Viewing Spearman's hypothesis from the perspective of multigroup PCA A comment on SchoÈnemann's criticism

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1 Intelligence 29 (2001) 231± 245 Viewing Spearman's hypothesis from the perspective of multigroup PCA A comment on SchoÈnemann's criticism Conor V. Dolan a, *, Gitta H. Lubke b,1 a Department of Psychology, University of Amsterdam, Roetersstraat 15, 1018 WB Amsterdam, Netherlands b Faculty of Psychology and Pedagogy, Vrije Universiteit, Van der Boechorstlaan 1, 1981 BT, Amsterdam, Netherlands Received 25 October 1999; received in revised form 15 March 2000; accepted 15 May 2000 Abstract Jensen's test of Spearman's hypothesis is meant to demonstrate the importance of general intelligence in Black ± White (B± W) differences in psychometric intelligence test scores. SchoÈnemann purports to demonstrate, through an analysis of real and simulated data, and the presentation of a theorem, that Spearman correlations are artifacts. We discuss the theorem and conclude that the theorem cannot be advanced in support of the contention that Spearman correlations are positive and substantial by mathematical necessity. The theorem encompasses a multigroup principal component analysis (PCA), which is a viable model to investigate Jensen's proposition that Blacks and Whites differ mainly with respect to g. We view SchoÈnemann's simulation study in the light of the multigroup PCA model, and interpret it as a study of the specificity of Spearman correlations, given model violations. As such, we find it to be open to several criticisms. SchoÈnemann does raise an important issue, namely whether Spearman correlation can be trusted to prove the importance of g in B±W differences in psychometric IQ scores. Based on recent studies, we consider the Spearman correlation to be a suboptimal test of Spearman's hypothesis, and contend that an explicit model-based approach should be used. D 2001 Elsevier Science Inc. All rights reserved. Keywords: Spearman's hypothesis; General IQ; Black ± white differences; Jensen's test; Specificity * Corresponding author. address: conor@psy.uva.nl (C.V. Dolan). 1 address: gh.lubke@psy.vu.nl (G.H. Lubke) /01/$ ± see front matter D 2001 Elsevier Science Inc. All rights reserved. PII: S (00)

2 232 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231± Introduction Jensen has proposed that the differences between Blacks (B) and Whites (W) in psychometric intelligence test scores are attributable mainly to a difference in general intelligence, or g (Jensen, 1985, 1998). To test this proposition, Jensen formulated Spearman's hypothesis. There are two versions of this hypothesis, a weak version and a strong version. The strong version states that ``variation in the size of the mean W±B difference across various tests is solely a positive function of variation in the tests' g loadings'' (Jensen, 1998, p. 372). The weak version states that the variation is mainly a positive function of variation in the tests' g loadings. The rationale is that a test, which is strongly influenced by g (i.e., characterized by a large factor loading on g) will display a large B±W difference in mean. Jensen has investigated Spearman's hypothesis by calculating the correlation between the vector of differences in means and the g loadings. These correlations are referred to as Spearman correlations. This use of Spearman correlations has been criticized extensively by SchoÈnemann (1989, 1992, 1997a, 1997b). In a recent special issue of Cahiers de Psychologie Cognitive (CPC), SchoÈnemann (1997a, 1997b) purports to demonstrate that Spearman correlations are positive and substantial by mathematical necessity. To this end, he presented various results including a theorem and a simulation study. These are based on principal component analysis (PCA). Although the special CPC issue included a large number of invited commentaries, none of the contributors actually identified the problematic aspects of SchoÈnemann's criticism. In view of the importance of the issue of B±W differences in IQ test scores in psychology (and beyond), we believe that criticisms of Spearman correlations should be considered carefully. In the present paper, therefore, we discuss SchoÈnemann's theorem. We show that the theorem cannot be advanced in support of the thesis that Spearman correlations are necessarily positive and substantial. The theorem comprises a scenario in which Spearman correlation will equal unity, but whether this scenario holds in reality is an empirical issue. In addition, we discuss the simulation study in the light of the theorem. We interpret this study as an investigation of the specificity of Spearman correlation given certain model violations. As such, this simulation is open to several criticisms. Remaining within the PCA model implied by the theorem, we suggest and illustrate a different way of investigating the specificity of Spearman correlations given model violations. Our own simulations show that Spearman correlations are not positive by mathematical necessity. We note that SchoÈnemann does raise an important issue, viz., is the Spearman correlation a good method to demonstrate the centrality of g in B±W differences. Based on our research, we contend that an explicit modeling approach is preferable to Spearman correlations (Dolan, 2000; Lubke, Dolan, & Kelderman, 2000). In the interest of clarity and to ease presentation, we introduce the central terms that feature in this paper. SchoÈnemann's theorem and SchoÈnemann's simulation are pinpointed below. Jensen's theory pertains to the centrality of the hypothetical construct, g, in B±W differences in psychometric intelligence scores. Jensen proposes that g is the only (strong version), or the main (weak version) source of B±W differences, i.e., B±W differences in psychometric intelligence scores are due (mainly) to differences in g (Jensen, 1998). The consequences of the centrality of g may be considered in different ways. Here we are concerned solely with the implication for mean and covariance structures. Spearman's hypothesis concerns a single aspect of the theory, namely the correlation between

