ABSTRACT
Up to now, our emphasis has been upon determining the n × r factor-pattern matrix, L, either in the fundamental equation of common-factor analysis, Y = LX + YE, or in the fundamental equation of component analysis, Y = LX. The factor-pattern matrix is useful because it provides a basis for relating the n observed variables in Y to the r underlying common factors or to the n underlying components (as the case may be) in X. But, given a particular observation on the n observed variables, represented by the n × 1 observation vector yi, we might wish to estimate or determine a corresponding observation vector x which relates the ith observation to either the r common factors or the n components (as the case may be). Doing this would be desirable in a common-factor analysis because presumably from a theoretical point of view the common factors have a more fundamental importance than the observed variables and therefore we wish to relate the observations to the common factors. For example, one may construct a personality test of many self-description items, administer this test to a large number of persons, intercorrelate the items, and perform a common-factor analysis on the correlations among the items. The resulting common factors may then be interpreted as fundamental, underlying (latent) variables which in uence performance across collections of items in the test. These factors may then assume importance for a theory of personality, and as a consequence one may wish to relate the people measured on the personality test to these factors, so that one could say that person i has score x1i on factor 1, score x2i on factor 2, and so on.
