FPA provides a data theoretic foundation to several procedures used very often in the social and behavioral sciences. Among these procedures are Sociometry and the structural analysis of social relations, Scalogram and Parallelogram Analysis in their one- and multidimensional versions, Osgood's Semantic Differential, Likert Scaling, and Kelly's Grid Techniques. These procedures were not treated in the now classic three volumes of the "Foundations of Measurement". We therefore provide detailed examples showing how FPA can be applied and what the practical benefits are of using this approach. Furthermore, we introduce a Three-- Mode FPA extension. For polytomous items, we demonstrate a technique to reduce the computational effort considerably, and briefly illustrate the treatment of missing data.
Key words: Sociometry, Scalogram, Parallelogram Analysis, Semantic Differential, Likert Scaling, Kelly Grid
The main purpose of this paper is to demonstrate the wide range and flexibility of FPA as a model to study contingencies. We first turn to a variant of sociometry which aims at revealing some of the social relations between members of a small group. Our example also requires the treatment of "missing data" which will be illustrated. Furthermore, we introduce the concept of a contrast zero cell, useful for limiting the computational effort and to analyze the uniqueness of solutions. Section 2 first illustrates the combination of two rating scales into one FPA solution. An interpretation of this kind of representation is offered as a monotone correlation, existing in various qualitative variants. The analysis of a Likert scale also discusses ways to evaluate the model fit. The next example studies the Kelly Grid of a Drug addict. A new method to analyze polytomous items with ordered categories is introduced, and the multidimensional scalogram structure of the responses to this grid are revealed. A repertory grid of social relations points to different aspects of the new method. Section 3 discusses Three-Mode FPA in the context of the Semantic Differential. Only some basic concepts are introduced, and the example provides merely some hints for future developments. The main idea of Three-Mode FPA is to pile up the various FPA solutions, found for different data sources, with respect to their similarity. In this context, we offer a method to quantify similarity for configurations of points in the plane. The final section provides some ideas how to find a "latent structure" in a sample so that this sample can be decomposed into several subgroups while a simple solution exists for every subgroup.
1. Structural analysis of sociometric and similar matrices
In the social sciences, quite often (square) matrices are analyzed with the property that the rows are defined by the same elements (as senders) as the columns (in the role of receivers). But not always is it meaningful to collect data on whether a source communicates with itself, dominates itself or likes itself. Then the main diagonal of this land of structure matrix remains empty. We will show how to analyze this kind of data, and at the same time provide a mode how to handle systematically or randomly missing data.
Bock (1952, his Tab. 1) provided data from 16 children in a ninth grade class. Each child selected three others "with whom he would most wish to work" (p. 264). The teacher announced that groups would be arranged on the basis of these choices. We report all choices in Fig. 1; the rows a ... p represent the pupils as senders, the columns A ... P as receivers of sociometric choices. The natural order of the letters refers to the arrangement in Bock's report. Our representation of Bock's data in our Fig. 1 reports the participants already as they are ordered by our two one-dimensional FPA solutions, one solution ordering the rows, the other solution ordering the columns (to be discussed below). …