3 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231± differences in means and factor loadings. Such correlations are referred to as Spearman correlations. Jensen's test of Spearman's hypothesis is based on such correlations. The specificity of Spearman correlations concerns the probability that a Spearman correlation assumes a low value, when aspects of Spearman's hypothesis are violated. Jensen's test has low specificity if Spearman correlations assume large values, even though aspects of Spearman's hypothesis are false (e.g., g is absent). Jensen's procedure of calculating such correlations involves the following steps: (1) Exploratory factor analyses of psychometric data are carried out in representative samples of Blacks and Whites, separately, to extract factor loadings of the tests on g (Jensen suggests that this can be done in a number of ways, see Jensen & Weng, 1994); (2) Factorial invariance over groups is established by calculating measures of factorial congruence of the factor loadings in the White and the Black samples (congruence should be high); (3) Variation in factor loadings is required to be appreciable; (4) Differences in means are standardized by dividing by the pooled standard deviations; (5) The standardized differences in means are correlated with the g factor loadings. A mean (Spearman's rho) correlation of.59 (S.D. = 0.12), obtained using this procedure, has been reported based on 11 studies (Jensen, 1985). 2. SchoÈnemann's theorem SchoÈnemann (1997a) offers two interpretations of Spearman's hypothesis, the Level I and Level II interpretations. The Level I interpretation concerns PCA of a pooled (over the two groups) covariance matrix without prior within-group centering (subtraction of means). In this case, the presence of between-group differences will distort the results of the within-group PCA. Specifically, the dominant eigenvalue will reflect the between-group variance and the associated eigenvector will approximately equal the vector of differences in means of the groups. The Level I interpretation is not relevant, given Jensen's procedure of testing Spearman's hypothesis. In recent discussions, Jensen (1992, 1998) does not advocate pooling without prior within-group centering. Rather, one is advised to carry out separate factor analyses or PCAs in the two groups. We therefore shall not dwell on the Level I interpretation. SchoÈnemann's Level II interpretation is more relevant. This interpretation runs as follows: `... the mean difference vector correlates positively with the regression weights of the PC1s of both the within sample covariance matrices' (SchoÈnemann, 1997a, p. 668). SchoÈnemann's theorem is closely related to the Level II interpretation, in that it provides a scenario in which the Level II interpretation holds: If the range Re p of a p-variate normal random vector y N p (0,S), where S is positive with diag(s)=i, is partitioned into a High set (H) and a Low set (L) by the plane orthogonal to the PC1, and containing the origin, and both within-covariance matrices S H, S L, remain positive, then (a) the mean difference vector d:=e[ yjh] E[ yjl] will be collinear with the PC1 of the pooled S, and (b) d will also be collinear with both PC1 of S L, S H (SchoÈnemann, 1997a, p. 685; this is a slightly revised version of the theorem presented in SchoÈnemann, 1992 and SchoÈnemann, 1989; see Dolan, 1997).

