Variabili ordinali e trasformazioni di scala, con particolare riferimento alla stima dei parametri dei modelli interpretativi con variabili latenti

[Autom. eng. transl.] The present work aims to study in depth the measurement, that is to say the typical tools for the quantification of results of phenomenal manifestations, whether they are of a physical, psychometric or simply of attitude and opinion, and the problems connected to the use of the scales of measurement and related transformations. "Measurement" is defined as a numerical or empirical process of assigning numbers and properties to elements or experimental events or situations in order to describe and qualify them. In particular we want to highlight how the measurement process actively determines the construction of the measurement scales, a necessary tool for the qualification, and sometimes quantification, of the phenomena under examination. The focus is on an important problem in the methodology and applications, most recent and current. In fact, the recent statistical studies of the "structure" underlying a "latent" notion, assessed through the conventional and arbitrary use of score scales, typical of the sociological and psychological area, are gaining great interest also for the economic, financial and company, especially to describe the notions of customer satisfaction and system quality. In these contexts the use of structural linear models with latent variables (known by the acronym LISREL), has become a current and widely used methodology. In these models the manifest variables play a fundamental role, that is to say the variables that refer to "empirical manifestations" linked to the latent notions of interest. It is common (and erroneous) practice in the usual statistical analyzes of LISREL models to assume that the manifest variables, related to known empirical phenomena, are measured on a metric scale, typically at intervals, while often the values of these variables, typically obtained from responses to a questionnaire from a group of respondents, they are expressed on a conventional scoring scale, which is only an ordinal scale. The present work therefore moves in this context, and it is proposed to rigorously and formally define the nature of the measurement scales, in order to be in a position to be able to establish exactly the nature of a scale linked to a manifest variable, and once defined from formal belonging, to provide criteria and methodologies to be able to make scale transformations, typically from sorting scales to metric scale at intervals. The work is basically divided into three parts. IV In the first part the concept of Representational Theory of Measurement is specified. Proceeding along the axiomatic path, a formal system is constructed for the definition of a measurement scale, and in this context the different measurement scales are configured, with an important monographic treatment for the ordinal scale. This part of the work draws inspiration and inspiration from the typically mathematical context of Measurement Theory, and is presented here in the statistical-probabilistic field. The second part is devoted to the study of ordinal scales: a critical bibliographic review is presented of the literature works carried out on the subject. An ad hoc survey was also implemented for the empirical verification of the inadequacy of certain hypotheses often assumed in the literature and in current practice regarding the nature of the measurement scales used in the evaluation of goods and services. This is followed by a discussion of the scale transformation techniques present in the literature (proposed by Herzel, Marbach, Del Vecchio) and recent research developments with a series of innovative proposals (Zanella). In particular, the substantial innovative contribution concerns some implications and a de extension