The present analysis is an extension of Spitz’s earlier (**1988**) study on the relationship between mental retardation (MR) lower score and subtest heritability (h2) and g-loadings. These relationships were found to be positive. But Spitz himself haven’t tested the possibility that MR (lower) score could be related with shared (c2) or nonshared (e2) environment. I use the WAIS and WISC data given in my earlier **post**, and have found that MR is not related with c2 and e2 values. These findings nevertheless must be interpreted very carefully because the small number of subtests (e.g., 10 or 11) is a very critical limitation.

**Introduction**. It is well known (or should be) that blacks and whites differ in the rates of mental retardation. Some researchers contend that it is closely related with environmental factors, most likely, family environments (e.g., Yeargin-Allsopp et al., **1995**). On the other hand, Jensen (**1998**, pp. 368-369, 405) noted the following :

In familial retardation there are no detectable causes of retardation other than the normal polygenic and microenvironmental sources of IQ variation that account for IQ differences throughout the entire range of IQ. Although persons with familial retardation are, on average, lower in IQ than their parents and siblings, they are no lower than would be expected for a trait with normal polygenic inheritance. For example, they score (on average) no lower in IQ compared with their first-order relatives than gifted children (above +2σ) score higher than their first-order relatives. Parent-child and sibling correlations for IQ are the same (about +.50) in the families of familial retardates as in the general population. …

Organic retardation, on the other hand, comprises over 350 identified etiologies, including specific chromosomal and genetic anomalies and environmental prenatal, perinatal, and postnatal brain damage due to disease or trauma that affects brain development. … The IQ of organically retarded children is scarcely correlated with the IQ of their first-order relatives, and they typically stand out as deviant in other ways as well. In the white population, for example, the full siblings of familial retarded persons have an average IQ of about ninety, whereas the average IQ of the siblings of organic retardates is close to the general population mean of 100.

Statistical studies of mental retardation based on the white population find that among all persons with IQ below seventy, between one-quarter and one-half are diagnosed as organic, and between one-half and three-quarters are diagnosed as familial. As some 2 to 3 percent of the white population falls below IQ seventy, the population percentage of organic retardates is at most one-half of 3 percent, or 1.5 percent of the population. Studies of the percentage of organic types of retardation in the black population are less conclusive, but they suggest that the percentage of organic retardation is at most only slightly higher than in the white population, probably about 2 percent. [21] However, based on the normal-curve statistics of the distribution of IQ in the black population, about 16 percent fall below IQ seventy. Assuming that organic retardation has a 2 percent incidence in the entire black population, then in classes for the retarded (i.e., IQ < 70) about 2%/16% = 12.5 percent of blacks would be organic as compared to about 1.5%/3% = 50 percent of whites — a white/black ratio of four to one. …

21. Nichols (

1984), reporting on the incidence of severe mental retardation (IQ < 50) in the white (N = 17,432) and black (N = 19,419) samples of the Collaborative Perinatal Project, states that at seven years of age 0.5 percent of the white sample and 0.7 percent of the black sample were diagnosed as severely retarded. However, 72 percent of the severely retarded whites showed central nervous system pathology (e.g., Down’s syndrome, posttraumatic deficit, Central Nervous System malformations, cerebral palsy, epilepsy, and sensory deficits), as compared with 54 percent of the blacks.A recent sociodemographic study by Drews et al. (

1995) of ten-year-old mentally retarded children in Metropolitan Atlanta, Georgia, reported (Table 3) that among the mildly retarded (IQ fifty to seventy) without other neurological signs the percentages of blacks and whites were 73.6 and 26.4, respectively. Among the mildly retarded with other neurological conditions, the percentages were blacks = 54.4 and whites = 45.6. For the severely retarded (IQ < 50) without neurological signs the percentages were blacks = 81.4 and whites = 18.6, respectively; for the severely retarded with other neurological conditions the percentages were blacks = 50.6 and whites = 49.4.

I would not try to explain this curious pattern, since I’m not so sure about the causes behind it. In any case, Spitz (**1988**) findings that MR scores are correlated with g-loadings and heritability is interesting in light of the B-W difference in mental retardation prevalence.

