To What Extent can Communalities Reduce Rank?

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Abstract
The question is raised as to whether the null hypothesis concerning the number of common factors underlying a given set of correlations should be that this number is small. Psychological and algebraic evidence indicate that a more appropriate null hypothesis is that the number is relatively large, and that smallness should be but an alternative hypothesis. The question is also raised as to why approximation procedures should be aimed primarily at the observed correlation matrix R and not at, say, R−1. What may be best for R may be worst for R−1, and conversely, yet R−1 is directly involved in problems of multiple and partial regressions. It is shown that a widely accepted inequality for the possible rank to which R can be reduced, when modified by communalities, is indeed false.