Factor analysis. Fact or fabrication
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In this survey paper the author critically examines all aspects of factor analysis: the data structure it is intended to describe; the methods of analysis aimed at estimating that structure; and the interpretations placed on the results of the analysis. The first part of the paper deals with the theory underlying the technique. It is shown that, in general, the estimation problem of factor analysis is insoluble. In particular, if there are p variables, and m factors are to be identified, there are indeterminacies in the model itself unless p is at least 2m+1; even when the condition is satisfied, the solution is determinate only up to a group of rotations. After a brief discussion of principal component analysis, principal factor analysis is described. It is shown by examples that "principal factor analysis may give factor estimates that bear no relationship whatever to the true values''. Even with maximum likelihood methods, which are briefly described, there are indeterminacies. It is shown with synthetic data that a restriction usually imposed to achieve uniqueness does not necessarily retrieve the originally applied parameters. The situation is further complicated by the rotation of factors to achieve either a simple structure or a psychologically meaningful solution. The second part of the paper describes the testing of factor analysis programs using artificial data. The general conclusion from this practical study is that, while the numerical accuracy of the programs is satisfactory, the results obtained do not correspond to the original structure of the data, but produce "factors'' that were not initially present. The examples also show that, when matrices of estimated loadings are subjected to an orthogonal transformation, "all resemblance to the true loadings disappeared''. The author makes various pungent comments on the findings from this study. The paper concludes with a short section on the clustering of variables into uncorrelated sets.