**Results**. As usual, my EXCEL file can be **accessed**. I use more generally the same data sets given in my **earlier blog post** on the Flynn effect, with one exception. I changed the Jacobs (2001) estimates. Initially I have used the MZ-DZ correlations to compute the h2, c2, and e2 values. This time, I use Jacobs own estimates reported in the same paper. This may have an impact, because when I use my earlier WISC_c2 values calculated with Falconer’s formulas, I noticed that c2 correlates negatively with MR (lower) scores although at a lesser degree than did WISC_h2 with MR (lower) scores. However, recomputing the N-weighted average c2 values using Jacobs’ Table III estimates under their best-fitting model, we see in my **EXCEL** spreadsheet that c2 is not related with MR (lower) scores anymore. Among possible reasons for this, is the fact that Jacobs’ (2001) sample is much larger (N=451) than the other samples having c2 values (e.g., Segal 1985 N=105, Owen & Sines N=42, Luo et al. 1994 N=283, LaBuda et al. 1987 N=143). Regardless, the earlier estimates based on MZ-DZ correlations could be less accurate since it is based on a smaller sample size. In any case, Jacobs h2, c2 and e2 values using either MZ-DZ correlations or Table III estimates did not correlate as much as I expected, respectively r=0.630, r=0.751, r=0.953. As usual, e2 ‘reliability’ is extremely high. I discussed this matter in my previous **post** on Jensen effects in h2, c2 and e2 vectors.

With regard to WAIS h2, c2 and e2 correlations with MR scores, there is no clear relationship between c2 and MR. Taken together, the WISC and WAIS data showed that while h2 and MR are always (highly) correlated, c2 and MR showed either a weaker and modest correlation, or a null correlation. And this holds true whether we use Pearson or Spearman correlations (see my **EXCEL** file).

The reason why I also use fluid g this time along with the usual g-loadings is because Flynn (**2000**, pp. 202-214) and Nisbett (Intelligence and How to Get It, 2009, pp. 240-241, footnote 33) have argued that fluid g and crystallized g are somewhat different and that fluid g could be more representative of g. As Must (**2003**, p. 470) explained, however, this distinction does not make any sense. Either way, we will see that fluid g and crystallized g are somewhat related. Crystallized g simply denotes the usual g-loadings among the Wechsler subtests, but Flynn named these g-loadings the crystallized g-loadings because he argues that Wechsler subtests are positively biased toward the more crystallized subtests. In other words, Wechsler’s g-loadings would be a function of subtests crystallization. What he terms fluid g is, in fact, the correlation of Wechsler subtests with Raven’s progressive matrices. Such correlations, as Jensen (1998) mentioned, could be interpreted and used as a proxy of g-loadings. Jasper Repko (**2011**), for instance, uses the same procedure in his meta-analytic test of Spearman effect in the Raven’s test.

Now, we begin with WAIS data. Below, I displayed the correlations between the relevant variables. Then I submit this correlation matrix to a principal component analysis. But before doing this, it should be noted that, just like reaction time, the sign of MR (lower) scores correlations with other variables must be reversed, so that negative signs become positive and vice versa. A negative sign, here, means that MR is correlated positively with the other variable. However, when the matrix is submitted, my SPSS output gave me the following message :

The matrix is not positive definite. This may be due to pairwise deletion of missing values.

Sadly. I don’t have any idea why it behaves like this. I encountered this problem many times. There is an interesting and old **discussion** I found recently, although it gives no clue. Anyway, the second part of the error message is ambiguous. I don’t know what it means by “missing values” and the phrase “due to pairwise deletion” is a complete nonsense because I never used pairwise in PC or PAF analyses, not even once. But most likely, given the number of times I encountered this message, it seems to appear when the matrix had a lot of very high correlations, or a lot of negative correlations. For example, when the signs of MR correlations were reversed, I noticed that e2 had only negative correlations with all other variables. First, I doubt it would be suitable for PC analysis. Second, when I removed e2 correlations from the matrix intercorrelations, PC analysis is finally working.

Now we see clearly that mental retardation, crystallized g, fluid g, h2 and BW difference form a common cluster. On the other hand, c2 bears little relationship with the other variables.

Next, I repeat the analyses using this time WISC data. Again, I came across the same problem of not positive definite matrix. And once again, e2 column vector had only (very) significant negative correlations with all other variables. I removed that one. And the PC analysis produces the following numbers.

Here again, mental retardation, crystallized g, fluid g, h2, inbreeding depression and BW difference tend to form a common cluster, while c2 had highly positive loadings on different components, namely, PC2 and PC3. It should be recalled, as evident, that the clustering of the variables depends on the pattern of their correlations.

In my **earlier post** on the Flynn effect, I used Spearman rho rather than Pearson’s, but I think now that Spearman is way too sensitive when the number of subtests is low. Also, all of the above correlations have not been corrected for attenuation using the subtest reliabilities.