4 234 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231± The meaning of the theorem We attempt to convey the theorem in less technical terms, as we want to make it clear that the consequences (a) and (b) follow trivially from the selection described in the theorem. Whether this type of selection is consistent with observed B±W differences, and so whether (a) and (b) hold, is an empirical issue. In discussing the meaning of the theorem we employ elementary matrix algebra. Many statistics books provide accessible introductions to such material (e.g., see Lawley & Maxwell, 1971, Appendix I; Morrison, 1990, pp. 36±44; Stevens, 1992, Chapter 2). Imagine a population in which realizations of a p-dimensional random vector y, distributed N p (0, S), are observed in a random sample (we do not employ subject subscripts). The notation N p (0, S) implies that y is multivariate normally distributed with zero mean and covariance matrix S. The random vector y may represent the scores of a given subject of an intelligence test (e.g., Jensen & Reynolds, 1982, for a good example). The theorem specifies that S contains positive elements, but that otherwise the covariance structure of S is not an issue. For instance, the covariance matrix S is not required to be consistent with a common factor model. We consider the eigenvalue decomposition of the covariance matrix S. This decomposition is usually carried out as a form of factor analysis, namely PCA. The aim is to reduce the p observed variables to a smaller number of uncorrelated linear combinations of the p variables. The eigenvalue decomposition of S is expressed as t (e.g., Morrison, 1990, Chapter 8), where ( p p) contains the orthonormal (column) eigenvectors ( =[ 1, 2,..., p ]; t equals the identity matrix, I), ( p p; diagonal) contains the positive eigenvalues, and superscript t denotes transposition. We assume that the eigenvalues are distinct and ordered in descending order: diag()=[d 1 > d 2 >...>d p 1 >d p ]. We denote the ( p 1) eigenvector associated with d i as i, and call the principal component scores, calculated as i t y, i scores. In the population, the i scores are distributed N(0, d i ), i.e., as zero-mean normal variables. The eigenvalues d i can be viewed as the variance of the i scores and the matrix may be viewed as a covariance matrix. Note that this matrix is diagonal because the i scores are mutually uncorrelated. According to the theorem, the population is divided into a high (H) and a low (L subpopulation. The H (L) subpopulation consists of individuals whose 1 scores are greater (less) than zero. The collinearity, which features in the theorem, follows naturally from the manner in which the subpopulations are formed by selection on 1 scores. In the H subpopulation we have (approximately) yjh N p (m H, S H ), where S H = H t, m H = H, and yjh denotes the vector y in the H subpopulation (similar results hold in the L subpopulation). The vector H ( L ) equals the means of the i scores in the H (L) subpopulation. The fact that S H = H t and m H = H is due to the nature of the selection variable (see Dolan, 1997). The consequence of selection on 1 scores is that the eigenvectors in the population equal those in the two subpopulations. Furthermore, because selection is on the 1 scores, and these are uncorrelated by definition with the i scores (i =2,..., p), we find that diag( H )=[d 1H, d 2 > d 3 >...> d p ] and diag( L )=[d 1L, d 2 > d 3 >...> d p ]. Thus the variances of the i scores (i =2,..., p) are unchanged, but the variances of the 1 scores do change due to the selection. The eigenvalues d 1H and d 1L

5 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231± are not necessarily the largest eigenvalues in the subpopulations. The means of yjh and yjl are a function of the means of the 1 scores. The ( p 1) vectors H and L equal [ H1,0,0,...,0] t and [ L1,0,0,...,0] t, respectively, where H1 ( L1 ) is the mean of the 1 scores in the H (L) subpopulation. Note that the means in the subpopulations of the other components are zero, because selection has taken place on the 1 scores, which are uncorrelated with the other i scores (i =2,..., p). Consequently, given present selection, we may be more explicit and write m H = H = 1 H1 and m L = L = 1 L1. For a detailed derivation of the effects on means and covariance matrices in groups selected on 1 scores the reader is referred to Dolan (1997). We view the theorem as a possible scenario in which the Level II interpretation will hold. SchoÈnemann expresses this as follows: ``...under the stated assumptions, the cosines between the mean difference vector and the largest eigenvectors and the two within-sample covariance matrices are not just positive in general, but, except for sampling error, are unity in all multinormal distributions with positive within-covariance matrices'' (SchoÈnemann 1997a, p. 685). 1 As it stands, however, the theorem is trivial. Given the selection variable ( 1 scores), the vector d (E[ yjh] E[ yjl]) is necessarily collinear with 1. In addition, the theorem is too restrictive: given selection on the 1 scores, the collinearity will hold in the absence of positiveness of S, S H or S L. Furthermore, the standardization of S is not required. The consequences (a) and (b) of the theorem follow necessarily from the premises contained in the theorem. With respect to SchoÈnemann's contention that Spearman correlations are artifacts, it is important to note the following. It is an empirical issue whether the premises of the theorem are, by reasonable approximation, tenable in reality. Hence, the theorem cannot be advanced in support of the thesis that the Spearman correlations are positive and substantial by mathematical necessity. As mentioned, SchoÈnemann presents a simulation study in addition to his theorem. To properly judge this simulation study, we require the means and covariance structure implied by the theorem. 4. The model implied by the theorem As we have seen above, the covariance structure model, after the selection on the 1 scores, is as follows: X ˆ PD H HP t 1a X ˆ PD L LP t ; 1b where, as defined above, the p p matrix contains the eigenvectors (equal across the groups), and the diagonal matrices containing the eigenvalues H and L, as defined above. 1 This `and' presumably should read `of': the eigenvectors of the two within-sample covariance matrices.

