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Doing it all Bass-Ackwards: The development of hierarchical factor structures from the top down

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Abstract

A simple method is presented for examining the hierarchical structure of a set of variables, based on factor scores from rotated solutions involving one to many factors. The correlations among orthogonal factor scores from adjoining levels can be viewed as path coefficients in a hierarchical structure. The method is easily implemented using any of a wide variety of standard computer programs, and it has proved to be extremely useful in a number of diverse applications, some of which are here described.

Introduction

Traditionally, structures that include the hierarchical arrangement of variables are developed from the bottom-up, starting with individual elements, which are then grouped into clusters, which in turn are then further grouped into larger categories of variables, until one reaches a highest level in which all of the variables are subsumed under the one most general attribute. For example, in hierarchical analyses of cognitive tests, individual test items are grouped to form sub-tests, which in turn are grouped to form total tests, which in turn are grouped to form highly specific factors, which in turn are grouped to form large group factors (such as fluid and crystallized intelligence), which in turn are the building blocks leading to a most general level (such as “g” for general intelligence).

There are two kinds of such bottom-up approaches to the development of hierarchical structures, one based on cluster analysis and the other on factor analysis. Perhaps the most popular of these are the various algorithms for constructing “tree” representations available in hierarchical cluster-analysis programs. However, although cluster analysis may be a useful method for analyzing the structure of objects (as labeled by nouns), it is not as useful for analyzing the structure of attributes (as labeled by adjectives). One reason is that attribute clusters may have complex relations with other such clusters, and these are not easily comprehended in the unidimensional arrangement of variables across the bottom of a tree diagram. Another reason is that negative relations among variables are not easily accommodated in conventional clustering algorithms.

In contrast, factor analysis permits one to examine the relations between any variable and each of many factors, and bipolarity is easily expressed in the signs of the factor loadings. When factors are rotated obliquely, the correlations among the factors can then be further analyzed, until one has reached a level when all of the factors are orthogonal or there is only one factor remaining. At that point, it is possible to use the procedure described by Schmid and Leiman, 1957, Yung et al., 1999 to provide a hierarchical structure in which the factors at each level are completely unrelated to those of all levels above and below it. Indeed, the use of such orthogonalized hierarchical solutions has now become the received wisdom in analyses of cognitive abilities (e.g., Carroll, 1993). But, what exactly can one make of factors that are independent of all factors at other levels? How is one to think about a general ability that is unrelated to any of its constituent parts? For readers who may wonder about the usefulness of such purified factors, this article provides a more transparent alternative. Moreover, this top down technique permits investigators to develop hierarchical representations that encompass far more levels than the two- or three-level factor structures that are most frequently used to represent cognitive abilities.

Section snippets

A simple top-down procedure

In principal factors or principal components analyses, the first factor to be extracted is the largest one possible, and the next one is therefore smaller than the first. Each subsequent principal factor extracted is completely orthogonal to all of those extracted before it. That is, the first principal factor provides a measure of whatever is most in common to the variables that have been included in the analysis, and each subsequent factor is again as large as possible after the influence of

Some illustrations from analyses of personality-descriptive terms

The principal components displayed in Fig. 1 are based on an analysis of self-descriptions to 1710 personality-related adjectives (Goldberg, 1982) reported by Ashton, Lee, and Goldberg (2004). In the article by Ashton et al. (2004), the hierarchy is extended to seven levels, but for simplicity only a truncated five-level version is displayed here. When compared to the size of the first principal component (set to 1.00), the relative sizes of the other four unrotated components are .93 (2), .72

Some other illustrations

An appealing characteristic of these top down factor representations is that one need not commit oneself in advance to the optimal number of factors to extract and rotate. Instead, one can continue down into the hierarchy until one reaches a level at which no new interesting factors appear. One criterion for stopping the process is that no variables have their highest loadings on a factor, in which case one should surely stop at the level above that one. However, in analyses of very large

Some caveats and conclusions

Although the illustrations in this article have come from studies conducted by the author or his former students, variations on these ideas have been proposed by others. One early example of a partial representation of this sort was provided by Zuckerman, Kuhlman, and Camac (1988). A more complete structural representation was included in Ostendorf‘s (1990) now-classic account of his German lexical taxonomy project. And, one of the most influential uses of the method may have been Saucier’s

Lewis R. Goldberg is a senior scientist at the Oregon Research Institute and an emeritus professor of psychology at the University of Oregon. A past president of the Society of Multivariate Experimental Psychology and the current president of the Association for Research in Personality, he has served on both the Cognition, Emotion, & Personality and the Personality & Cognition research review committees of the National Institute of Mental Health. His contributions to the scientific literature

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    Lewis R. Goldberg is a senior scientist at the Oregon Research Institute and an emeritus professor of psychology at the University of Oregon. A past president of the Society of Multivariate Experimental Psychology and the current president of the Association for Research in Personality, he has served on both the Cognition, Emotion, & Personality and the Personality & Cognition research review committees of the National Institute of Mental Health. His contributions to the scientific literature in personality and psychological assessment have included articles on judgment and decision-making, the comparative validity of different strategies of test construction, the measurement of situational vs. dispositional attributions in self and peer descriptions, the characteristics of personality traits and states, and the development of taxonomies of personality-descriptive terms in diverse languages. To provide public-domain measures of the most important personality attributes, he has developed an Internet-based scientific collaboratory, the International Personality Item Pool (IPIP: http://ipip.ori.org/). His current research interests are well described by the titles of the two NIH grants on which he serves as principal investigator: “Mapping Personality Trait Structure” and “Personality and Health—A Longitudinal Study.”

    Funds for this project have been provided by Grant MH49227 from the National Institute of Mental Health, US Public Health Service. The author is enormously grateful to those who have helped him think about the issues discussed in this article, including G. Scott Acton, Michael C. Ashton, Jack Block, Shawn Boles, Roy D’Andrade, M. Lynne Cooper, Samuel Gosling, Sarah E. Hampson, John Horn, John A. Johnson, Robert F. Krueger, Patrick Kyllonen, Daniel Levitin, John Loehlin, John McArdle, Roderick P. McDonald, Robert McGrath, Matthias Mehl, Gregory J. Meyer, Boris Mlačić, Fritz Ostendorf, William Revelle, Gale Roid, Leonard G. Rorer, Gerard Saucier, Oya Somer, Auke Tellegen, Krista Trobst, Niels Waller, Keith Widaman, Michelle Yik, and Richard Zinbarg. Niels Waller has developed a set of procedures, and a computer program, for reproducing these structures without having to compute factor scores; for information, contact him at [email protected]. In addition, Daniel Levitin has developed a computer program for drawing the figures, with box widths corresponding to factor sizes; for information, contact him at [email protected].

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