---
title: "Facet theory"
chunk: 3/8
source: "https://en.wikipedia.org/wiki/Facet_theory"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T09:54:10.375224+00:00"
instance: "kb-cron"
---

Receiving as input (or computing from input data) a matrix of similarity coefficients, specifying, for each pair of items how similar they are. A common example is the computation of a correlation-coefficient matrix from input data, where the size of a correlation coefficient between two variables reflects the degree of similarity between them.
Mapping the items (variables) as points in a geometric space of a given dimensionality while preserving as well as possible the condition:  If rij>rkl then dij<dkl  for all i,j,k,l   where rij is the similarity measure (e.g., correlation coefficient) between variables i,j and dij is the distance between their points in the space. Most often, Euclidean distance function (Minkowsky distance of order 2) is used. But other distance functions, especially the Manhattan distance function (Minkowsky distance of order 1) are called for. (See Subsection Relating POSAC Measurement Space to the SSA Concept Space below.) The goodness-of-fit of the resulting mapping may be assessed by a loss function– Kruskal's Stress coefficient or Guttman's Coefficient of Alienation.
Partitioning the space as well as possible, into simple regions (stripe, sectors or concentric rings) whose variables are in 1-1 correspondence with a pre-conceived content-facet. To run this option, content facet(s) must be specified as Faceted SSA input.
Step 3 of Faceted SSA incorporates the idea that observed variables included in the Faceted SSA procedure, typically constitute a small subset from the countless items that define the attribute content-universe. But their locations in space may serve as clues that guide the partitioning of the space into regions, in effect classifying all points in space, including those pertaining to unobserved items (had they been observed). 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. The Shye-Kingsley Separation Index (SI) assesses the goodness-of-fit of the partition to the content-facet.
The spatial scientific imagery suggested by Facet Theory has far reaching consequences that set Facet Theory apart from other statistical procedures and research strategies. Specifically, it facilitates inferences concerning the structure of the entire content-universe investigated, including unobserved items.

=== Example 1. The structure of intelligence ===
Intelligence testing has been conceived as described above, with Mapping Sentence 2 as a framework for its
observation. In many studies, different samples of variables conforming to Mapping Sentence 2 have been analyzed confirming two regional hypotheses:

The Material Content Facet corresponds to a partition of the Faceted SSA map of intelligence into sectors, each containing the items of a single material — verbal, numeric, and figural (spatial).
The Cognitive Operation Facet corresponds to a partition of the Faceted SSA map of intelligence into concentric rings, with the innermost ring containing  inference items; the middle ring containing the rule-application items; and the outermost ring containing the rule-recall items.
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", has been found also in the study of color perception as well as in other domains of research.
Faceted SSA has been applied in a wide variety of research areas including value research social work  and criminology and many others.

=== Example 2. The structure of quality of life ===
The Systemic Quality of Life (SQOL) has been defined as the effective functioning of human individuals in four functioning subsystems: the cultural, the social, the physical and the personality subsystems. The axiomatic foundations of SQOL suggest the regional hypothesis that the four subsystems should be empirically validated (i.e., item of each would occupy a distinct region) and that they be mutually oriented in space in a specific 2x2 pattern topologically equivalent to the 2x2 classification shown in Figure 2 (i.e., personality opposite cultural, and physical opposite social). The hypothesis has been confirmed by many studies.

=== Types of partition patterns ===
Of the many possible partitions of a 2-d concept space, three stand out as especially useful for theory construction:

The Axial Partition Pattern: Partitioning of the space into stripes by parallel lines.
The Angular (a/k/a polar) Partition Pattern: Partitioning of the space into sectors by radii emanating from a point in space.
The Radial (a/k/a modular) Partition Pattern: Partitioning of the space into concentric rings by concentric circles.
The advantages of these partition patterns as likely models for behavioral data are that they are describable by a minimal number of parameters, hence avoid overfitting; and that they are generalizable to partition in spaces of higher dimensionalities.
In testing regional hypotheses, the fit of a content-facet to any one of these three models is assessed the Separation Index (SI), a normalized measure of the deviation of variables from the region assigned to them by the model.
Concept spaces in higher dimensionalities have been found as well.