**Limitations**. As mentioned in **earlier posts**, MCV and PC analysis have some shortcomings. The correlation between two variables could depend mostly on one or two subtests, which means that our observed correlations can markedly differ due to an even slight deviation in one number or subtest. The same thing could be said with regard to MGCFA. As Dolan & Hamaker (**2001**) pointed out in their measurement invariance test of the black-white IQ difference : “Dolan found that strict factorial invariance was tenable and that certain models would be rejected quite readily (including the model incorporating the strong version of Spearman’s hypothesis). However, it proved very difficult to distinguish between other competing models. These included models compatible with the weak version of Spearman’s hypothesis and models which did not include the g-factor. This failure to distinguish between models may be due in part to the fact that the WISC-R comprises only 13 tests. To detect the often subtle differences between the models, one may require a larger number of observed tests.” (p. 19). So it is not necessarily a defect of the MCV or PC/PAF analyses themselves but because in the real world we generally don’t have IQ batteries with more than 10 or 11 subtests. A more reliable test battery than what we generally have is urgently needed.

SPSS syntax :

MATRIX DATA VARIABLES=WAIS_Mretard WAIS_R_Mretard WAIS_g WAIS_gFluid WAIS_h2 WAIS_c2 WAIS_BW

/contents=corr

/N=5000.

BEGIN DATA.

1

.422 1

.265 .808 1

.442 .843 .905 1

.558 .710 .582 .683 1

-.438 .405 .315 .114 -.049 1

-.045 .305 .506 .496 .561 .090 1

END DATA.

EXECUTE.

FACTOR MATRIX=IN(COR=*)

/MISSING LISTWISE

/PRINT UNIVARIATE INITIAL CORRELATION SIG DET KMO EXTRACTION

/PLOT EIGEN

/CRITERIA MINEIGEN(1) ITERATE(25)

/EXTRACTION PC

/ROTATION NOROTATE

/METHOD=CORRELATION.

MATRIX DATA VARIABLES=WISC_Mretard WISC_R_Mretard WISC_g WISC_gFluid WISC_h2 WISC_c2 WISC_inbreeding_D WISC_BW

/contents=corr

/N=5000.

BEGIN DATA.

1

.735 1

.501 .677 1

.229 .628 .425 1

.749 .799 .487 .368 1

-.067 -.039 -.181 .057 -.215 1

.370 .574 .633 .327 .322 -.293 1

-.039 .316 .650 .187 .277 -.139 .470 1

END DATA.

EXECUTE.

FACTOR MATRIX=IN(COR=*)

/MISSING LISTWISE

/PRINT UNIVARIATE INITIAL CORRELATION SIG DET KMO EXTRACTION

/PLOT EIGEN

/CRITERIA MINEIGEN(1) ITERATE(25)

/EXTRACTION PC

/ROTATION NOROTATE

/METHOD=CORRELATION.

**Addendum**. I am adding below the correlation matrices displayed in the above tables and to which I add Kees-Jan Kan (**2011**) data on the so-called cultural_load. WISC_c2 does not correlate with either g_loading or cultural_loading. WAIS_c2, on the other hand, correlates somewhat with these two variables, but still a PC analysis would show that WISC_c2 and WAIS_c2 have a relatively high loading on PC2, while the other variables were clustered in PC1.

MATRIX DATA VARIABLES=WAIS_Mretard WAIS_R_Mretard WAIS_g WAIS_gFluid WAIS_h2 WAIS_c2 WAIS_BW Culture_load

/contents=corr

/N=5000.

BEGIN DATA.

1

.422 1

.265 .808 1

.442 .843 .905 1

.558 .710 .582 .683 1

-.438 .405 .315 .114 -.049 1

-.045 .305 .506 .496 .561 .090 1

.251 .753 .871 .730 .617 .449 .420 1

END DATA.

EXECUTE.

FACTOR MATRIX=IN(COR=*)

/MISSING LISTWISE

/PRINT UNIVARIATE INITIAL CORRELATION SIG DET KMO EXTRACTION

/PLOT EIGEN

/CRITERIA MINEIGEN(1) ITERATE(25)

/EXTRACTION PC

/ROTATION NOROTATE

/METHOD=CORRELATION.

MATRIX DATA VARIABLES=WISC_Mretard WISC_R_Mretard WISC_g WISC_gFluid WISC_h2 WISC_c2 WISC_inbreeding_D WISC_BW Culture_load

/contents=corr

/N=5000.

BEGIN DATA.

1

.735 1

.501 .677 1

.229 .628 .425 1

.749 .799 .487 .368 1

-.067 -.039 -.181 .057 -.215 1

.370 .574 .633 .327 .322 -.293 1

-.039 .316 .650 .187 .277 -.139 .470 1

.642 .596 .811 .230 .547 .033 .655 .447 1

END DATA.

EXECUTE.

/MISSING LISTWISE

/PRINT UNIVARIATE INITIAL CORRELATION SIG DET KMO EXTRACTION

/PLOT EIGEN

/CRITERIA MINEIGEN(1) ITERATE(25)

/EXTRACTION PC

/ROTATION NOROTATE

/METHOD=CORRELATION.