6 236 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231±245 Eqs. (1a) and (1b) represent a multigroup PCA (see Flury, 1988). The model for the means is: m H ˆ n P H m L ˆ n P L ; where the vector contains constant intercepts and the vectors H and L contains the means of the i scores (see Dolan, 1997). It is interesting to note that Flury, Pienaar, and Nel (1995) previously presented just this model to investigate group differences. Note that the model implied by the theorem (Eqs. (2a) and (2b)) comes very close to the idea behind Jensen's theory: i.e., if the groups differ with respect to g (strong version of Spearman's hypothesis), and if 1 scores are a good approximations of scores on g, the collinearity should indeed hold. The ( p 1) vector H ( L ) equals [ H1,0,0,...,0] t ([ L1,0,0,...,0] t ), where H1 ( L1 ) is the mean of the 1 scores in the H (L) subpopulation. Collinearity between the eigenvector and the difference in mean vectors follows as m H m L equals 1 ( H1 L1 ) (see Millsap, 1997, for a similar discussion of Spearman's hypothesis in the context of the factor model). As mentioned, the degree to which Spearman's hypothesis is represented by the theorem hinges on the proposition that 1 scores are a good operationalization of g. Within the confines of PCA, this is quite difficult to establish by formal goodness of fit testing. Usually the eigenvalue >1 rule (given that S is a correlation matrix) and the screeplot are used to establish that a given S is compatible with the common factor model (e.g., Stevens, 1992). We return to this issue below. With the model implied by the theorem in place, we now evaluate SchoÈnemann's simulation study. First, we describe this study briefly. 2a 2b 5. SchoÈnemann's simulation In addition to the theorem, SchoÈnemann presents an extensive simulation study to investigate the effects of sample size on his level II interpretation. SchoÈnemann demonstrates that Spearman correlations assume substantial values (about.6) under circumstances that the covariance matrix S is positive, but is not consistent with a single common factor model. In the simulation, the correlation S is calculated by standardizing the matrix TT 0 +I, where I is the p p identity matrix and the matrix T contains elements drawn randomly from the [0,1] uniform distribution. The correlation matrix S constructed in this fashion does not fit a single factor model (as we demonstrate below). The mean vector is set to equal zero, m = 0. Next, SchoÈnemann simulated data from N(m,S). Using these data, he carried out selection to form a high group and a low group. In this simulation, the selection is carried out on the basis of the selection variable w t y, where w is the p-dimensional unit vector, w = 1. Members of the high (low) group have scores w t y > c (w t y < c). Note that this selection variable differs from the selection variable in the theorem. In the theorem, selection is on the 1 scores. For various choices of c and p, SchoÈnemann's carries out Jensen's test of Spearman's hypothesis. For instance, for c = 0 and p = 10, he finds a correlation of about.6, i.e., about the value observed in empirical studies (Jensen, 1985). SchoÈnemann purports to show that the Spearman correlation may assume substantial values,

7 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231± even though S is not consistent with the single factor model, i.e., the simplest case of the factor-analytic g model. This suggests that Spearman correlations cannot be trusted to demonstrate the importance of g in B±W differences. We judge SchoÈnemann's simulation to be open to several criticisms, which render it unconvincing. 6. Evaluation of the simulation Working within the context of PCA, one could investigate the soundness of Jensen's procedure by violating the premises of the theorem and gauging their effects on the Spearman correlation. In effect, SchoÈnemann introduces two violations: (1) the correlation matrix S is created such that it does not fit the single common factor model, and (2) in creating the selection variable w equals 1, the unit p column vector, rather than 1. To ease presentation, we consider the normalized vector 1/ p, which we denote 1n, instead of 1. As we interpret SchoÈnemann's simulation study, it essentially addresses the specificity of Spearman correlations, given these violations. The persuasiveness of SchoÈnemann's simulation depends on the degree to which the violations represent a departure from the theorem. If the violations are severe and the Spearman's correlations remain substantial, this would demonstrate the lack of specificity of Spearman correlations. To investigate the violations mentioned, we constructed five correlation matrices, S, in the manner outlined by SchoÈnemann. We created a matrix T consisting of elements drawn randomly from the [0,1] uniform distribution. The S is calculated by standardizing the covariance matrix TT 0 +I, where I is identity matrix. The mean vector is set to equal the zero vector, m = 0. Next we calculated the expected covariance matrices, S L and S H, and mean vectors, m L and m H, in samples selected on the criterion 1 t n y >0or1 t n y < 0. We need not simulate data, as both covariance matrices and mean vectors can be derived analytically: 2 S L ˆ S Sws 2 s 2 L s2 s 2 w t S S H ˆ S Sws 2 s 2 H s2 s 2 w t S m L ˆ m Sws 2 m L m H ˆ m Sws 2 m H 3a 3b 4a 4b 2 Dolan (1997) shows that Eqs. (3a)±(4b) reduce to Eqs. (1a)±(2b) if 1 is substituted for w and t is substituted for S. If1 n is substituted for w, however, this is not the case. As mentioned, the model in Eqs. (1a)± (2b) is a possible model for the strong version of Spearman's hypothesis. Assuming 1 scores are a good proxy for g scores, selection on 1 scores create group differences that are consistent with Jensen's theory. Other selection variables, such as 1 n y, do not give rise to data that are strictly consistent with the model in Eqs. (1a)±(2b), and so consistent with Jensen's theory. The question remains, however, how well Eqs. (1a) ±(2b) can approximate Eqs. (3a)±(4b). This issue of statistical power is addressed in the text.

