Radex theory of complex interactions
Radex theory as explored by Louis Guttman in the study of intelligence and intelligence testing in the 1950s and 60s, represented one of the forerunners of complexity theory. As stated in (9:392), with references 1-8:
- 1 Radex theory of Guttman
- 2 Kind and degree of complexity
- 3 Complex multivariate interaction versus reductionist bivariate scaling
- 4 Stardex or radial correlation patterns
- 5 Single-factor structure
- 6 Complex fields of multiple stardex structures
- 7 Cognitive/affective complexity
- 8 Cultural complexity and autocorrelation
- 9 Dynamics
Radex theory of Guttman
- Two General Facets
- Radex theory recognizes that mental tests may vary among themselves as to content on at least two facets: differences in kind of, and differences in degree of, complexity. A simplex may result when the first of these facets is held constant and only the second is varied. Another special case, called a circumplex, may result when the second facet is held constant but the first is varied. The general radex case is when both facets vary simultaneously over the given battery of tests. An algebraic basis for radex theory has been developed in (4), with some further extensions for the simplex in (6). Some empirical examples have been worked out in some detail in (I, 4, 7) to illustrate the theory, and to verify that some current mental tests reveal the hypothesized simplex and circumplex structures. The relation of radex analysis to other approaches to factor analysis has also been worked out in (4), as well as the important implications it has for parsimonious prediction of external criteria.
- Gabriel, Reuben K. "The Simplex Structure of the Progressive Matrices Test." British Journal of Statistical Psychology, VII (I954), 9-I4.
- Guilford, J. P. "Dimensions of Intellect." International Colloquium on Factor Analysis, Paris, 1955 (to be published). [L'Analyse factorielle et ses applications. 1955. Centre National de la Recherche Scientifique. Paris, France.]
- Guttman, Louis. Two New Approaches to Factor Analysis. Technical Report on Project Nonr-731 (00), Office of Naval Research, Washington, D. C., submitted jointly by International Public Opinion Research, Inc., and the Israel Institute of Applied Social Research, June, 1953 stencilled).
- Guttman, Louis. "A New Approach to Factor Analysis: The Radex." In Lazarsfeld, P. F. Mathematical Thinking in the Social Sciences. Glencoe, Illinois: Free Press, I954.
- Guttman, Louis. "An Outline of Some New Methodology for Social Research." Public Opinion Quarterly, XVIII (954-55), 395-404.
- Guttman, Louis. "A Generalized Simplex for Factor Analysis." Psychometrika, XX (I955), I73-I92.
- Guttman, Louis. "The Radex Approach to Factor Analysis." International Colloquium on Factor Analysis Paris, 1955 (to be published). L'Analyse factorielle et ses applications. 1955. Centre National de la Recherche Scientifique. Paris, France.]
- Guttman, Louis. 1958. "What Lies Ahead for Factor Analysis?" Educational and Psychological Measurement 18(3): 497-515. http://epm.sagepub.com/cgi/content/citation/18/3/497
- Guttman, Louis. 1957. "Empirical Verification of the Radex Structure of Mental Abilities and Personality Traits." Educational and Psychological Measurement 17:391-407. http://epm.sagepub.com/cgi/reprint/17/3/391
Kind and degree of complexity
The Guttman-Lingoes approach to measurement via nonmetric scaling represented a rebellion against metric multidimensional scaling and the assumption reflected in most scaling methods today, namely, that all meaningful statistical interaction is bivariate. That is, all higher order interactions among quantitative variables can be ignored. This is somewhat like saying that no three or more bodies interact, other than dyadically. In developing alternatives such as discrete structure approaches (including Guttman scaling), Guttman had developed the idea of recombinatory facets of intelligence, and then discovered the two very general facets described above: "differences in kind of, and differences in degree of, complexity". "The general radex case is when both facets vary simultaneously over the given battery of tests."
Complex multivariate interaction versus reductionist bivariate scaling
Analyzing patterns of responses to individual items on intelligence tests, Guttman and Lingoes were trying to see through the forest of correlations into patterns in which complexities showed up in multivariate interactions. One way this shows up in modeling today is in entailment structures (items 10, 12, in Topical publications: Douglas R. White and in Concept Lattice or Galois Lattice analysis, as employed, for example, by Camille Roth  or John Mohr [ref]. See Entailment analysis
Stardex or radial correlation patterns
But within the radex theory of Guttman, in the analysis of patterns of correlation, one of the common patterns that Guttman was referring to a set of items (possibly must one) that is central, and if multiple then highly correlated, and a large number of other items that are correlated with the central set or item but not correlated amongst themselves, which is what network analysts would call a star or core-periphery structure. For correlations, I will call this a stardex pattern. More specifically, if there are three or more items in the central set of variables, they would pass the single-factor test of principal components analysis, while the peripheral items are less correlated with each other than would be expected from having a common central variable or factor with which each is correlated pairwise. That is, the peripheral items are oblivious of each other but occuppy statistically independent orbits around a common center. This is not, then, a case where the axioms of independent bivariate interaction applies. The interesting question here, as with planets rotating around the sun, is whether these orbits are really independent, or in the case bivariat correlations among variables, are they nonindependent just because their partial correlations are negative controlling for their correlations to the central factor? The analogy to the orbits of planets is apt in the sense that planetary sizes and orbits are not statistically independent but spaced, as a result of evolutionary selection, in such a way that they do not interfere with one another in ways that would cause cataclysmic crashes.
The single factor model in principal components analysis is this: only the first factor has an eigenvalue greater than than 1, which are a eigenvalues indexing randomness or independence, and the eigenvalue of the first factor is 3 to 10 times greater than that of the second. A stardex pattern is more evident the more items in the central single-factor and the more items in the periphery of items correlated to the center but whose inclusion in the statistical factor analysis violates the single-factor structure.
