядерной гамма-спектрометрии. К таким задачам относятся: определение нуклидного и элементного состава вещества; разработка системы определения метрологических коэффициентов регистрации гамма-излучения объемных образцов; разработка системы стабилизации и понижения фона; разработка системы внутренней защиты. Показано, что эти системы выступают в качестве автономных исследовательских стендов, которые обычно существуют как независимые подразделения., Розглянуто багатофункціональність типових гамма-спектрометричних комплексів, які повинні вирішувати різноманітні задачі прикладної ядерної гамма-спектрометрії. До таких задач відносять: визначення нуклідного та елементного складу речовини; розробка системи визначення метрологічних коефіцієнтів реєстрації гамма-випромінювання об'ємних зразків; розробка системи стабілізації та пониження фону; розробка системи внутрішнього захисту. Показано, що ці системи виступають у ролі автономних дослідницьких стендів, які зазвичай існують як незалежні підрозділи.

Рассматривается метод адаптивного многозонального управления энерготехнологическими объектами в порядковых шкалах. Метод основан на введении порядковых оценок качества распределенных выходных характеристик многозонального энерготехнологического объекта. Оценки определяются на основе использования дискриминантных функций качества, которые выделяют на множестве распределенных характеристик объекта области их допустимых значений. Предложен алгоритм идентификации дискриминантных функций объекта на основе данных эксплуатации. Для устойчивого решения задачи идентификации использована регуляризация в виде минимизации нормы уклонения текущих оценок структурных параметров дискриминантных функций от их номинальных значений. Нахождение оптимальных значений режимных параметров энерготехнологического объекта осуществляется на основе двух этапов принятия решений. На первом этапе решается задача построения области эффективных значений режимных параметров в порядковых шкалах. Данная задача сводится к нахождению максимально совместных подсистем взвешенных неравенств, выделяющих область эффективных значений режимных параметров объекта. На втором этапе в построенной области определяются оптимальные значения параметров по критерию минимума потребления энергии. В целом на основе указанных задач решается нестационарная задача оперативного принятия решений по управлению режимами энерготехнологического объекта при изменениях его состояния в процессе эксплуатации. An adaptive polyzonal control method in ordinal scales for energotechnological objects is considered. The method is based on using ordinal scales for quality estimation of the object output characteristics. Discriminant functions are used for quality grades estimation which determine the corresponding output characteristics domains. A discriminant functions identification algorithm based on object operation data is offered. Robust solution of identification problem is achieved using a regularization method based on minimizing the deviation norm of the estimates of disctiminant function parameters from its nominal values. Solving for optimum values of work-process parameters is based on successive solutions of two subproblems. At the first stage a construction problem of efficient work-process parameters values domain in ordinal scales is solved. The problem is brought to determination of maximum consistent inequalities set of discriminant functions. At the second stage optimum values of work-process parameters are determined using minimum energy consumption criterion. As a whole, a non-stationary on-line decision- making problem for energotechnological objects with variable parameters using the above-considered subproblems is solved. Вставская Елена Владимировна, канд. техн. наук, доцент кафедры автоматики и управления, Южно-Уральский государственный университет (г. Челябинск); elena_vstavskaya@mail.ru. Казаринов Лев Сергеевич, д-р техн. наук, профессор, декан приборостроительного факультета, Южно-Уральский государственный университет (г. Челябинск); kazarinov@ait.susu.ac.ru. E.V. Vstavskaya, South Ural State University, Chelyabinsk, Russian Federation, elena_vstavskaya@mail.ru, L.S. Kazarinov, South Ural State University, Chelyabinsk, Russian Federation, kazarinov@ait.susu.ac.ru

The paper presents a new method for statistical process control when ordinal variables are involved. This is the case of a quality characteristic evaluated by an ordinal scale. The method allows a statistical analysis without exploiting an arbitrary numerical conversion of scale levels and without using the traditional sample synthesis operators (sample mean and variance). It consists of a different approach based on the use of a new sample scale obtained by ordering the original variable sample space according to some specific ‘dominance criteria’ fixed on the basis of the monitored process characteristics. Samples are directly reported on the chart and no distributional shape is assumed for the population (universe) of evaluations. Finally, a practical application of the method in the health sector is provided. Copyright © 2005 John Wiley & Sons, Ltd.

This paper will give a description of all continuous functions which are comparison meaningful in the sense of measurement theory. Copyright 2001 Academic Press.

We present a model allowing to aggregate decision criteria when the available information is of qualitative nature. The use of the Sugeno integral in this model is justified by an axiomatic approach. An illustrative example is also provided.

The Sugeno integral, for a given fuzzy measure, is studied under the viewpoint of aggregation. In particular, we give some equivalent expressions of it. We also give an axiomatic characterization of the class of all the Sugeno integrals. Some particular subclasses, such as the weighted maximum and minimum functions are investigated as well.

Measurement Theory has been proposed as an approach for providing software measurement with a sound basis. However, Measurement Theory has hardly ever been used for building new measures. Much more often, it has been used to analyze the properties of existing measures, but there has been little agreement on the compliance of existing measures with Measurement Theory's strict requirements. This paper introduces a modified version of Measurement Theory, called Weak Measurement Theory. Because it has weaker requirements than Measurement Theory, Weak Measurement Theory may be better suited for the needs of the state of the art of software measurement. We provide an extension of the theory of the levels of measurement and we focus on ordinal scales and extensive measurement for Weak Measurement Theory. In addition, we show how Weak Measurement Theory can be used for widening the application scope of mean values.