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Cited by 99Psychometric Methodologies and TestingCognitive Abilities and TestingEducation, Achievement, and Giftedness

Investigating Spearman's Hypothesis by Means of Multi-Group Confirmatory Factor Analysis

Open access
Dolan, Conor V.·2000·Multivariate Behavioral Research
DOI 10.1207/s15327906mbr3501_2
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Abstract

Differences between blacks and whites on cognitive ability tests have been attributed to a fundamental difference between these groups in general intelligence (or g, as it is denoted). The hypothesized difference in g gives rise to Spearman's hypothesis, which states that the differences in the means of the tests are related to the tests' factor loadings on g. Jensen has investigated this hypothesis by correlating differences in means and tests' g loadings. The aim of the present article is to investigate B-W differences using multi-group confirmatory factor analysis. The advantages of multi-group confirmatory factor analysis over Jensen's test of Spearman's hypothesis are discussed. A published data set is analyzed. Strict factorial invariance is tested and judged to be tenable. Various models are tested, which do and do not incorporate g. It is observed that it is difficult to distinguish between several hypotheses, including and excluding g, concerning group differences. The inability to distinguish between competing models using multi-group confirmatory factor analysis makes it difficult to draw clear conclusions about the exact nature of black-white differences in cognitive abilities. The implications of the results for Jensen's test of Spearman's hypothesis are discussed.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper applies multi-group confirmatory factor analysis (MGCFA) to Jensen and Reynolds’s (1982) WISC-R data to test Spearman’s hypothesis that black–white differences in cognitive test scores are mainly due to differences in g. It finds strict factorial invariance tenable but concludes that models with and without g cannot be clearly distinguished, so Jensen’s Spearman correlations cannot safely be interpreted as support for the central role of g.

Key findings

  • Strict factorial invariance was judged tenable; model A4 provided an acceptable fit, with χ² = 327.7, df = 148, χ²/df = 2.21, RMSEA = .033, NNFI = .966.
  • The strong version of Spearman’s hypothesis (model B2) did not fit well (χ² = 392.7, df = 155, χ²/df = 2.53), while several models including and excluding g fit comparatively well; models A4, B4, and C5 were identified as best descriptions, and a preference among them was hard to justify.
  • In model B4, the percentage of black–white mean differences attributable to g ranged from 49% (OA) to 100% (I, A, V, DS, TS), with 92% for S, 93% for C, 62% for PC, 67% for PA, 56% for BD, 76% for CO, and 58% for MA; the weak version of Spearman’s hypothesis was judged questionable for five variables (PC, PA, BD, OA, MA).
  • Jensen’s test yielded correlations of .75 and .64 for this data set, but because models without g also fit well, the paper concludes these correlations cannot safely be interpreted as support for g’s central role in black–white differences.

Methods

  • MGCFA models were specified as first-order, second-order, and alternative first-order models, with nested comparisons and constraints for strict factorial invariance; analyses used LISREL 8.20 with normal maximum likelihood estimation and fit indices including χ², χ²/df, CAIC, AIC, ECVI, RMSEA, and NNFI.
  • Data were from Jensen and Reynolds (1982): 1,868 whites and 305 blacks aged 6 to 16.6 years, with 13 WISC-R subscales (Information, Similarities, Arithmetic, Vocabulary, Comprehension, Digit Span, Tapping Span, Picture Completion, Picture Arrangement, Block Design, Object Assembly, Coding, Mazes); age effects were removed by age-standardization.
  • Exploratory factor analyses using CEFA 1.01 with quartimax oblique rotation expected three correlated first-order factors—verbal (VERB), performance (PERF), and memory span (MEM)—and the confirmatory loading pattern was based on the union of significant loadings in the white and black samples.

Limitations

  • The paper states that it was difficult to distinguish between competing models because differences between models were essentially small, unequal sample sizes adversely affect power in tests of factorial invariance, and the common factor space had low dimensionality; better discrimination would require more varied and numerous indicators and a larger sample more equally distributed across groups.
  • The paper notes that specific variance terms may include systematic specific effects rather than pure error; squared multiple correlations ranged from 22% (CO) to 70% (V), meaning much variance was not accounted for by common factors, and such specific effects could contribute to means differently across groups, requiring repeated measures to identify.
  • The paper reports that the Comprehension subtest may be biased with respect to group: it had the largest modification indices for its intercept (10.8, 12.52, and 12.18 in models A4, B4, and C5), and its PERF loading was not significant in whites (.040, SE .025) but was significant in blacks (.138, SE .065).

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1207/s15327906mbr3501_2

Cites

58 of 58 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 20 cited sources checked · 0 supported · 0 flagged

  • Findingown data

    The standardized residuals and modification indices suggest that models A4, B4, and C5 fit the observed means well.

