Facet theory

Facet theory is a metatheory for the multivariate behavioral sciences that posits that scientific theories and measurements can be advanced by discovering relationships between conceptual classifications of research variables and empirical partitions of data-representation spaces. For this purpose, facet theory proposes procedures for Constructing or selecting variables for observation, using the mapping sentence technique, and Analyzing multivariate data, using data representation spaces, notably those depicting similarity measures, or partially ordered sets, derived from the data.
Facet theory is characterized by its direct concern with the entire content-universe under study, containing many, possibly infinitely many, variables. Observed variables are regarded just as a sample of statistical units from the multitude of variables that make up the investigated attribute. Hence, Facet theory proposes techniques for sampling variables for observation from the entire content universe; and for making inferences from the sample of observed variables to the entire content universe. The sampling of variables is done with the aid of the mapping sentence technique ; and inferences from the sample of observed variables to the entire content universe are made with respect to correspondences between conceptual classifications and partitions of empirical geometric representation spaces obtained in data analysis.
Of the many types of representation spaces that have been proposed, two stand out as especially fruitful: Faceted-SSA for structuring the investigated attribute ; and POSAC for multiple scaling measurements of the investigated attribute.
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 and has been further developed and applied in a variety of disciplines of the behavioral sciences including psychology, sociology, and business administration.

The mapping sentence

Definition and properties of the mapping sentence

Definition. A mapping sentence is a verbal statement of the domain and of the range of a mapping including connectives between facets as in ordinary language.
In the context of behavioral research, a mapping sentence is essentially a function whose domain consists of the respondents and of the stimuli as arguments, and whose image consists of the cartesian product of the ranges of responses to the stimuli, where each response-range is similarly ordered from high to low with respect to a concept common to all stimuli. When stimuli are classified a priori by one or more content criteria, the mapping sentence facilitates stratified sampling of the content-universe. A classification of the stimuli by their content is called a content facet; and the pre-specified set of responses to a stimulus is called a range facet.
The mapping sentence defines the system of observations to be performed. As such, the mapping sentence provides also the essential concepts in terms of which research hypotheses may be formulated.

An example from intelligence research

Suppose members p_i of a population P are observed with respect to their success in a written verbal intelligence test. Such observations may be described as a mapping from the observed population to the set of possible scores, say, R = : P_q1 → R, where q1 is the sense in which a specific score is assigned to every individual in the observed population P, i.e., q1 is "verbal intelligence" in this example. Now, one may be interested in observing also the mathematical or, more specifically, the numerical intelligence of the investigated population; and possibly also their spatial intelligence. Each of these kinds of intelligence is a "sense" in which population members p_i may be mapped into a range of scores R = . Thus, 'intelligence' is now differentiated into three types of materials: verbal, numerical and spatial. Together, P, the population, and Q = , the set of types of intelligence, form a cartesian product which constitutes the mapping domain. The mapping is from the set of pairs to the common range of test-scores R = : P × Q → R.
A facet is a set that serves as a component-set of a cartesian product. Thus, P is called the population facet, Q is called a content facet, and the set of scores obtainable for each test is a range facet. The range facets of the various items need not be identical in size: they may have any finite number of scores, or categories, greater or equal to 2.

The Common Meaning Range (CMR)

The ranges of the items pertaining to an investigated content-universe – intelligence in this example – should all have a Common Meaning Range ; that is, they must be ordered from high to low with respect to a common meaning. Following Guttman, the common meaning proposed for the ranges of intelligence-items is "correctness with respect to an objective rule".
The concept of CMR is central in facet theory: It serves to define the content-universe being studied by specifying the universe of items pertaining to that content-universe. Thus, the mapping-definition of intelligence, advanced by facet theory is:
"An item belongs to the universe of intelligence items if and only if its domain requires performance of a cognitive task concerning an objective rule and its range is ordered from high correctness to low correctness with respect to that rule."
An initial framework for observing intelligence could be Mapping Sentence 1.
The mapping sentence serves as a unified semantic device for specifying the system of intelligence test items, according to the present conceptualization. Its content facet, the material facet, may now serve as a classification of intelligence test items to be considered. Thus, in designing observations, a stratified sampling of items is afforded by ensuring an appropriate selection of items from each of the material facet elements; that is, from each class of items: the verbal, the numerical and the spatial.

Enriching the mapping sentence

The research design can be enriched by introducing to the mapping sentence an additional, independent classification of the observations in the form of an additional content-facet, thereby facilitating systematic differentiations of the observations. For example, intelligence items may be classified also according to the cognitive operation required in order to respond correctly to an item: whether rule-recall, rule-application, or rule-inference. Instead of the three sub-content-universes of intelligence defined by the material facet alone, we now have nine sub-content-universes defined by the cartesian multiplication of the material and the mental-operation facets. See mapping sentence 2.
Another way of enriching a mapping sentence is by adding an element to an existing content facet; for example, by adding Interpersonal material as a new element to the extant material facet. See Mapping Sentence 3.

Content profiles

A selection of one element from each of the two content facets defines a content profile which represents a sub-content-universe of intelligence. For example, the content profile represents the application of rules for performing mathematical computations, such as performing long division. The 3x4=12 sub-content-universes constitute twelve classes of intelligence items. In designing observations, the researcher would strive to include a number of varied items from each of these 12 classes so that the sample of observed items would be representative of the entire intelligence universe. Of course, this stratified sampling of items depends on the researchers' conception of the studied domain, reflected in their choice of content-facets. But, in the larger cycle of the scientific investigation, 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.

Cartesian decomposition and completion: an example

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. And for each content facet, attempt is made to specify its elements so that they be exhaustive and exclusive of each other. Thus, the element 'interpersonal' has been added to the incumbent 3-element material facet of intelligence by a two-step facet-analytic procedure. Step 1, cartesian decomposition of the 3-element material facet into two binary elementary facets: The Environment Facet, whose elements are 'physical environment' and 'human-environment'; and the Symbolization Facet whose elements are 'symbolic', and 'concrete'. Step 2, cartesian completion of the material facet is then sought by attempting to infer the missing material classifiable as 'human environment' and 'concrete'.
In facet theory, this 2×2 classification of intelligence-testing material may now be formulated as an hypothesis to be tested empirically, using Faceted Smallest Space Analysis.

Complementary topics concerning the mapping sentence

Despite its seemingly rigid appearance, the mapping sentence format can accommodate complex semantic structures such as twists and recursions, while retaining its essential cartesian structure.
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.