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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.

Background Despite the vagueness and uncertainty that is intrinsic in any medical act, interpretation and decision (including acts of data reporting and representation of relevant medical conditions), still little research has focused on how to explicitly take this uncertainty into account. In this paper, we focus on the representation of a general and wide-spread medical terminology, which is grounded on a traditional and well-established convention, to represent severity of health conditions (for instance, pain, visible signs), ranging from Absent to Extreme. Specifically, we will study how both potential patients and doctors perceive the different levels of the terminology in both quantitative and qualitative terms, and if the embedded user knowledge could improve the representation of ordinal values in the construction of machine learning models. Methods To this aim, we conducted a questionnaire-based research study involving a relatively large sample of 1,152 potential patients and 31 clinicians to represent numerically the perceived meaning of standard and widely-applied labels to describe health conditions. Using these collected values, we then present and discuss different possible fuzzy-set based representations that address the vagueness of medical interpretation by taking into account the perceptions of domain experts. We also apply the findings of this user study to evaluate the impact of different encodings on the predictive performance of common machine learning models in regard to a real-world medical prognostic task. Results We found significant differences in the perception of pain levels between the two user groups. We also show that the proposed encodings can improve the performances of specific classes of models, and discuss when this is the case. Conclusions In perspective, our hope is that the proposed techniques for ordinal scale representation and ordinal encoding may be useful to the research community, and also that our methodology will be applied to other widely used ordinal scales for improving validity of datasets and bettering the results of machine learning tasks.

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.

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.

The handling of ordinal variables presents many difficulties in both the measurements phase and the statistical data analysis. Many efforts have been made to overcome them. An alternative approach to traditional methods used to process ordinal data has been developed over the last two decades. It is based on a fuzzy inference system and is presented, here, applied to the student evaluations of teaching data collected via Internet in Modena, during the academic year 2009/10, by a questionnaire containing items with a four-point Likert scale. The scores emerging from the proposed fuzzy inference system proved to be approximately comparable to scores obtained through the practical, but questionable, procedure based on the average of the item value labels. The fuzzification using a number of membership functions smaller than the number of modalities of input variables yielded outputs that were closer to the average of the item value labels. The Center-of-Area defuzzification method showed good performances and lower dispersion around the mean of the value labels.

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.

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

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

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.