The Scientific Use of Factor Analysis in Behavioral and Life Sciences

The student of psychology lives today in that exciting phase of the sociobiological sciences wherein genuine quantitative laws have at last begun to emerge. The prequantitative phase of shrewd literary observation and of penetrating clinical intuition, gave us some useful theories; but the age of quantitative experiment places psychology with the more mature sciences. It promises deeper understanding and the practical effectiveness of precise models and equations.

Typically, the factor analyst takes a careful choice of variables in the area of behavior in which he is out to test a theory or to explore structure, e.g., primary ability variables; marker variables for a supposed extravert temperament; measures which psychologists have thought to be indicators of strength of motivation; or representatives of an hypothesized learning gain pattern. As he looks at the square matrix of all correlations among, say, thirty such variables, he wants to know how many distinct independent influences—presumably decidedly fewer than thirty—can be considered responsible for the observed covariations in this domain of behavior.

It has been pointed out in the last chapter that in factor analysis as in most scientific analysis systems, e.g., the mathematical structure of the atom, the same model can be presented either algebraically or geometrically. Having given in Chapter 2 the algebraic and arithmetic meaning of a factor common to several variables we now propose to reinforce it by a geometrical, graphic presentation and to use this as the clearest approach to the next step—that of rotation.

Before proceeding to the art of rotation, which gives the unique resolution ultimately needed as the solution to a factorial investigation, we need to return to base and tidy up some by-passed technical details of the factor extraction process itself. One may hope that the reader has, through the combined algebraic and geometrical approaches, got the conception of a factor as a common source and direction of variance contributing to the observed covariation of several psychological measures. He has, further, become acquainted with the computational process of extracting a series of successively smaller factor variances from the R matrix. The general outline has thus been given for the extraction process but there still remains thorny practical technique details about how we decide the number of factors, how we guess (or, to use a better term “estimate”) the communalities, like those inserted in Table 2.1 and how we finally get to the most accurate form of the unrotated factor matrix. Also we need to note the slight distinctions between the factors from weighted and unweighted summation methods.

The psychologist eager to get to the use of factor analysis may have fretted a little at the almost philosophical perspective of the discussion in the last chapter concerning what we mean by factors. But neglect of the firm basis necessary for the scientific model itself would have landed us here only in benighted squabbles among various trivial technical rules of thumb.

Having determined the number of factors, and holding in his hands an unrotated factor matrix V0 with consistent communalities, the investigator stands at the most perilous phase of a factor analysis. However, excellent the work up to this point, the way in which the next step is carried out—that of finding the uniquely meaningful rotation—will decide whether a conclusion is drawn that is enlightening, or somewhat misleading, or positively absurd.

The preceding chapter has stated the general aims and styles of rotation. In this chapter we shall concentrate entirely on the practical techniques of obtaining a resolution on the simple structure principle, because, until confactor rotation is perfected, it is likely to remain the sole dependable principle in hypothesis-producing factor analysis. In any case, where the obtaining of suitable mutually related groups for confactor is not possible, and wherever for any reason one is confined to a single experiment, simple structure remains the one dependable everyday method of reaching the hidden unique resolution.

The last chapter has “blocked in” the main concepts in simple structure rotation: the calculation of transformation and shift matrices; the reasons in the model for freedom for oblique factors and reference vectors; and the use of automatic and visual Rotoplot rotation programs. But it has by no means exhausted the conceptual intricacies, or completed the necessary formulas, or reached out into the tactical skills needed to discover simple structure with greatest precision.

The researcher has now achieved a unique solution as regards primary factors. He possesses a factor pattern matrix of oblique factors at a position of demonstrated maximum simple structure, or confactor resolution, and hopefully of sufficient statistical significance in loadings, hyperplanes, etc. He should preserve in file, as the hallmarks of his solutions, the Rυ matrix, the V0 (with a column of communalities), the L matrix, which transformed from the V0 to the Vrs , and the correlations among the oblique factors Rf . For all of these may be needed; for further steps or for examination later, by others or by himself, of the adequacy of his solution; for searches for the possibility of alternative solutions; and for theory checking and development in the next stage of experiment.

Behind us lies exploration and definition of the varieties of scientific factor models, and the grasp of practical procedures for extracting and uniquely rotating factors in conformity with those models. The trunk of the tree of factor analytic procedures is completed and we stand at a juncture where we may branch out in a variety of directions. Among these directions are included: (1) the identification and interpretation of factors; (2) the calculation of individual factor scores; (3) the calculation of factor battery validities; (4) the application in this area of psychometric concepts that go with scaling; (5) the examination of the outcome of using various covariation indices in the correlation matrix; and (6) further pursuit of the model, from traits, into the domain of states and processes, etc.

Once having reached unique resolution of factors in a domain and—by matching and interpretation—established them as replicated and meaningful concepts, the psychologist can enter many useful directions of pure and applied research. Indeed, it is here that multivariate and bivariate experimentalists can most profitably come together, for the latter can now avail themselves, in their ANOVA designs, of meaningful factor scores, as independent and dependent variables. To take these steps the next requirement is to set up measuring batteries and scales appropriately scored and of definable reliability and validity. To serve that need we turn in this chapter to the principles of scoring, to the salient issues of battery construction in the light of those principles, and to the evaluation and maximizing of psychometric properties of factor scales, as reflected in validity and consistency coefficients.

Our choice of direction at this point lies between exploring the rich and diverse roles of factor analysis in psychological, social, and biological experiment, on the one hand, and scrutinizing, on the other, purely psychometric problems and statistical requirements. Both research and applied science must certainly in the end study intelligent application of sophisticated formulations in the latter area. But it has seemed best to us to continue at this juncture the momentum of our expanding general view of factor analysis. Perspective must precede the detailed technical pursuit of ultimate accuracy, for we should not be doing justice to accuracy unless we viewed it also in the context of these wider approaches. And there may be some readers who desire the broad methodological overview without intending to progress to meticulous computing issues.

It is desirable at this point to reiterate the distinctions between a scientific model, an analysis model, and a factor extraction procedure. A scientific model is by no means confined to the factor domain. It can be any thing that we can set up in definite, testable form, couched in mechanical, or mathematical, or chemical or biological, etc., form. It is some precisely defined set of relations to which we suppose there is an analogy in the working relations of the data.

As the statistical interests of psychologists advanced, with expanding mathematical models, beyond those of psychophysics, reaction times, paired comparisons, stochastic processes in learning, and the like, they found themselves in domains which lacked the statisticomathematical landmarks to which they had been accustomed. ANOVA provided a tight, dependable system for examining the significance of experimental results in simple models; probability theory offered mathematically firm answers in other directions, and so on. It was largely in the factor analytic developments out of CORAN that psychologists found themselves having to make shrewd, even “artistic” judgments, and the more “precise” investigators found these rough pioneer customs not easily tolerable.

It is proposed here to run through the essential steps of a factor analytic research experiment in the context of the general strategy of investigation. This will permit integrating the application of the principles and concepts treated in preceding chapters, with the touch of a summary.