Facet theory
Facet theory is characterized by its direct concern with the entire content-universe under study, containing many, possibly infinitely many, variables.
The sampling of variables is done with the aid of the mapping sentence technique (see Section 1); and inferences from the sample of observed variables to the entire content universe are made with respect to correspondences between conceptual classifications (of attribute-variables or of population-members) and partitions of empirical geometric representation spaces obtained in data analysis (see Sections 2 & 3).
Of the many types of representation spaces that have been proposed,[1] two stand out as especially fruitful: Faceted-SSA (Faceted Smallest Space Analysis)[2][3] for structuring the investigated attribute (see Section 2); and POSAC (Partial Order Scalogram Analysis by base Coordinates)[4] for multiple scaling measurements of the investigated attribute (see Section 3).
Inasmuch as observed variables in a behavioral study form in fact but a sample from the content-universe of interest, facet theory's procedures and principles serve to avoid errors that may ensue from incidental sampling of observed variables, thus meeting the challenge of the replication crisis in psychological research and in behavioral research in general.
Facet Theory was initiated by Louis Guttman[5] and has been further developed and applied in a variety of disciplines of the behavioral sciences including psychology, sociology, and business administration.
The range facets of the various items (variables) need not be identical in size: they may have any finite number of scores, or categories, greater or equal to 2.
The mapping sentence serves as a unified semantic device for specifying the system of intelligence test items, according to the present conceptualization.
But, in the larger cycle of the scientific investigation (which includes Faceted SSA of empirical data, see next section), this conception may undergo adjustments and remolding, converging to improved choices of content-facets and observations, and ultimately to robust theories in research domain.
In general, mapping sentences may attain high levels of complexity, size and abstraction through various logical operations such as recursion, twist, decomposition and completion.
In drafting a mapping sentence, an effort is made to include the most salient content-facets, according to the researcher's existing conception of the investigated domain.
[7] In addition to guiding the collection of data, mapping sentences have been used to content-analyze varieties of conceptualizations and texts—such as organizational quality, legal documents and even dream stories.
In facet-theoretical data analysis, the attribute (e.g., intelligence) is likened to a geometric space of suitable dimensionality, whose points represent all possible items.
This procedure, then, tests the regional hypothesis that the sub-content-universes defined by a content-facet elements exist each as a distinct empirical entity.
[13] In many studies, different samples of variables conforming to Mapping Sentence 2 have been analyzed confirming two regional hypotheses: The superposition of these two partition patterns results in a scheme known as the Radex Theory of Intelligence, see Figure 1.
The radex structure, which originated earlier as "a new approach to factor analysis",[14] has been found also in the study of color perception[15] as well as in other domains of research.
Facet Theory proposes a way of transcending accidental clustering of variables by focusing on a robust and replicable aspect of the data, namely the partitionability of the attribute-space.
Besides analyzing a data matrix of N individuals by n variables, as discussed above, Faceted SSA is usefully employed in additional modes.
For example, intercorrelations between members of a multidisciplinary team of experts were computed based on their human quality-of life value assessments.
The resulting Faceted SSA map yielded a radex of disciplines, supporting the association between social institutions and human values.
Facet Theoretical measurement consists in mapping points a(pi) of A' into a coordinate space X of the lowest dimensionality while preserving observed order relations, including incomparability: Definition.
multiple scaling is facilitated by partial order scalogram analysis by base coordinates (POSAC) for which algorithms and computer programs have been devised.
Recent developments include the algorithms for computerized partitioning of the POSAC space by the range facet of each variable, which induces meaningful intervals on the coordinate scales, X, Y.
LSA1 procedure, within 2-dimensional POSAC/LSA program, is a special version of SSA with E* as the similarity coefficient, and with lattice ("city block") as the distance function.
Concerned with the entire cycle of multivariate research – concept definition, observational design, and data analysis for concept-structure and measurement, Facet Theory constitutes a novel paradigm for the behavioral sciences.
Guttman's SSA, as well as Multidimensional Scaling (MDS) in general, were often described as a procedure for visualizing similarities (e.g., correlations) between analyzed units (e.g., variables) in which the researcher has specific interest.
(See, for example, Wikipedia, October 2020: "Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a dataset").
(c) Faceted SSA is a useful preliminary procedure for performing meaningful non arbitrary measurements by Multiple Scaling (POSAC).
[32] Indeed, missing in Facet Theory are statistical significance tests that would indicate the stability of discovered or hypothesized partition patterns across population samples.
For example, it is not clear how to compute the probability of obtaining a hypothesized partition pattern, assuming that in fact the variables are randomly dispersed over the SSA map.
Moreover, Facet Theory adds a stringent requirement for establishing scientific lawfulness, namely that the hypothesized partition-pattern would hold also across different selections of variables, sampled from the same mapping sentence.
