# Spearman’s Hypothesis Tested on Black Adults: A Meta-Analysis

^{*}

## Abstract

**:**

## 1. Introduction

- The samples should not be selected on any highly g-loaded criteria.
- The variables should have reliable variation in their g loadings.
- The variables should measure the same latent traits in all groups. The congruence coefficient of the factor structure should have a value of >0.85.
- The variables should measure the same g in the different groups; the congruence coefficient of the g values should be >0.95.
- The g loadings of the variables should be determined separately in each group. If the congruence coefficient indicates a high degree of similarity, the g loadings of the different groups should be averaged.
- To rule out the possibility that the correlation between the vector of g loadings (V
_{g}) and the vector of mean differences between the groups or effect sizes (V_{ES}) is strongly influenced by the variables’ differing reliability coefficients, V_{g}and V_{ES}should be corrected for attenuation by dividing each value by the square root of its reliability. - The test of Spearman’s hypothesis is the Pearson correlation (r) between V
_{g}and V_{ES}. To test the statistical significance of r, Spearman’s rank order correlation (r_{s}) should be computed and tested for significance.

## 2. Method

#### 2.1. Meta-Analysis

#### 2.2. Inclusion Criteria

#### 2.3. Searching and Screening Studies

#### 2.4. Description of Available Data

#### 2.5. Method of Correlated Vectors

#### 2.6. Calculating d

#### 2.7. Choice of SD Used in Calculating the Difference Scores (d)

#### 2.8. Selecting g Loading for Calculating r (d × g)

#### 2.9. Correcting for Unequal Group Sizes in a Datapoint

_{i}is the size of each individual group [49]. The advantage of this formula is that, for a datapoint with samples of 100 and 900, the value of the harmonic N = 180, which is quite close to the value of the smallest sample, indicating a quite strong sampling error (see Table 2). However, the disadvantage of this formula is that for a datapoint with samples of 15 and 15, the total sample size is only 15 and that, for a datapoint with samples of 500 and 500, the total sample size is only 500 (see Table 2).

_{i}is the size of each individual group. For a datapoint with samples of 100 and 900, the value of the harmonic N then becomes 360, which is quite conservative, but not as strict as the value of only 180 for the first formula (see Table 2). For data points with samples of 15 and 15, the total sample size now becomes 30, and for a datapoint with samples of 500 and 500 the total sample size now becomes 1000 (see Table 2), which is in line with the reasoning in Hunter and Schmidt [39] mentioned above. We therefore continue to use this formula, which is based on sound reasoning, namely that data points consisting of samples with widely differing Ns receive a substantially reduced weight in a meta-analysis, and that data points based on samples with highly comparable weights receive a weight based on the total number of research participants in these samples.

## 3. Results

_{r}). The last column presents the percentage of variance explained by sampling error (%VE). The analysis of all 15 data points yields a mean vector correlation of 0.57, with 0.6% of the variance in the observed correlations explained by sampling error. This percentage is very low and suggests the presence of a strong moderator or several moderators.

## 4. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Original Publication | Previously Used to Test Spearman’s Hypothesis | Data Availability | Sample Background |
---|---|---|---|

Carretta [41] | No | Group Level Only | Air Force applicants |

Murray [42] | No | Group Level Only | Varied (Nationally representative samples) |

Kaufman et al. [43] | No | Group Level Only | Varied (Stratified sample) |

Sternberg [44] | No | Group Level Only | Mostly college students with some high school students |

Department of Defense [45] ^{1} | Yes; In: Jensen [37] | Group Level Only | Varied (Representative sample) |

Department of Labor [46] ^{1} | Yes; In: Jensen [37] | Group Level Only | Varied (33 different occupational samples) |

Hennessy and Merrifield [47] ^{1} | Yes; In: Jensen [37] | Group Level Only | High School seniors |

National Longitudinal Study ^{1,2} | Yes; In: Jensen [37] | Group Level Only | Varied (Stratified sample) |

Nyborg and Jensen [38] | Yes | Group Level Only | Males in the Armed Forces |

Veroff et al. [48] ^{1} | Yes; In: Jensen [37] | Group Level Only | Cross section of population Detroit |

