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A recurring question is whether rating scales should be considered metrically scaled or merely ordinally scaled. This has direct implications for the permissible statistical procedures for significance testing. Based on the results of a simulation study, it is shown that the use of parametric procedures for rating scales has distinct advantages over the nonparametric alternatives. It is also shown that the parametric procedures are robust to violations of the assumption of normality, which only result in a modest loss of power compared with continuous variables. This loss should be taken into account when calculating the optimal sample size. The results suggest that sample sizes about 25% larger should be chosen for discrete rating scales than for continuous variables. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Agent-based models (ABMs) often rely on psychometric constructs such as 'opinions', 'stubbornness', 'happiness', etc. The measurement process for these constructs is quite different from the one used in physics as there is no standardized unit of measurement for opinion or happiness. Consequently, measurements are usually affected by 'psychometric distortions,' which can substantially impact models' predictions. Even if distortions are well known in psychometrics, their existence and nature is obscure to many researchers outside this field. In this paper, we introduce distortions to the ABM community. Initially, we show where distortions come from and how to observe them in real-world data. We then show how they can strongly impact predictions, qualitative comparison with data and the problem they pose for validation of models. We conclude our analysis by discussing how researchers may mitigate this problem and highlight possible future modelling trends that will address this problem. [ABSTRACT FROM AUTHOR]

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 article aims to address some of the ensuing empirical challenges. It suggests an analytical way to approach the scientific complications of ordinal data. The article 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, this article shows 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. [ABSTRACT FROM AUTHOR]

The article discusses descriptive statistics of data measured in ordinal and quantitative scales, and criteria for determining the statistical significance of differences between samples when it is impossible to analyze using parametric methods. Special attention is paid to the problem of multiple comparisons of this type of data. For each method, examples of processing data obtained in pharmacological studies are given.

Social research is rich in methods for analysing societal differences. Yet, although qualitative characteristics are a key component to understanding such differences, the analysis of qualitative data remains a major methodological challenge in most social research, particularly when aiming to compare more than a few cases. The article proposes an innovative method for systematic comparison of qualitative data (SCQual), applying Weber's original idea of ideal types as a tool to measure and compare societal differences. This method allows for the quantification of qualitative data, thereby enabling systematic analysis and repeatable comparison of the qualitative characteristics of societal differences for a relatively large number of cases. Developed for the comparative analysis of essential characteristics of social orders, particularly comparative welfare state analysis, SCQual may also be applicable to other areas of social research. [ABSTRACT FROM AUTHOR]

Abstract 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.

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.

Data collected in many biology laboratory classes are on ratio or interval scales where the size interval between adjacent units on the scale is constant, which is a critical requirement for analysis with parametric statistics such as t-tests or analysis of variance. In other cases, such as ratings of disease or behavior, data are collected on ordinal scales in which observations are placed in a sequence but the intervals between adjacent observations are not necessarily equal. These data can only be interpreted in terms of their order, not in terms of the differences between adjacent points. They are unsuitable for parametric statistical analyses and require a rank-based approach using nonparametric statistics. We describe an application of one such approach, the Kruskal-Wallis test, to biological data using online freeware suitable for classroom settings. [ABSTRACT FROM AUTHOR]

The purpose of the present study was to (1) examine the internal validity of the Parental Involvement Survey in their Children's Elementary Studies (PISCES) and (2) illustrate how survey instruments can be validated using modern psychometric techniques. The PISCES was developed by the present authors by adopting items from the Hoover-Dempsey and Sandler Revised Model of Parent Involvement and the Parent Reading Belief Inventory. The PISCES is comprised of 49 new items and 35 modified items that measure parental beliefs about education, reading with children, self-efficacy, and involvement in school activities. Data were collected from 230 parents of kindergarten students enrolled in a major Midwest school district. We utilized modern psychometric techniques to validate the instrument, including ordinal parallel analysis, ordinal exploratory factor analysis (EFA), Rasch modeling, second-order EFA, and reliability analysis. Our findings revealed the PISCES attained a very high level of internal validity although some of its subscales could benefit from the addition of more items. Tables for converting the sum of individual item scores to Rasch scores are provided. We advise readers to use the whole instrument if they want a holistic measure of parental involvement and the individual scales if they are only interested in a particular domain of parental involvement. We also advise readers to adopt our conversion tables to facilitate comparisons across studies. Finally, we recommend that survey researchers utilize ordinal parallel analysis and ordinal EFA to investigate the dimensionality of survey instruments and Rasch modeling to further explore and refine them. [ABSTRACT FROM AUTHOR]