8 238 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231±245 where w = 1 n, m = 0, and s 2 represents the variance of 1 n t y in the population, i.e., 1 n t S1 n.as m = 0, the mean of 1 n t y, i.e., 1 n t m, is zero. Expressions for the variance (s 2 L and s 2 H) and mean (m L and m H )of1 n t y in the subpopulations can be found by considering the moment of the truncated normal distribution (Kendall & Stuart, 1968). Greene (1993, p. 685) provides the relevant expressions. Note that given the present selection (symmetric about the mean), s 2 L = s 2 H and m H = m L. Eqs. (3a)±(4b) follow from the application of the Pearson±Lawley selection rules (Lawley, 1943). These rules are often used to derive the means and covariance matrix of one set of variable given conditioning (or selection) on a second set of variables (Greene, 1993, p. 76; Morrison, 1990, p. 90 ff.). Here we investigate the means and covariance matrix of y, given selection on the variable 1 n t y. See Meredith (1964) and MutheÂn (1989) for important applications of these rules within the common factor model. We investigated the effects of the violations in the following way: (1) We calculated the scalar 1 n t 1. The more this scalar deviates from 1, the greater the violation. Note that 1 t 1 = 1. (2) We fitted the model (Eqs. (1a), (1b) and (2a), (2b)) to the summary statistics in the L and the H subpopulations (Eqs. (3a), (3b) and (4a), (4b)) using LISREL (JoÈreskog & SoÈrbom, 1993; for details see Dolan, Bechger, & Molenaar, 1999). 3 In order to fit the model, however, we parameterized the model for the means as follows (Eqs. (5a) and (5b)): m H ˆ n m L ˆ n PQ ˆ n 1 H1 L1 ; 5b where 0 equals [ H1,0,0,...,0] t [ L1,0,0,...,0] t,or H L. In this parameterization, the difference in the means of the 1 scores is estimated rather than the means themselves. The parameterization is required for reasons of identification, but is otherwise innocuous. SoÈrbom (1974) discusses this parameterization in the context of the common factor model. If selection based on 1 t n y, rather than on t 1 y, represents a serious violation, this model should not fit the population matrices. 2 In carrying out these analyses, we set N(L) = N(H) = 200, total N = 400. We estimated parameters by minimizing the loglikelihood ratio function. These analyses provide a c 2 goodness of fit index that we use to assess the degree of misfit. 4 For the five covariance matrices the scalar 1 t n 1 equaled 0.997, 0.997, 0.998, 0.998, and This suggests that selection of 1 t n y is not very different from selection on t 1 y. Fitting the model (df = 64) to the population matrices produced the following goodness of fit indices, which may be interpreted as noncentrality parameters (NCPs): , 0.339, 0.182, 0.108, and As these observed indices are extremely small (fitting the true model, would result in NCPs of zero). This again suggests that there is very little difference between selection of 5a 3 A MATLAB m-script to calculate the summary statistics are available on request, as are LISREL input files for all the analyses reported. 4 The goodness of fit index is a function of the loglikelihood ratio (Saris & Satorra, 1993). Fitting the true model to population statistics yields an index of zero. Fitting the true model to sample statistics yields an index that is central c 2 distributed (given that the sample is large and the data multinormally distributed). The expected value of the c 2 equals the degrees of freedom (df ) of the model. Fitting the false model to population statistics yields the noncentrality parameter (NCP) of the noncentral c 2 distribution, and fitting the false model to sample statistics yields a test statistic that has a noncentral c 2 distribution. This distribution is characterized by df and the NCP. The expected value of the index equals df + NCP.