**Table 2.**Various Values for the Harmonic N of Data Points with Two Samples Using Two Different Formulas.

Size of Group 1 (x1) | Size of Group 2 (x2) | Formula 1 $\frac{\mathit{N}}{\frac{\mathbf{1}}{\mathit{x}\mathbf{1}}\mathbf{+}\frac{\mathbf{1}}{\mathit{x}\mathbf{2}}\mathbf{+}\mathbf{\xb7}\mathbf{\xb7}\mathbf{\xb7}\frac{\mathbf{1}}{\mathit{x}\mathit{n}}}$ | Formula 2 $\frac{\mathit{N}\mathbf{\xb7}\mathit{N}}{\frac{\mathbf{1}}{\mathit{x}\mathbf{1}}\mathbf{+}\frac{\mathbf{1}}{\mathit{x}\mathbf{2}}+\mathbf{\xb7}\mathbf{\xb7}\mathbf{\xb7}\frac{\mathbf{1}}{\mathit{x}\mathit{n}}}$ |
---|---|---|---|

15 | 15 | 15 | 30 |

500 | 500 | 500 | 1000 |

100 | 900 | 180 | 360 |

Study | Test | r (d × g) | N_{subtests} | N_{Whites} | N_{Blacks} | N_{harmonic} | Mean Age ^{1} (Range) |
---|---|---|---|---|---|---|---|

Carretta [41] | AFOQT | 0.56 | 16 | 212,238 | 12,647 | 47,743 | 21 (18–27) |

Murray [42] | WJ-I | 0.35 | 6 | 3329 | 436 | 1542 | (6–65) |

WJ-II | 0.50 | 7 | 3573 | 807 | 2633 | (6–65) | |

WJ-III | 0.72 | 7 | 2592 | 426 | 1463 | (6–65) | |

Kaufman et al. [43] | WAIS-R | 0.59 | 11 | 344 | 50 | 175 | (16–19) |

0.67 | 11 | 440 | 50 | 180 | (20–34) | ||

0.64 | 11 | 443 | 51 | 183 | (35–54) | ||

0.48 | 11 | 437 | 41 | 150 | (55–74) | ||

Sternberg [44] | Various | 0.46 | 11 | 348 | 47 | 83 | (18–22) ^{1} |

Department of Defense [45] ^{3} | ASVAB | 0.30 | 10 | 5533 | 2298 | 6495 | 19.5 ^{2} (16–23) |

Department of Labor [46] ^{3} | GATB Aptitudes | 0.71 | 8 | 4001 | 2416 | 6025 | 40 (16–70) |

Hennessy and Merrifield [47] ^{3} | CGP | 0.66 | 10 | 1818 | 431 | 1394 | 18 (17–19) |

National Longitudinal Study ^{3,4} | CGP, SAT, ACT | 0.78 | 12 | 12,275 | 1938 | 6695 | 18 (16–23) |

Nyborg and Jensen [38] | Various | 0.81 | 16 | 3535 | 502 | 1758 | 19.9 (17–25) |

Veroff et al. [48] ^{3} | Various | 0.46 | 6 | 179 | 186 | 365 | (18–49) |

_{harmonic}is computed using the formula $\frac{4}{\frac{1}{n1}+\frac{1}{n2}}$ where n1 and n2 are the amount of participants in group n1 and n2, respectively.

^{1}Mean age not known for all groups;

**Estimated;**

^{2}^{3}These studies were taken from Jensen [37];

^{4}Reference not given in Jensen [37]. AFOQT: Air Force Officer Qualifying Test; WJ-I/II/III: Woodcock-Johnson I/II/III; WAIS-R: Wechsler Adult Intelligence Scale—Revised; ASVAB: Armed Services Vocational Aptitude Battery; GATB: General Aptitude Test Battery; CGP: Comparative Guidance and Placement Program’s test battery; SAT: Scholastic Aptitude Test; ACT: American College Testing.

**Table 4.**Exploratory Bare Bones Meta-analytical Results for Correlations between g Loadings and Adult Black/White Differences.

K | N_{h} | Mean r | SD_{r} | %VE |
---|---|---|---|---|

15 | 76,884 | 0.57 | 0.12 | 0.6 |

_{r}= standard deviation of observed correlation; %VE = percentage of variance accounted for by sampling errors.

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**MDPI and ACS Style**

Te Nijenhuis, J.; Van den Hoek, M.
Spearman’s Hypothesis Tested on Black Adults: A Meta-Analysis. *J. Intell.* **2016**, *4*, 6.
https://doi.org/10.3390/jintelligence4020006

**AMA Style**

Te Nijenhuis J, Van den Hoek M.
Spearman’s Hypothesis Tested on Black Adults: A Meta-Analysis. *Journal of Intelligence*. 2016; 4(2):6.
https://doi.org/10.3390/jintelligence4020006

**Chicago/Turabian Style**

Te Nijenhuis, Jan, and Michael Van den Hoek.
2016. "Spearman’s Hypothesis Tested on Black Adults: A Meta-Analysis" *Journal of Intelligence* 4, no. 2: 6.
https://doi.org/10.3390/jintelligence4020006