Guttman, Louis (Eliyahu)
Born: February 10, 1916, in Brooklyn
Died: October 25, 1987, in Minneapolis, Minnesota.
Contributed to: Scaling theory (including “Guttman Scale”), factor analysis,reliability theory,
methodology & theory construction in social and psychological research (“Facet Theory”).
Louis Guttman was one of the most influential psychometricians of the 20th century and a promoter of formalization in the social and psychological sciences. He was born in Brooklyn, New York on February 10, 1916, and grew up in Minneapolis Minnesota. He received his Ph.D. in Sociology in 1942 from the University of Minnesota with a thesis on the algebra of Factor Analysis. With his academic base at Cornell University at Ithaca, New York, Guttman served during World War II as an expert consultant at the Research Branch of the Information and Education Division of the War Department, where he developed the Scale bearing his name . In 1947 he immigrated to Israel where he founded and directed The Israel Institute of Applied Social Research (later: The Guttman Institute of Applied Social Research). From 1955 he served also as Professor of Social and Psychological Assessment at the Hebrew University of Jerusalem, until his death on October 25, 1987. Guttman published in numerous journals and books of sociology, psychology and statistics covering half a century from 1938, both as a sole author and in collaboration with others. Many of these earlier papers are still quoted as relevant to current statistical and mathematical advances. The development of scaling theory by Louis Guttman and Clyde Coombs is one of 62 “major advances in social science” identified and analyzed in Science  for the period 1900 - 1965.
While still a graduate student, Guttman undertook the clarification and formalization of intuitive -- and often erroneous -- techniques of data analysis that were beginning to pervade sociology and psychology . Here lay the foundations of much of his later work on scale analysis, reliability theory, factor analysis and nonmetric data analysis.
Guttman’s early work focused on linear algebra of factor analysis frequently in relation to multiple regression [e.g. 5, 7, 9-13, 15, 16]. It ranged from developing computational formulae (for example, for the inverse of a correlation matrix  and for lower bounds for the number of factors ) to investigating fundamental issues concerning the logic of factor analysis (especially the indeterminacy of factor scores , see also ). His works were highly valued by psychometricians, both practitioners and theoreticians.
Later on, however, in the early 1950’s Guttman proposed a new way of looking at factor analysis, initiating a radical shift in the way multivariate data can be analyzed and interpreted. Revising Spearman’s notion of a hierarchy among intelligence tests, Guttman conceived of tests that increase in complexity, as if they successively activate a sequence of “bonds” in the human mind: as tests become more complex additional centers, along a given path, are employed. This consideration resulted in the number of factors being equal to the number of tests, in seeming contradiction to the very purpose of factor analysis which seeks to identify a small number of underlying factors. However, a simple (unidimensional) ordering among tests is in itself a parsimonious representation of a new kind, which holds regardless of the number of factors [10, 25, 28]. This representation, termed the (parametrized) simplex, implies certain relationships among correlation coefficients: tests may all be mapped (as points) into a straight line so that the larger the correlation coefficient between any two tests, the closer they are on the line [10, 29]. Similarly, the circumplex configuration (tests circularly ordered in the plane) has been formulated and discovered in psychological data (e.g. of color perception ). The Radex -- a two-dimensional concatenation of simplexes and circumplexes, wherein each test is a member of a simplex and of a circumplex -- was introduced in its parametrized form by Guttman  and then corrected and developed further by others (see  for an excellent review of the parametrized formulation of some basic test configurations). In 1968 Guttman published in Psychometrika his much quoted paper “A general nonmetric technique for finding the smallest coordinate space for a configuration of points”, an algorithm for mapping tests in the space of the smallest dimensionality capable of reflecting pair-wise similarity (e.g. correlations) between them . This data analytic procedure has become known as Smallest Space Analysis (SSA) and has been computer programmed and included in statistical packages as a Multidimensional Scaling (MDS) technique.
Guttman’s true concern, however, was not with statistical or data analytic procedures as such. Rather, he sought to combine such procedures, and the aspects of empirical data they invoke, with conceptual-definitional framework of a substantive domain of research, in order to discover lawfulness, and contribute to theory construction in social and psychological domains of research. Indeed, in order to clarify his research strategy, Guttman felt he should propose a definition for the very term theory (in the context of the empirical sciences):
An hypothesis of a correspondence between a definitional system of observations and an aspect of the empirical structure of those observations, together with a rationale for such an hypothesis (e.g. ).