    “The standardized residuals and MI's suggest that the three models fit the observed means well.”
    Quote in the paperSubstantial claim
  • Findingown data

    Model C2, in which the PERF factor is the only source of B-W differences, fits the data better than model B2.

    “It is notable that model C2, in which the PERF factor is the only source of B-W differences, fits the data better than model B2 ( 2 of 363.7 vs. 392.7,both models have 155 df).”
    Quote in the paperSubstantial claim
  • Findingown data

    Model C6 does not fit the data.

    “Model C6 (MEM & VERB) does not fit the data ( 2 = 420.8,df = 152)”
    • Also stated in: § Analysis Based on Multi-Group Confirmatory Factor Models 3 · plain
      “On the basis of the results in Table 4, we conclude that models C1, C3, and C6 do not fit the data.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    Model A4 provides an acceptable fit to the Jensen and Reynolds (1982) data.

    “On the basis of these, we conclude that model A4 provides an acceptable fit.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The specific scores are normally distributed with a diagonal covariance matrix.
    • The authors state this as an assumption: The common factor scores are normally distributed.
  • Findingnone given

    The sample size of 86 within each group in Naglieri and Jensen (1987) affords insufficient power to detect the smaller effects.

    “The sample size of 86 within each group affords insufficient power to detect the smaller effects.”
    Quote in the paperNot a substantial claim
  • Classified as probably not a substantial claim. 14
  • May combine two claims. 13

  • Classified as probably not a substantial claim.

    Findingown data

    Model A2 has χ2 = 281.2 (df = 125).

    “we observe a 2 of 281.2 (df = 125; Model A2)”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Following rotation, the factor loading of C on the PERF factor is not significant in the white sample (.040, s.e.: .025), but it is significant in the black sample (.138, s.e.: .065).

    “Following rotation, the factor loading of C on the PERF factor is not significant in the white sample (.040, s.e.: .025),but it is significant in the black sample (.138, s.e.: .065).”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data

    Model B1 has χ2 = 310.7 (df = 143).

    “The 2 for this model is 310.7 (df = 143).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    In the white sample, standardized residuals range from -3.56 to 3.26 (median -.43); in the black sample, they range from -2.77 to 2.66 (median .07).

    “In the white sample the standardized residuals, which can be viewed as z-scores, range from -3.56 to 3.26 (median -.43); in the black sample, they range from -2.77 to 2.66 (median .07).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    Neither RMSEA nor NNFI discriminate well; they indicate that models A1 to A4 fit well.

    “Neither RMSEA and NNFI discriminate well; they indicate that models A1 to A4 fit well.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The differences in chi-square of models A4, B4, and C5 from model A3 are 22.0 (df=10), 31.7 (df=15), and 29.0 (df=14), respectively.

    “The differences in 2 of these models and model A3, are 22.0 (A4; df = 10; 2 /df = 2.20), 31.7 (B4, df = 15; 2 /df = 2.11), and 29.0 (C5; df = 14; 2 /df = 2.07).”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data

    Model A4 has χ2 = 327.7 (df = 148).

    “The 2 of this model equals 327.7 (df = 148).”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data

    Model A3 has χ2 = 305.7 (df = 138).

    “we observe a 2 of 305.7 (df = 138; Model A3)”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingown data

    The AIC is slightly better for model A3 than for A4, but the difference is very small.

    “The AIC is slightly better for model A3, but the difference is very small (438.6 vs. 439.8).”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The constraints that Ψw* = Ψb* and Γw = Γb are reasonable.

    “The difference in 2 equals 5.0 (df = 5), so we conclude that the constraints that ⌿ w * = ⌿ b * and ⌫ w = ⌫ b are reasonable.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The specific scores are normally distributed with a diagonal covariance matrix.
    • The authors state this as an assumption: The common factor scores are normally distributed.

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

    Study design and conductnone given

    The choice to equate the observed means in the white group with the intercept vector is arbitrary; equating the observed means in the black group instead does not substantively change the results.

    “The choice to equate the observed means in the white group with is arbitrary; equating and the observed means in the black group does not substantively change the results.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The first order factor model holds in the white population.
    • The authors state this as an assumption: The black population differs systematically from the white population with respect to some of the common factors.
    • The authors state this as an assumption: Differences between the two groups are modeled through a process of selection on the basis of factor scores.

  • May combine two claims.

    Study design and conductnone given

    The decision to model the white group data using Equations 7 and 8 and the black group data using Equations 11 and 12 is arbitrary; reversing the groups does not substantively change the results.

    “The decision to model the data of the white group using Equations 7 and 8, and the data of the black group using Equations 11 and 12 is arbitrary.Modeling the data of the whites using Equations 11 and 12, and the data of the blacks using Equations 7 and 8 (with the appropriate substitution of subscripts w and b), does not substantively change the results.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The first order factor model holds in the white population.
    • The authors state this as an assumption: The black population differs systematically from the white population with respect to some of the common factors.
    • The authors state this as an assumption: Differences between the two groups are modeled through a process of selection on the basis of factor scores.

