APPENDIX F A Note on Factor Analysis

Some statistically sophisticated readers may wonder why we did not make use of factor analysis, an alternative and much more common procedure. We must confess to some bias in this respect—not against factor analysis as a computational device, but against this form of analysis as an end in itself. The cluster analysis we have mentioned of the data in Chapter III was, for instance, carried out on the basis of a principal-axis solution, dropping the factors with negative average factor loadings, estimating revised communalities from the remaining factors, and repeating the process until the communality estimates stabilized and the excluded factors had average loadings of about zero. The distances between varied vector points in the multiple-factor space were then calculated from the final communalities and the correlations of the variables and variable clusters determined accordingly.

Our objections to factor analysis as an end in itself may be explained as follows:

(1) Unless the data permit a unique simple structure solution,1 the final selection of reference axes is arbitrary, whereas the relative positions of the variables in the multiple-factor space is always invariant with respect to a given set of data. We prefer to deal with the givens, rather than with arbitrary constructions on them. The latter can, in our view, be quite deceptive. In one instance, we cluster-analyzed a set of data from a published factor analysis which purportedly supported a series of hypotheses and found that every one of these hypotheses was flatly contradicted by the actual variable configurations. The deceptiveness of the factor analysis came from the arbitrarily chosen reference axes and the common practice of examining factor loadings for each factor separately. It is obvious, however, that variables which cluster in the multiple-factor space must have similar "factorial compositions," regardless of which reference axes are chosen—a fact which can be brought out only by examination of the pattern of factor loadings of each variable rather than by examination of the pattern of factor loadings for each factor, and this amounts to a cluster analysis.

(2) The general theory of factor analysis seems to us to be the indefensible view that the sole source of correlation between variables is the presence of common components. To be sure, such an assumption is not mathematically necessary, since one may think in purely actuarial terms of "accountable" portions of the variance, and one may think of the factors as hypothetical variables which are more-or-less predictable from the actual variables rather than as representing that which is common to them. On this basis, however, the reason for any special interest in such hypothetical variables becomes obscure. As long as one thinks of a factor as a common component, it is clear that it may be regarded as a real variable and, if the original matrix is large enough and encompasses a judicious selection of variables, as a probably unitary, or at most limitedly divisible, one. If one abandons the common-component notion, however, why think of it as a variable at all? And, even if one does, the usual practice of interpreting or "naming" such a variable—i.e., speculating on what can be common in variables with high loadings and missing from variables with low loadings—becomes indefensible. We can think of no reason for assuming that correlations with such hypothetical variables (i.e., the factor loadings) should have fewer possible sources than correlations among other variables. Suppose, for instance, that the following census-tract variables turn out to have high loadings on a common factor: percentage of Negroes, poverty rates, percentage excess of females over males, and index of disrupted family life. What sense would it make to look for a common component in these variables? Or a common cause? Or a common anything else? Would it not make better sense to assume that these variables have "got together" for a complex variety of reasons?

This is not to say that there may not be circumstances when factor analysis is justified as more than a convenient tool. Thus, if there are independent grounds (e.g., a theoretical schema) that require the postulation of a particular variable, it seems to us to be entirely legitimate to test relevant data for their compatibility with such a postulate.

1 See L. L. Thurstone, Multiple Factor Analysis (Chicago: University of Chicago Press, 1947). The criteria for unique simple structure are apparently so rigorous that one hardly hears of the concept. If a unique simple structure solution is possible, the outcome would necessarily be virtually identical to that of a cluster analysis.