On some properties of Cronbach’s α coefficient for interval-valued data in questionnaires
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Abstract Along recent years, interval-valued rating scales have been considered as an alternative to traditional single-point psychometric tools for human evaluations, such as Likert-type or visual analogue scales. More concretely, in answering to intrinsically imprecise items in a questionnaire, interval-valued scales seem to allow capturing a richer information than conventional ones. When analyzing data from given performances of questionnaires, one of the main targets is that of ensuring the internal consistency of the items in a construct or latent variable. The most popular indicator of internal consistency, whenever answers to items are given in accordance with a numerically based/encoded scale, is the well-known Cronbach α coefficient. This paper aims to extend such a coefficient to the case of interval-valued answers and to analyze some of its main statistical properties. For this purpose, after presenting some formal preliminaries for interval-valued data, firstly Cronbach’s α coefficient is extended to the case in which the constructs of a questionnaire allow interval-valued answers to their items. The range of the potential values of the extended coefficient is then discussed. Furthermore, the asymptotic distribution of the sample Cronbach α coefficient along with its bias and consistency properties, are examined from a theoretical perspective. Finally, the preceding asymptotic distribution of the sample coefficient as well as the influence of the number of respondents to the questionnaire and the number of items in the constructs are empirically illustrated through simulation-based studies.Keywords:
Interval data
Cooperative interval games and interval solution concepts are useful tools for modeling various economic, modern finance and Operations Research situations where payoffs are affected by interval uncertainty. In this study, we deal with mountain situations arising from connection situations where edge costs are closed intervals of real numbers. We try to give a fair allocation by using the extended obligation rule.
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In this paper, for performance measurement of decision making units (DMUs) with interval data, it is suggested to integrate the efficiencies obtained from optimistic and pessimistic views in the form of an interval. The upper bound of efficiency interval from the optimistic view is obtained based on the most favourable position of each DMU using a set of the most favourable weights. The lower bound of efficiency interval from the pessimistic view is obtained based on the most unfavourable position of each DMU using a set of the most unfavourable weights. The obtained efficiency interval shows all possible evaluations from different perspectives. Thus, the efficiency interval provides the decision-maker with all possible efficiency values, reflecting different views. Two numerical examples will show the proposed data envelopment analysis approach.
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This paper deals with the research area of cooperative interval games arising from airport situations with interval data. We also extend to airport interval games some results from classical theory.
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The basic assumption in data envelopment analysis patterns (DEA) (such as the CCR andBCC models) is that the value of data related to the inputs and outputs is a precise andpositive number, but most of the time in real conditions of business, determining precisenumerical value is not possible in for some inputs or outputs. For this purpose, differentmodels have been proposed in DEA for imprecise data over recent years and also severalresearches have been conducted on DEA that are able to evaluate efficiency with negativedata. The negative interval DEA pattern which has been introduced and used in the presentstudy, addresses uncertainty both in inputs and outputs and provides user with more stableand reliable results for decision making.Now, in this paper a model is presented that is able to compute efficiency interval of unitswith interval input and output that while some indicators can also be negative and then weprove that the efficiency interval that this model gives us is more precise compared toefficiency interval of models previously proposed and finally, ten decision making units(DMUs) with the negative imprecise (interval) data are investigated by the proposed modeland the results of the proposed model are compared with the results of the previous models.
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