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationreasoning

    The use of goodness of fit measures and explicit comparison of competing models provide a more transparent justification than a Spearman correlation for accepting or rejecting the hypothesis that g is fundamental to black-white differences.

    “The use of goodness of fit measures and the explicit comparison of competing models provide a more transparent justification than a Spearman correlation for the acceptance or rejection of the hypothesis that g is fundamental to B-W differences.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    Given the large total sample size, these likelihood ratios are not judged to be particularly large.

    “Given the large total sample size, these likelihood ratios are not judged to be particularly large.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Premisereasoning

    Decisions made on the basis of MGCFA concerning the nature of black-white differences are based on fit indices.

    “Decisions made on the basis of MGCFA concerning the nature of B-W differences are based on fit indices.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    MGCFA may be extended to several groups, including sub-groups of blacks and whites and other groups such as Asian-Americans.

    “MGCFA may be extended to several groups.For instance, one can include sub-groups of blacks and whites (e.g., urban and rural, high and low SES), as well as other groups, such as Asian-Americans.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    Although a number of these models are compatible with the weak version of Spearman's hypothesis (notably model B4), the correlations of .75 and .65 would appear to be possible with models that do not include g (model C5, A4).

    “Although a number of these models are compatible with the weak version of Spearman's hypothesis (notably model B4), the correlations of .75 and .65 would appear to be possible with models that do not include g (model C5, A4).”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationcitation

    Models for repeatedly measured groups can be tested, which allows one to investigate black-white differences in cognitive development.

    “In addition, models for repeatedly measured groups can tested (e.g., Jöreskog, 1979;Marsh & Grayson, 1994, Hanna & Lei, 1985), which allows one to investigate black-white (say) differences in cognitive development.”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites Jöreskog, 1979
    • cites Marsh & Grayson, 1994 source
    • cites Hanna & Lei, 1985 source

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    This large correlation was also observed in similar analysis of the Naglieri and Jensen (1987) data set.

    “This large correlation was also observed in similar analysis of the Naglieri and Jensen (1987) data set.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Naglieri and Jensen (1987) source

  • May combine two claims.

    Interpretationcitation

    The hypothesis of SFI is tested explicitly in MGCFA, whereas in Jensen's test it is not tested directly, rather some aspects are investigated piecemeal.

    “The hypothesis of SFI (Meredith, 1993), which we view as crucial to the meaningful interpretation of group differences within the factor model, is tested explicitly in MGCFA.In Jensen's test, this hypothesis is not tested directly, rather, some (but not all) aspects of SFI are investigated piecemeal in the prescribed procedure of Jensen's test (Jensen, 1992).”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites Meredith, 1993 source
    • cites Jensen, 1992 source

  • Classified as probably not a substantial claim.

    Interpretationnone given

    MGCFA allows one to fit a variety of competing models, which may or may not include g.

    “Second, MGCFA allows one to fit a variety of competing models, which may, or may not include g.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable). May combine two claims.

    Interpretationreasoning

    MGCFA allows one to investigate Spearman's hypothesis in an integrative, more elegant manner: the various steps involved in Jensen's procedure are included in a single model.

    “First, MGCFA allows one to investigate Spearman's hypothesis in an integrative, more elegant manner: the various steps involved in Jensen's procedure are included in a single model.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    If SFI is found to be tenable, this provides support for the hypothesis that within and between group differences are attributable to common causes.

    “However, if SFI is found to be tenable, we propose that this does provide support the hypothesis that within and between group differences are attributable to common causes.”
    Quote in the paperSubstantial claim
23 of 23

  • The paper relies on this without support.

    The single common factor model does not fit the covariance structure of multidimensional cognitive ability tests such as the WISC.

    “A problem with the single common factor model, however, is that it does not fit the covariance structure of multidimensional cognitive ability tests (e.g., the WISC).”

  • The authors state this as an assumption.

    The specific scores are normally distributed with a diagonal covariance matrix.

    “We assume that ε ε ε ε ε ij ~ N p (0, ⌰ i )”

  • The authors state this as an assumption.

    The common factor scores are normally distributed.

    “␩ ij ~ N q (␣ i ,⌿ i )”

  • The authors state this as an assumption.

    The first order factor model holds in the white population.

    “We assume that the first order factor model holds, as shown in Equations 7 and 8, in (say) the white population.”

  • The authors state this as an assumption.

    The black population differs systematically from the white population with respect to some of the common factors.

    “The black population is supposed to differ systematically from the white population with respect to some of the common factors.”

  • The authors state this as an assumption.

    Differences between the two groups are modeled through a process of selection on the basis of factor scores.