A disease severity index (DSI) is a single number for summarising a large amount of information on disease severity. The DSI has most often been used with data based on a special type of ordinal scale comprising a series of consecutive ranges of defined numeric intervals, generally based on the percent area of symptoms presenting on the specimen(s). Plant pathologists and other professionals use such ordinal scale data in conjunction with a DSI (%) for treatment comparisons. The objective of this work is to explore the effects on both of different scales (i.e. those having equal or unequal classes, or different widths of intervals) and of the selection of values for scale intervals (i.e. the ordinal grade for the category or the midpoint value of the interval) on the null hypothesis test for the treatment comparison. A two-stage simulation approach was employed to approximate the real mechanisms governing the disease-severity sampling design. Subsequently, a meta-analysis was performed to compare the effects of two treatments, which demonstrated that using quantitative ordinal rating grades or the midpoint conversion for the ranges of disease severity yielded very comparable results with respect to the power of hypothesis testing. However, the principal factor determining the power of the hypothesis test is the nature of the intervals, not the selection of values for ordinal scale intervals (i.e. not the mid-point or ordinal grade). Although using the percent scale is always preferable, the results of this study provide a framework for developing improved research methods where the use of ordinal scales in conjunction with a DSI is either preferred or a necessity for comparing disease severities. [ABSTRACT FROM AUTHOR]

We present a link between two types of logic systems for reasoning with graded if–then rules: the system of fuzzy logic programming (FLP) in sense of Vojtáš and the system of fuzzy attribute logic (FAL) in sense of Belohlavek and Vychodil. We show that each finite theory consisting of formulas of FAL can be represented by a definite program so that the semantic entailment in FAL can be characterized by correct answers for the program. Conversely, we show that for each definite program there is a collection of formulas of FAL so that the correct answers can be represented by the entailment in FAL. Using the link, we can transport results from FAL to FLP and vice versa which gives us, e.g., a syntactic characterization of correct answers based on Pavelka-style Armstrong-like axiomatization of FAL. We further show that entailment in FLP is reducible to reasoning with Boolean attribute implications and elaborate on related issues including properties of least models. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Background: Studies on how to better measure change have been published at least since the third decade of the last century, but no general indicator or strategy of measurement is currently agreed upon. The aim of this study is to propose a new indicator, the indicator of positive change, as an option for the assessment of change when ordinal scores are used in pretest and posttest designs. Methods: The basic idea is to measure the proportion of possible (positive) change inside a group that can be attributed to an intervention. The approach is based on the joint distribution of the before and after scores (differences), represented by the cells ( i, j) of a contingency table m ×  m ( m is the number of classes of the ordinal measurement scale; i and j are the lines and columns of the table, respectively). By convention, higher classes are the most unfavorable on the scale such that subjects that improve “migrate” from the higher to the lower classes as a result of an intervention and vice versa. Results: The introduced indicator offers a new strategy for the analysis of change when dealing with repeated measurements of the same subject, assuming that the measured variable is ordinal (e.g., clinician-rating scales). Conclusion: The presented approach is easily interpretable and avoids the problems that arise, for instance, in those cases where a large concentration of high/low scores is present at the baseline. [ABSTRACT FROM AUTHOR]

The need for scale conversion may arise whenever an attitude of individuals is measured by independent entrepreneurs each using an ordinal scale of its own with possibly different numbers of (arbitrary) ordinal categories. Such situations are quite common in the marketing realm. The conversion of a score of an individual measured on one scale into an estimated score of a similar scale with a different range is the concern of this paper. An inferential Bayesian approach is adopted to analyze the situation where we believe the scale with fewer categories can be obtained by collapsing the finer scale. This leads to inferences concerning rules for the conversion of scales. Further, we propose a method for testing the validity of such a model. The use of the proposed methodology is exemplified on real data from surveys concerning performance evaluation and satisfaction. [ABSTRACT FROM AUTHOR]

Scales of conclusion in forensic interpretation play an important role in the interface between scientific work at a forensic laboratory and different bodies of the jurisdictional system of a country. Of particular importance is the use of a unified scale that allows interpretation of different kinds of evidence in one common framework. The logical approach to forensic interpretation comprises the use of the likelihood ratio as a measure of evidentiary strength. While fully understood by forensic scientists, the likelihood ratio may be hard to interpret for a person not trained in natural sciences or mathematics. Translation of likelihood ratios to an ordinal scale including verbal counterparts of the levels is therefore a necessary procedure for communicating evidence values to the police and in the courtroom. In this paper, we present a method to develop an ordinal scale for the value of evidence that can be applied to any type of forensic findings. The method is built on probabilistic reasoning about the interpretation of findings and the number of scale levels chosen is a compromise between a pragmatic limit and mathematically well-defined distances between levels. The application of the unified scale is illustrated by a number of case studies. [ABSTRACT FROM AUTHOR]

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. 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 (α and β). 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. [ABSTRACT FROM AUTHOR]

The article offers an analysis of the Exercise Training for Claudication (ETC) Study conducted to model pain trajectory during treadmill (TM) testing in peripheral artery disease (PAD). Details related to the study which used hierarchical generalized linear model to describe pain trajectories during a peak walking test after treatment in participants in four random groups comprising TM, arm ergometry (AE), combination (TM and AE), and usual care (UC) as compared to baseline patterns.