Let us define a simple system of variables as one in which there is only bivariate interaction, analagous to homogeneous particles interacting according to gas laws. If the variables under study interact in a complex manner, and we take into account Guttman's idea of kinds and degrees of complexity as the two general facets of a complex system, then a stardex or radial correlation pattern seems to indicate a central single factor in which the variables are of the same kind but may or may not differ in complexity, while the peripheral variables in the stardex are of different kinds.
The stardex might also be a special case of an orthogonal stardex pattern in which there are multiple sets of items each of which passes the single-factor test, and each such factor is correlated with items in at least one other, but when any item from another orthogonal factor is factor analyzed with items from a different set, then the resultant factor structure is no longer singular. The factor analysis community looked long and hard for such structures -- large orthogonal components of interaction -- but did not find them.
If we fall back to the simpler stardex model, however, we have a very plausible model of causality: namely, it is the central factor that contributes to understanding the causes of variation in the peripheries, and not the other way around. This is not just a matter of the parsimony of one cause, many effects rather than many causes for singlular effects.
Complex fields of multiple stardex structures
We may find multiple stardex patterns, but pairs of stardexes need not be statistically independent. That is, if we compute the factor scores for each set of stardex variables that passes the single-factor test, these factor scores need not be independent.
For example, cross-cultural variables measuring internal warfare and intra-societal violence form a single factor, as do those measuring external warfare and inter-societal hostility, but the factor scores of their single factors are correlated. In this case the correlation is nonlinear, another possible signal of complexity: the embeddedness of human violence.
This leads to some hypotheses about how multiple stardex structures are coupled in complex fields of interaction. Each stardex has a core dimension in which concrete elements (nodes in a network) are scaled by bipolar complexity (zero to more, where more is different), negative to positive, simpler form to more differentiated form, for example. The peripheral correlations indicate interactions between these nodes, and the problem of dynamics is to find out through further investigation which sets of elements scaled in the core interact with other elements, whether peripheral (and often affected by multiple cores) or in other cores.
Cores must reproduce elements or effects that in turn, through complex interactions elsewhere in positive feedback circuits, sustain them, otherwise they would not continue to exist as cores. Negative feedback loops among cores create dynamical fluctuations and, rarely, stable equilibria.
It is now fairly well understood that the intelligence test scores that Guttman and others set out to understand do measure different facets of kinds of intelligence, like specialized mental, emotional, lunguistic, or combined skills that orbit around a set of rather closely connected practices. The practices that are focal to these kinds of intelligence are interlocked with others in the same set, but also with those from other sets. These interlocks represent what Guttman called not just kinds of intelligence, but kinds of complexity. It is also understood that within these sets there are levels of skill, also levels of complexity. But at some levels of complexity in one skill set, these skills spill over to enhance other skill sets.
Cultural complexity and autocorrelation
If we find multiple stardex patterns in cross-cultural correlates, such as separate single-factors for internal and external violence, it may be because these have different spillover effects in different contexts, and that these differences in context and spillover dynamics have a great many causal and unintended side effects. One cannot aim a shotgun whose stardex pellets fly in all directions. But often, the combination of causality on factors that remain independent in spite of connection to a common causal source -- but then either having different causal effects in each case or else having effects that are reshaped independently at each peripheral node -- might means that there is a sum of independent effects that may robust in some cases, tuning in different ways combinations of things whose aggregate effects into turn tend to cancel out. But a small amount of synchronous are asynchronous tuning of multiple effects may produce large effects unanticipated from aggreging random variates. The combination of these two possibilites implies that the dynamical variance in stardex and multiple stardex systems might be much greater than an assimption of statistical independence of cultures would imply. And in addition to his, we know that cultures have effects on each other, either as a result of stardex perturbations or as an added sources of nonindependent variance, known in cross-cultural studies as Galton's problem  of understimating variance, and in econometrics as autocorrelation, but basically the same problem, one of underspecification .
We cannot say that one culture is more complex than another, or that evolution is headed toward greater complexity. Complexity lies in the parts and their interrelations, not the whole, and spillover effects are dynamic, changing with context, and sometimes with changes in perspective or new ways of understanding.
There are an abundance of indicators of complexity in the stardex and more generally, radex, patterns of cross-cultural correlations. How do we turn to the discovery of dynamics?
- One is to test hypotheses about causal effects using both regression analysis and an analysis of the unpredicted residuals, testing the residuals for nonindependence. This means testing whether residuals are clustered in time, in space (by distance), along known routes and clusters of interaction, by common language, by common historical origin, by common mitochondrial or Y chromosom whose historical dispersals have been reconstructed. For example, Anthon Eff provides language and distance matrices for such tests, and Trevor Denton tests model specifications controlling for effects of neighbors.
- The second is to observe changes over time, and identify the "sufficient measures" at which conserved entities (e.g., populations, individuals) exist and causally interact with other entities. Peter Turchin, in his 2003 book on Complex Population Dynamics, describes kinds of models of dynamics that work for ecological and population systems. For systems of multiple time series variables that show recurrent kins of fluctuates the most effective of these models is the 2-equation time-lagged negative feedback model that he also employes in his work on historical dynamics.
- Turchin considers population in a region over a time period in which there are only small exogenous impacts from the outside to act as a "sufficient statistic" for treating population number as it interacts with other variables, such as resource scarcity measured against population, numbers and events involved in conflict, organized policies implemented by polities, etcetera. This general approach is also known as "structural demography" as developed by Jack Goldstone.