On some properties of Cronbach’s α coefficient for interval-valued data in questionnaires

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.