Three Faces Of Factor Analysis
David Bartholomew
Athens University of Economics and Business

Spearman's basic idea was that the mutual correlation of variables (such as test scores) might be explained by their common dependence on a latent variable, which he called a factor. Such factors were to be uncovered, therefore, by appropriate numerical analysis of the correlation matrix. Using the existing theory of partial correlation, Spearman was able to show that the presence of an underlying factor would be revealed by a particular pattern in the correlation matrix. Although others extended this idea to several factors, the analysis of the correlation matrix on these lines dominated factor analysis for the next half century.

The second face is most easily seen in Lawley and Maxwell's Factor Analysis as a Statistical Method which first appeared in 1963 but it would be difficult to identify its precise origin. It was part of a general shift in statistics to the use of probability models which became obvious around the 1950s. The starting point was now a linear model in which the observed variables were expressed as linear functions of the factors and random 'errors.' This enabled the theory of statistical inference to be brought to bear on estimation and hypothesis testing using the likelihood function. A subtle change of focus from the correlation matrix to the covariance matrix accompanied this move and paved the way for covariance structure analysis.

The third face went unrecognized for nearly half a century but it was already present in Lazarsfeld's work on latent structure analysis. Spearman's original insight here re-surfaced in the observation that association between categorical variables might be explained by latent variations in continuous or categorical 'factors.' In fact, the difference between factor and latent structure analysis is more apparent than real. The two can be brought together in a factor analysis framework by supposing that the normal distribution of classical factor analysis is replaced by a member of the one-parameter exponential family in which the canonical parameter, rather than the mean, is linear in the factors and errors.

Factor analysis, in its maturity, thus emerges as one method of studying the inter-dependence structure of random variables. It differs from recent work on graphical models, multivariate dependencies etc., principally by introducing latent variables. Spearman was handicapped by the limits of the statistical and computational technology a century ago but, even more, by his narrow focus on measuring intelligence. Such is the fate of truly original pioneers. Genuinely new ideas are rare in any field and Spearman should be honoured for one which virtually created psychometrics and, eventually, had much wider effects.