A Criterion for Significant Common Factor Variance

Up to the present only empirical methods have been available for determining the number of factors to be extracted from a matrix of correlations. The problem has been confused by the implicit attitude that a matrix of intercorrelations between psychological variables has a rank which is determinable. A table of residuals always contains error variance and common factor variance. The extraction of successive factors increases the proportion of error variance remaining to common factor variance remaining, and a point is reached where the extraction of more dimensions would contain so much error variance that the common factor variance would be overshadowed. The critical value for this point is determined by probability theory and does not take into account the size of the residuals. Interpretation of the criterion is discussed.