The set of all items -- observable variables -- that pertain to an investigated concept (such as marital adjustment of couples, or intelligence of individuals) concern all acts of that concept (i.e. acts of adjustment, or acts of intelligence). Guttman called that set “universe of content”. Indeed, as a research strategy, Guttman’s Facet Theory requires, first, the definition of the concept studied in terms of its content universe, i.e., its items. For example, he defined attitude items as those whose range (set of response categories, or values) is ordered from very positive to very negative behavior towards an object; and intelligence items, as those whose range is ordered from very correct to very incorrect performance with respect to an objective rule. The specification of such a common meaning to item ranges provides a rationale for “laws of monotonicity”. For example:
The law of Intelligence monotonicity: If any two items are selected from the universe of intelligence items, and if the population observed is not selected artificially, then the population regression between these items will be monotone with positive or zero sign. (e.g. , )
This law, on the one hand, summarizes in a formal fashion findings that have been noted for some time by intelligence researchers, and, on the other hand (as an incumbent hypothesis), tells us what to expect in the future observations. Since the law specifies a correspondence between a definitional framework (i.e. the common range of intelligence items, as defined) and an aspect of the empirical data (i.e. the correlation sign) it qualifies as a “theory” according to the above stated definition.
Guttman suggested that finer laws can be proposed by classifying items according to aspects of their content. One such classification for intelligence items may be the material facet -- i.e. whether the items deal with verbal, numerical, figural or some other kind of material. Another, independent classification is that of the task facet -- i.e. whether items require rule-recall, rule-application or rule-inference. According to the radex theory of intelligence [17, 23, 20, 28] this double classification corresponds to two specific ways of partitioning the SSA 2-space of intelligence items (see Figure 1): elements (classes) of the material facet (verbal, numerical, figural) are circularly ordered, i.e. items of each class (e.g. verbal) all fall within a sector. Elements (classes) of the task facet (recall, application, inference) are linearly ordered along the radius, i.e., items of each class (e.g. application) all fall within one of a concentric rings with rule-recall at the outermost ring and rule-inference as a central disk.
Figure 1. The Radex Theory of Intelligence
The material facet is angular, the task facet is radial. One implication: rule-recall differentiates among abilities better than rule-inference
The unique feature of Guttman’s facet theory as a methodological philosophy and a research strategy stems from the recognition that behavioral scientific concepts are typically manifested by an infinite number of items (observational variables) but only a finite sample of items can be actually observed. Facet theory copes with this challenge in both its aspects: the research design and the data analytic. For the research design Guttman proposes the mapping sentence technique [14, 25, 29, 1] which focuses the researcher’s attention on a finite number of relevant conceptual facets (content classifications), rather than on the infinite number of items.
Multivariate data analysis has coped with the question of item sampling by the gradual shift from traditional factor analysis, through parametrized test configuration and the contiguity principle , to regional partition patterns in continuous SSA spaces [26,29].
There remained, of course, the problem of scoring individuals with respect to the studied concept. Guttman insisted that single scores can be assigned to subjects only if observed profiles (rows in data matrix) turn out in fact to form a unidimensional (Guttman) scale (also known as cumulative scale). Most often they do not, and the methods of multiple scaling by partial order scalogram analysis must be used .
Guttman was strongly individualistic and creative, yet committed to the scientific tradition; kindly in his daily contacts, yet argumentative, sometimes to the point of arrogance, in insisting on the principles in which he so deeply believed. His conviction in the formalized integration of conceptual framework with data analysis has placed him in the seemingly dual position of “a statistican among psychologists” opposed to untestable “theories”, and a “psychologist among statisticians” opposed to routine “number crunching.” Colleagues and students who were not alienated by his style benefited from his seminal insights and provocative presentations.
Guttman’s work has had an impact on intelligence research, attitude research, environmental psychology, the study of general behavioral systems, and other fields. In recognition of his scientific contributions Guttman has been awarded the Rothschild Prize for Social Sciences, and the Israel Prize. He was elected to memberships in the Israel Academy of Science and Humanities, and to foreign honorary membership of the American Academy of Arts and Sciences. He held the Andrew White Professorship-at-Large at Cornell University, received the Outstanding Achievement Award of the University of Minnesota and the 1984 Educational Testing Service Award for distinguished service to measurement. The latter’s citation recognized that “a central theme in Guttman’s work [is] that measurement is not merely the assignment of numbers but the construction of structural theory.” 
 Borg, I. & Shye, S. (1995). Facet Theory: Form and Content. Thousand Oaks, CA: Sage. (A mathematical formulation of Facet Theory, and algorithms for computing optimal regions in SSA (MDS))
 Canter, D. (Ed.) (1985). Facet Theory: Approaches to Social Research. New York: Springer. (A selection of applications of Facet Theory to psychology)
 Deutsch, K.W., Platt, J. & Senghan D. (1971). Conditions favoring major advances in social sciences. Science, 171, 450-459.