9 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231± t n y and t 1 y. Power calculations indicate that given the NCP of 0.339, one would require over 50,000 subjects to reject the model given =.01 and power = The fact that the Spearman correlations in SchoÈnemann's simulation assume large positive values is not surprising given these findings. SchoÈnemann introduces a second violation in that S does not fit a single common factor model. Indeed, using LISREL to fit a single common factor to our five correlation matrices S (again using maximum likelihood estimation) produced the following NCPs: 273.4, 264.3, 237.9, 231.7, (df = 35; N was set to equal 400). Given the NCP of 231.7, power calculations reveal that one would require less than 100 subjects to reject the model with =.05 and power = Clearly, SchoÈnemann's method of constructing correlation matrices that do not fit a single factor model is effective. Still a problem that remains is that, within PCA, it is difficult to establish that a single common factor model does not fit. We have mentioned that the eigenvalue >1 and screeplots are usually employed to determine the numbers of principal component to retain in a PCA. Fig. 1 contains the plot of the eigenvector associated with the dominant eigenvalue of S and the plots of the eigenvalues of S and S L (we do not consider those in S H as they equal those in S L ). This figure is instructive for two reasons. First, it shows why the scalar 1 t n 1 assumes values that are so close to 1: given the manner in which the matrices are constructed, there is very p little variation in 1. The components of 1 equal about 0.32, while those in 1 n equal 1= 10 ( 0.316). Second, the screeplots of the eigenvalues of the correlation matrix S and the covariance matrix S L indicate that both appear to be consistent with the single factor Fig. 1. Top left panel: 1 (eigenvector associated with the dominant eigenvalue). Top right panel: screeplot of S. Bottom left panel: screeplot of S L (S L is not standardized).

10 240 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231±245 model. Clearly SchoÈnemann's method of creating the covariance matrix S does not represent a severe violation, at least given the current heuristics as applied in PCA to determine the number of factors. The first eigenvalue is clearly dominant (see Fig. 1). Again, the fact the Spearman correlations assume substantial values is unsurprising. 7. Three specific criticisms of the simulation When viewed in the light of the theorem, SchoÈnemann's simulation, as a study of specificity, is not convincing or informative. It approximates the scenario of the theorem quite closely, and it is limited and idiosyncratic in the violations of the model (Eqs. (1a)± (2b)). Given the manner of constructing S, selection based on 1 n t y does not differ greatly from selection based on 1 t y. More generally, as long as w contains positive elements, and S L and S H are required to remain positive, selection on w t y will resemble selection on the 1 scores, 1 t y. Satisfying these requirements makes the model associated with selection on w t y closely resemble the model associated with selection on 1 t y. In addition to this general point, we note that the simulation study is inconsistent with Jensen's theory and his procedure on three counts. (1) Jensen does not actually require that covariance matrices be positive. Positiveness can be relinquished, e.g., removed by simple transformation, with no ill effects to Jensen's test. (2) Jensen does not propose that Blacks and Whites arise through any selection mechanism based on the variable 1 t y. This mechanism is implausible (see Loehlin, 1992). Jensen (1998) proposes that Blacks and Whites differ with respect to the genetic and environmental causes of individual differences in g. (3) Jensen emphasizes that the factor loadings are required to display ample variation (Jensen, 1985, 1992, 1998). It is not clear just how much variation should be present, but it is reasonable to conclude, from Fig. 1 and our results, that this requirement is not met. In conclusion, we do not believe that SchoÈnemann's simulation provides convincing support for the proposition that Spearman correlations are positive by mathematical necessity. 8. Returning to the model implied by the theorem As mentioned, working within the confines of PCA, Jensen's theory that Blacks and Whites differ with respect to g, may accurately be conveyed by the model implied by the theorem (Eqs. (1a),(1b) and (2a), (2b)). This model is consistent with the idea that g is the source of group differences (insofar as these differences are expressed in means and covariance structures). One can investigate Spearman correlations without relying on artificial selection by simply introducing violations of this model and carrying out Jensen's procedure. In so doing, one may create covariance matrices and mean vectors that differ from expectation (i.e., Eqs. (1a)±(2b)) in a controllable and informative manner. In addition, by staying with a model-fitting approach, one can formally assess the power to detect violations. Such an investigation was carried out by Lubke et al. (2000). Working with the multigroup confirmatory factor model, Lubke et al. (2000) found that the Spearman correlation lacks specificity. Although a detailed study is beyond the present scope, we do

