Attention has been drawn to the lack of standards for evaluating the degree of goodness of fit of patterns resulting from a principal components analysis of two data sets. An empirical sampling distribution of the statistic average trace (E'E) as E is obtained in the orthogonal procrustes problem, for various orders of A matrices was developed through a Monte Carlo approach. A method is presented which can be used as a guideline in determining whether factor structures obtained from two data sets are congruent. (Author)