Peter H. Schonemann
Extension of Guttman's result from g to PC1
Multivariate Behavioral Research, 1992, 27, 219-224
In a Target Article, The irrelevance of factor analysis for the study of group differences, Multivariate Behavioral Research, 1992, 27, 175-204, Louis Guttman had shown that perfect collinearity of the mean difference vector d with all three g factors extracted from both within covariance matrices and the pooled covariance matrix follows if one assumes that Spearman's (one-) factor model holds in all three populations, thereby undermining Jensen's (1980) claims that his finding of positive correlations between d and any of the 3 g's corroborates the empirical realty of the g factor (See Famous Artefacts for more details and background on "Spearmans's Hypothesis").
Guttman was well aware that in practice Spearman's one factor model virtually never fits any data:
"Any reader of these lines can himself easily disprove g by looking at almost any mental test correlation matrix at his disposal and checking for proportionality [i.e. hierarchy, a necesssary condition for the existence of g, PHS]" (Guttman, 1992, p.182).
In the present paper it is shown that a similar result can be proved for principal components (rather than g), without any need to invoke the unrealistic factor model: All one needs is
(a) that the total (pooled) distribution is multivariate normal, and
(b) that all subtest correlations remain positive in both subpopulations defined by a bisecting plane orthogonal to PC1 and containing the centroid.
In this case, positive (Level II) Spearman correlations emerge as artifacts because the mean difference for the Hi and Lo subpopulations will be perfectly collinear with the PC1s of all three subpopulations.
(a) It should be noted that Jensen and most other workers in this field actually use principal components, not factors.
(b) For the bizarre history of the whole Spearman Hypothesis saga see Famous Artefacts and Schonemann (unpublished).
(c) This particular paper is relevant to Herrnstein and Murray's rendition of Spearman's Hypothesis in their Bell Curve:
"Another commentator suggested that Jensen had inadvertently built into his own analysis the very correlation between g loadings and black-white differences that he purported to discover (Schonemann, 1985). In the next round ... after being apprised of a response by physicist William Shockley (Shockley, 1987) he withdrew his argument" (Herrnstein and Murray, The Bell Curve, 1994, p. 726).
As the above paper shows, the exact opposite is true: Far from retracting my claim that said correlations are artifacts, I extended it to the stronger Level II case, asserting not just approximate but perfect collinearity with all three PC1 under the stated conditions. It is interesting to note that Herrnstein and Murray cite a number of other MBR commentators more to their taste, while studiously avoiding mention of my Commentary in the same issue which refutes their claim.
(d) The paper contains a minor slip in defining the partitioning plane in the proof, as has been noticed by Dolan, Multivariate Behavioral Research, 1997, 32, 319-325. As I pointed out in my Reply (Schonemann, 1998, 16, 788-812), the practical effect of this oversight would have been miniscule, because the first centroid and the PC1 are virtually collinear for positive symmetric matrices.