11 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231± want to illustrate this method of investigating Spearman correlations within the context of PCA (i.e., Eqs. (1a)±(2b)). 9. Illustrative simulation Blacks and Whites may differ in many ways that are not compatible with the strong version of Spearman's hypothesis (Lubke, Dolan, & Kelderman, 2000). We distinguish two classes of violation, not mutually exclusive. (1) Blacks and Whites differ with respect to principal components in addition to the dominant one. This violation includes instances of the weak version of Spearman's hypothesis. Because Jensen formulation of the weak version is ill defined (Dolan, 2000; Lubke, Dolan, & Kelderman, 2000), it is difficult to tell when we may conclude that the weak version holds (rather than reject Spearman's hypothesis). (2) Blacks and Whites differ with respect to variables that have nothing to do with the principal components. We illustrate instances of both classes of violation within the PCA model. The first violation involves the introduction of nonzero components into the vectors of means of the i scores (i =2,..., p). According to the strong version of Spearman's hypothesis these should equal B =[ B1,0,0,...,0] t and W =[ W1,0,0,...,0] t (subscripts B and W denote Blacks and Whites, respectively). The second violation involves setting both B1 and W1 to equal zero, and adding a vector of random negative values to the mean vector m B (means of the observed y). These violations differ in an interesting manner. In the first, a central aspect of Jensen's theory is retained: the groups do differ with respect to principal components, but not in the manner predicted by the strong version of Spearman's hypothesis. The mean structure can still be modeled in a manner that is consistent with Eqs. (2a) and (2b). The second violation is more serious in theory as it implies that the groups differ in a manner that cannot be modeled by Eqs. (2a) and (2b). Here a crucial aspect of the model, namely, the fact that withinand between-group variance is attributable to the same principal components, is lost. We create 500 (10 10) covariance matrices S according to SchoÈnemann's method. 3 As above, the eigenvalue decomposition is denoted S = t. We assign identical S to both the Black (S B ) and the White group (S W ). Although this is not realistic (variance in White samples is larger), it is of little consequence here. Trial and error revealed that W1 = 0 and B1 = 3.25 produces observed mean differences of about one standard deviation. We introduce the following violations of the strong version of the hypothesis: Violation A: B ˆ 3:25; 0:2; 0;...; 0Š t : Violation B: B ˆ 3:25; 0:2; 0:2; 0; 0;...; 0Š t :

12 242 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231±245 Violation C: B ˆ 3:25; 0:2; 0:2; 0:2; 0;...; 0Š t : Violation D: B ˆ 3:25; 0:2; 0:2; 0:2; 0:2; 0;...; 0Š t : In all cases, we retain W =[0, 0,...,0] t. Using these vectors and n=[0,...,0] t,we construct mean vectors according to Eqs. (2a) and (2b). In addition, we introduced the following violation: Violation E: m W ˆ n and m B ˆ n 1 j; where again is the ( p 1) zero vector and the components of X are drawn randomly from the uniform [.05,.05] distribution. Given Violations A±E, the means of the Spearman correlation (based on 500 replications) equal.65 (S.D. = 0.20),.52 (0.24),.44 (0.27),.40 (0.27),.002 (0.34), respectively. Histograms of the Spearman correlations are shown in Fig. 2. We do not draw definite conclusions about Spearman correlations based on these results, because Violations A±D are quite arbitrary. However, the drop in mean Spearman correlation Fig. 2. Histograms of Spearman correlations given violation A (top histogram) to E (bottom histogram) based on 500 replications. The nature of the violation is explained in the text.