 Foa, U.G. (1958). The contiguity principle in the structure of interpersonal behavior. Human Relations, 11, 229-238.
 Guttman, L. (1940). Multiple rectilinear prediction and the resolution into components. Psychometrika, 5, 75-99.
 Guttman, L. (1941). The quantification of a class of attributes: A theory and method of scale construction. In The Prediction of personal Adjustment (with P. Horst and others). New York: Social Science Research Council.
 Guttman, L. (1944). General theory and methods for matric factoring. Psychometrika, 9, 1-16.
 Guttman, L. (1950). Measurement and Prediction (with S.A. Stouffer and others). Studies in Social Psychology in World war II, Vol.4 Princeton, N.J.: Princeton University Press.
 Guttman, L. (1952). Multiple-Group Methods for Common-Factor Analysis. Psychometrika, 17, 209-222.
 Guttman, L. (1954). A new approach to factor analysis: The radex. In P.F. Lazarsfeld (Ed.) Mathematical Thinking in the Social Sciences. New York: Free Press.
 Guttman, L. (1954). Some necessary conditions for common-factor analysis. Psychometrika, 19, 149-161.
 Guttman, L. (1955). The determinacy of factor score matrices with implications for five other basic problems of common-factor theory. British Journal of Statistical Psychology, 8, 65-81.
 Guttman, L. (1956). ‘Best Possible’ systematic estimates of communalities. Psychometrika, 21, 272-28.
 Guttman, L. (1957). Introduction to facet design and analysis. In Proceedings of the Fifteenth International Congress of Psychology, Brussels. Amsterdam: North-Holland.
 Guttman, L. (1957). Simple proofs of relations between the communality problem and multiple correlation. Psychometrika, 22, 147-157.
 Guttman, L. (1958) To what extent can communalities reduce rank? Psychometrika, 23, 297-308.
 Guttman, L. (1965) A faceted definition of intelligence. In Studies in Psychology. Scripta Hierosolymitana 14, 166-181.
 Guttman, L. (1968). A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 33, 469-506.
 Guttman, L. (1971). Measurement as structural theory. Psychometrika, 36, 329-347.
 Guttman, L. and Levy, S. (1991). Two structural laws for intelligence tests. Intelligence, 15, 79-103.
 Levy, S. (1985). Lawful roles of facets in social theories. In D. Canter (Ed.). Facet theory: Approach to social research (pp.59-96). New York: Springer.
 Levy, S. (Ed.) (1994) Louis Guttman on Theory and Methodology: Selected Writings. Aldershot, England: Dartmouth. (A good selection of Guttman’s original papers.)
 Schlesinger, I.M. & Guttman, L. (1969). Smallest space analysis of intelligence and achievement tests. Psychological Bulletin, 71, 95-100.
 Shepard, R.N. (1978). The circumplex and related topological manifolds in the study of perception. In Shye, S. (Ed.) Theory Construction and Data Analysis in the Behavioral Sciences. San Francisco: Jossey-Bass.
 Shye, S. (1978). On the search for laws in the behavioral sciences. In Shye, S. (Ed.) Theory Construction and Data Analysis in the Behavioral Sciences. San Francisco: Jossey-Bass.
 Shye, S. (1978b). Facet analysis and regional hypothesis. In Shye, S. (Ed.) Theory Construction and Data Analysis in the Behavioral Sciences. San Francisco: Jossey-Bass.
 Shye, S. (1985). Multiple scaling: The theory and application of partial order scalogram analysis. Amsterdam: North Holland. (Scalogram algebra extending the Guttman scale to spaces of higher dimensionalities, with examples.)
 Shye, S. (1988). Inductive and deductive reasoning: A structural reanalysis of ability tests. Journal of Applied Psychology, 73, 308-311 (with appendix comparing SSA and factor analysis).
 Shye, S. & Elizur, D. (1994). Introduction to Facet Theory: Content Design and Intrinsic Data Analysis in Behavioral Research. Thousand Oaks CA: Sage. (A basic text book.)
 Steiger, J. & Schonemann, P. (1978). A history of factor indeterminacy. In S. Shye (Ed.) Theory Construction and Data Analysis in the Behavioral Sciences. San Francisco: Jossey-Bass.
 Van den Wollenberg, A.L. (1978) Nonmetric representation of the radex in its factor pattern parametrization. In Shye, S. (Ed.) Theory Construction and Data Analysis in the Behavioral Sciences. San Francisco: Jossey-Bass.
Adapted from: Shye, S. (1997). Guttman, Louis. In N. L. Johnson and S. Kotz (Eds.) Leading Personalities in Statistical Sciences. New York: Wiley.