    “Differences between the two groups are modeled through a undefined process of selection on the basis of the factor scores.”

  • The authors state this as an assumption.

    The specific variance terms (components of the vector ε) do not contribute significantly to the means.

    “Here they are, because we have assumed that the specific variance terms, components of the vector ε ε ε ε ε, do not contribute significantly to the means.”

  • The paper relies on this without support.

    The weak version of Spearman's hypothesis is ill-defined.

    “This judgement is necessarily subjective, because the weak version of Spearman's hypothesis is ill-defined.”
8 of 8

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Dolan's finding that small samples in some cases yield unreliable outcomes comes as no surprise.

    “Dolan's [25] finding that small samples in some cases yield unreliable outcomes comes as no surprise [26].”

    From Spearman’s Hypothesis Tested on Black Adults: A Meta-Analysis (2016) · cites this paper as [25]

    The Discussion (page unavailable) reports difficulty distinguishing between models, says unequal group sample sizes can reduce statistical power, and calls for a larger, more evenly distributed sample. This supports a sample-size-related limitation in this analysis, but not the broader claim that Dolan found small samples generally yield unreliable outcomes. (LLM, conf=0.83)

    Experimental — accuracy not yet measured

  • All data in this study are only available at the group level, so it is not possible to use MGCFA to analyze these data.

    “As noted above, Dolan [25] suggests using MGCFA to test Spearman's hypothesis, but since all data in this study are only available at the group level, it simply is not possible to use MGCFA to analyze these data.”

    From Spearman’s Hypothesis Tested on Black Adults: A Meta-Analysis (2016) · cites this paper as [25]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Black-sample g correlation does not match the reported covariances

    Minor

    For model B4, the black-sample covariance between VERB and g and the variance of g are both 5.11. A correlation of .95 with VERB variance 5.57 would imply a g variance of .952 × 5.57 = 5.026925, not 5.11; the tabulated values imply a correlation of about .958.

    Reported 5.11 · recomputed 5.026925

  • Unconstrained model A1 uses the white specific-variance matrix for both groups

    Minor

    The displayed equation assigns ⌰ w to the black-group covariance as well as the white-group covariance, imposing an equality constraint despite the stated absence of any cross-group equality constraints in this model.

  • Factor covariance matrix is given the wrong dimensions

    Minor

    The factor scores are q-dimensional, so their covariance matrix must be q × q, not p × q as stated.

  • The reported Spearman correlation changes from .64 to .65

    Minor

    The paper reports .64 for the correlation based on black-sample loadings, but refers to the same pair of correlations later as .75 and .65.

4 of 4

Cites

Flagged first, then by citations

Research works printed but not linked: 5

  1. Horn J., 1997, Cahiers de Psychologie Cognitive, 16, 721
  2. Millsap R. E., 1997, Cahiers de Psychologie Cognitive, 16, 750
  3. Schonemann P. H., 1997, Cahiers de Psychologie Cognitive (Current Psychology of Cognition), 16, 665
  4. Schonemann P. H., 1997, Cahiers de Psychologie Cognitive (Current Psychology of Cognition), 16, 788
  5. Sorbom D., 1975, British Journal of Mathematical and Statistical Psychology, 27, 229, 10.1111/j.2044-8317.1974.tb00543.x · DOI

Cited by

Sorted by citations

Related

Authors
Dolan, C. V.
Affiliations
University of Amsterdam (NL)
Corresponding author
Dolan, Conor V.
ORCID
Dolan, Conor V.
Venue
Multivariate Behavioral Research
Publisher
Informa UK Limited
Published
2000-01
Volume
35
Issue
1
Pages
21-50
ISSN
  • 0027-3171 (ISSN-L)
  • 1532-7906
DOI
10.1207/s15327906mbr3501_2
OpenAlex
W2052275277
Cited by
99
References
57
Citations by year
2026: 3 · 2025: 4 · 2023: 4 · 2022: 1 · 2021: 2 · 2020: 2 · 2019: 8 · 2018: 3 · 2017: 4 · 2016: 7 · 2015: 3 · 2014: 5 · 2013: 1 · 2012: 3
FWCI
4.23
Type
journal article
Language
en
Open access
Locations
  • Multivariate Behavioral Research (publishedVersion)
  • PubMed (publishedVersion)
Topics
  • Social Sciences › Decision Sciences › Management Science and Operations Research › Psychometric Methodologies and Testing
  • Social Sciences › Psychology › Experimental and Cognitive Psychology › Cognitive Abilities and Testing
  • Social Sciences › Psychology › Experimental and Cognitive Psychology › Education, Achievement, and Giftedness
Keywords
Spearman's hypothesis, general intelligence, black-white disparities, cognitive ability tests, multi-group confirmatory factor analysis, measurement invariance, factor loadings

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