13 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231± reflects the increasing severity of the violation. More importantly, the mean correlation in the case of Violation E is about zero (.002). Here the between-group differences have nothing to do with the within-group differences, in contrast to Violations A±D, which may be viewed as compatible with the weak version of the hypothesis. Although limited in scope, these results show clearly that Spearman correlations are not positive by mathematical necessity. 10. Discussion The aim of the present paper is to consider SchoÈnemann's criticism of Spearman correlations from the perspective of multigroup PCA. SchoÈnemann's theorem concerns a scenario in which the Spearman correlation will equal unity by mathematical necessity. As explained, the theorem comprises a multigroup PCA-based model that provides a quite accurate representation of Jensen's hypothesis concerning B±W differences. The theorem however does not prove the contention that Spearman correlations are substantial by mathematical necessity. The consistency between the model implied by the theorem (Eqs. (1a), (1b) and (2a), (2b)) and a given data set is an empirical issue and should be treated accordingly. Our discussion of SchoÈnemann's simulation study identified several limitations, which render it quite unconvincing. These concern the manner of creating S and the creation groups based on the selection variable 1 t y. These aspects of the simulation give rise to covariance and mean structures that closely approximate the structures expected given SchoÈnemann's theorem. In addition the variation in the components of 1 (as observed in the simulation Ð see Fig. 1) is necessarily small. Jensen clearly emphasizes the importance of variable factor loadings. We believe that a central problem is the creation of groups based on the variable 1 t y. SchoÈnemann does not provide any rationale or justification for this manner of data simulation. Ultimately, we view SchoÈnemann's simulation as a study of specificity of Spearman correlations given model violations. Of course, demonstrating a lack of specificity is not the same as demonstrating that Spearman correlations are positive by mathematical necessity. All tests are characterized by a degree of specificity and sensitivity: this does not render their outcome trivial. Our own simulation study, although limited in scope, shows that Spearman correlations vary with the degree of model violation, and are not positive by necessity. Regardless of the details of his simulation study, it is clear that SchoÈnemann raises an important issue, viz., are Spearman correlations a good test of the role of g in B±W differences in IQ scores? Many commentators on Spearman's hypothesis appear to accept that Jensen's procedure is OK (for an overview, see SchoÈnemann, 1997a, pp. 666±667), and apply it accordingly (for recent applications, see Lynn & Owen, 1994; Nyborg & Jensen, 2000; Rushton, 1999; Te Nijenhuis, Evers, & Mur, 2000; Te Nijenhuis & van der Flier, 1997). Dolan (2000) and Lubke, Dolan, & Kelderman, (2000) have identified several weaknesses of Jensen's procedure. Perhaps the most obvious problem with Spearman correlations is that they address a single aspect of the model implied by Spearman's hypothesis concerning the centrality of g in B±W differences. In addition, Jensen's procedure does not involve any goodness-of-fit testing or any comparison or investigation of competing models (see Dolan,

14 244 C.V. Dolan, G.H. Lubke / Intelligence 29 (2001) 231± ). Spearman's hypothesis embodies a specific multigroup means and covariance structure model. Several commentators (Dolan, 1997; Gustafsson, 1992; Horn, 1997; Millsap, 1997) have pointed out that the issue of B±W differences in intelligence scores should be investigated using multigroup confirmatory factor analysis (MGCFA; SoÈrbom, 1974). The conditions for a meaningful comparison of groups within the factor model are known (Meredith, 1993). Such analyses can be carried out readily using widely disseminated software. Dolan (2000) presented an extensive discussion and application to the data published in Jensen and Reynolds (1982). In the present paper, we have considered the multigroup PCA model implied by SchoÈnemann's theorem and previously presented by Flury et al. (1995). However, we do not believe that this model is preferable to MGCFA. MGCFA has several advantages that include scale invariance and flexibility in model specification (Dolan, 2000). An important aspect of the model-based approach is that the relevant models capture a central aspect of Jensen's theory. This aspect concerns the relationship between within- and betweengroup variance. Jensen essentially argues that factors that cause within-group (individual) differences also cause between-group (mean) differences. An often repeated response to Jensen's theory is that within and between-group variance may be independent (e.g., Jensen, 1973, p. 134). However, the present PCA-based model, as well as the MGCFA-based models presented in Dolan (2000), provides a viable way to investigate the proposition that withingroup and between-group variance are attributable to common causes. The often-repeated proposition that these two classes of variance may be independent is thus amenable to empirical investigation. Lubke, Dolan, Kelderman, and Mellenbergh (2000) discuss the usefulness of MGCFA in investigations of both B±W differences and the Flynn effect. Acknowledgments The research of Conor Dolan has been made possible by a fellowship of the Royal Netherlands Academy of the Arts and Sciences. References Dolan, C. V. (1997). A note on SchoÈnemann's refutation of Spearman's hypothesis. Multivariate Behavioral Research, 32, 319 ±325. Dolan, C. V. (2000). Investigating Spearman's hypothesis by means of multi-group confirmatory factor analysis. Multivariate Behavioral Research, 35, 21± 50. Dolan, C. V., Bechger, T., & Molenaar, P. C. M. (1999). Using structural equation modeling to fit models incorporating principal components. Structural Equation Modeling, 6, 233 ±261. Flury, B. (1988). Common principal components and related multivariate models. New York: Wiley. Flury, B. D., Nel, D. G., & Pienaar, I. (1995). Simultaneous detection of shift in means and variances. Journal of the American Statistical Association, 90, 1474 ±1481. Greene, W. H. (1993). Econometric analysis (2nd ed.). New York: Macmillan. Gustafsson, J. E. (1992). The relevance of factor analysis for the study of group differences. Multivariate Behavioral Research, 27, 319 ± 325. Horn, J. (1997). On the mathematical relationship between factor or component coefficients and differences between means. Cahiers de Psychologie Cognitive, 16, 750 ± 757.

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