Note On a Criterion for the Number of Common Factors

approach to the number of factors problem involving the use of random normal deviates as additional variables in correlational matrices from which principal components are extracted. Horn (1966) has made use of an independent component analysis of random normal deviates, parallel to the analysis of the dquo;realdquo; variables, in order to correct the Kaiser criterion (see Horn) which assumes population values of the correlations, for capitaliza-tion on chance in the sample. This note briefly presents some results obtained from another, and more promising, variation on their procedures. Linn found that his mean square ratios, analogous to F-ratios with data from the latent roots of real variables in the numerator and of random variables in the denomination, fluctuated widely as ratios were formed from successive roots. The lack of indepen-dence arising from the fact that both the numerator and denomi-nator were derived from the same analysis, though from independent sets of variables, is the probable cause of this variability. Horn’s use of a parallel analysis thus is indicated. Horn, however, was interested in the Kaiser rule which involves the use of unities in the diagonal of the correlational matrix. The automatic use of unity for a real variable whose population com-munality in any particular set of measures may be zero, and for random variables whose population communalities are zero, follows