Many theoretical constructs of interest to psychologists are multidimensional and derive from the integration of several input variables. We show that input variables that are measured on ordinal scales cannot be combined to produce a stable weakly ordered output variable that allows trading off the input variables. Instead a partial order is obtained in which the amount of ordering depends on the number and nature of the input variables and the relationship between them. However, if trade-offs are excluded, it is still possible to obtain a weak order using lexicographic ordering of the input variables. An implication is that psychological processes that integrate information from different input variables and that produce consistent output require that the input variables be measured on more than ordinal scales. A further implication is that the level of measurement of the input variables affects the kind of psychological model that can be applied to the process. [ABSTRACT FROM AUTHOR]

Recently, the data analysts pay more attention to the assessment of rater agreement, especially in areas of medical sciences. In this context, the statistical indices such as kappa and weighted kappa are the most common choices. These indices are simple to calculate and interpret, although, they fail to describe the structure of agreement, particularly when the available outcome has an ordinal nature. In the previous decades, statisticians suggested more efficient statistical tools such as diagonal parameter, linear by linear association and agreement plus linear by linear association models for describing the structure of rater agreement. In these models, the equal interval scores are the common choice for the levels of the ordinal scales. In this manuscript, we show that choosing the common equal interval scores does not necessarily lead to the best fit and propose a modification using a power transformation for the ordinal scores. We also use two different data sets (IOTN and ovarian masses data) to illustrate our suggestion more clearly. In addition, we utilize the category distinguishability concept for interpreting the model parameter estimates. [ABSTRACT FROM AUTHOR]

The article investigates the psychological distance (D) between response categories at varying scale lengths and the impact of response styles. The variables considered include the generalized partial credit model (GPCM), Likert-type scales with only the endpoints labeled (END) and fully labeled (ALL) scales, the scale format for extroversion and neuroticism, and net acquiescence. The correlation between extreme response styles (ERS) and greater D is evaluated.

This paper presents theoretical foundations for the description of preferences in multiple criteria decision analysis models using quantitative estimates of the relative importance of criteria. It is assumed that not only the values of criteria but also the differences of such values are ordered according to the preference of the decision maker. Simple decision rules that define the preference and indifference relations induced by the information about the quantitative importance of such criteria are developed. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Likert scales are widely used in survey studies for attitude measuring. In particular, the questionnaires propagated by the SERVQUAL approach are based on Likert scales. Though the problem of attitude suggests an ordinal interpretation of Likert scales, attitude survey data are often evaluated with techniques designed for cardinal measurements. The present paper discusses the interpretation of scales for attitude measuring and gives a survey of data analysis techniques under the proper ordinal understanding. [ABSTRACT FROM AUTHOR]

Grade averaging (by arithmetic mean) is often performed as an attempt to assess overall student performance. In the case of grade comparison originating in non-equivalent scales, rank errors and absurd averaging may result. As averages are sometimes used for candidate selection, the paper dicusses how decisions based on arithmetic mean interpretation may be true, false, or fuzzy, according to different categories of participating candidates. A two stage selection process is analyzed from the perspective of candidate categories. The impact of the choice of asessment scale on the decision-making process is also evaluated and statistical biases are identified. The relevance of using a uniformity criterion is demonstrated. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The scales used in schools for the purpose of student assessment are ordinal. The average of ordinal values is often used for the evaluation and comparison of overall student performance. We demonstrate a theorem for the selection of scales invariant with respect to rank of average and compare scales according to this property. A uniformity criterion is also defined for the choice of the scale on which to calculate the average. Concatenated sets of grades from scales not belonging to the same category may bring about errors of rank and absurd averaging, which may have a heavy impact on related decision-making processes. [ABSTRACT FROM AUTHOR]

We quantify the information loss incurred by categorizing an unobserved continuous variable (X) into an ordered categorical scale (Z). The continuous variable is conceptualized as true score (r) (which varies across individuals) plus random error (epsilon), with both components assumed to be normally distributed. The index of metric quality is operationalized as r[sup 2](Z, tau) /r[sup 2] (X, tau), where r[sup 2], the squared correlation coefficient, is a descriptive measure of the power of X or Z to predict r. The index is useful in defining limits on explanatory power (population R[sup 2]) in multiple regression models in which an ordered categorical variable is regressed against a set of predictors. The index can also be used to correct correlations for the effects of ordered categorical measurement. The index of metric quality is extended to the case when several ordered categorical scales are averaged as in the multi-item measurement of a construct. We prove theoretically that as long as the error variance is "large," the index of metric quality for the average Z of ordered categorical scales goes to 1 as the number of scales becomes "large." The index for averaged data is useful in answering questions such as whether the measurement of a construct by averaging three 5-point scales is better or worse than the measurement obtained by averaging five 3-point scales. The results indicate that the loss of information by marketing researchers' ad hoc use of Z as opposed to the more refined X is small (<10%) when the Z scale is well designed with at least five categories. The loss would be even smaller when a multi-item based Z is employed. [ABSTRACT FROM AUTHOR]