# A Meta-Analysis of Spearman’s Hypothesis Tested on Latin-American Hispanics, Including a New Way to Correct for Imperfectly Measuring the Construct of g

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Meta-Analysis

#### 2.2. Rules for Inclusion

#### 2.3. Searching and Screening Studies

#### 2.4. g Loadings

#### 2.5. Calculating Glass’ d

#### 2.6. Studies Supplying Multiple Effect Sizes

#### 2.7. Correcting for Sampling Error, Reliability of the g Vector, and Reliability of the Glass’ d Vector

#### 2.8. Correction for Restriction of Range in g Loadings

_{study}/SD

_{ref}. To give an example, suppose the observed correlation is r = 0.50, the SD

_{study}= 0.150, and the SD

_{ref}= 0.192, then the value of u is 0.150/0.192 = 0.78, which results in a correction factor of 1.28, which yields a corrected correlation with a value of rho = 0.64.

#### 2.9. Correction for Deviation from Perfect Construct Validity

^{2}

_{sg}/(1 − r

^{2}

_{sg})]

^{−1})}

^{−0.5}

^{2}

_{sg}= each subtest’s squared g loading. The formula implies that longer test batteries in general are more g-loaded than shorter test batteries, with g-loadedness being an asymptotic function of the number of subtests. Using this formula resulted in a g loading of 0.92–0.95 for the various Wechsler Full Scale scores based on 10–12 subtests. It may be that having about fifteen subtests from one or more test batteries gives one a total score with perfect g-loadedness. The next step is to argue that when using datasets with many cognitive tests the larger the collection of subtests becomes, the more the resulting g score approaches Jensen’s concept of “true” g. The final step is to compute a g score based on, for instance, six subtests from a large collection of cognitive tests and to correlate this g score with a g score based on, say, 25 subtests yielding an estimate of the correlation of the sum score based on six subtests with “true” g. Various combinations of six subtests from a larger collection are possible and their correlations with g based on a large number of subtests yield an estimate of the distribution of the value necessary for the correction for imperfectly measuring the construct of g when using a battery consisting of six cognitive tests.

#### 2.9.1. Research Participants

#### 2.9.2. Psychometric Variables

- Closure; the child is given very incomplete pictures and has to figure out the complete picture. According to Carroll’s [47] taxonomy, this subtest is a measure of Closure Speed at stratum I, which makes this subtest a measure of Broad Visual Perception at stratum II.
- Exclusion; out of four abstract figures the child has to select the one that is different from the other three. The child has to detect the necessary rule to solve the task. This subtest measures Induction at stratum I, which makes it a measure of Fluid Intelligence at stratum II.
- Memory Span; the child has to memorize figures put on cards and the sequence in which they are presented. After five seconds the card is turned and the child has to reproduce the figures in the right sequence using blocks on which the figures are printed. The subtest contains a series with concrete figures and a series with abstract figures. Both series measure (Visual) Memory Span at stratum I. Both series fall under General Memory and Learning at stratum II.
- Verbal Meaning; words are presented to the child in an auditory fashion and from four figures the child has to choose the one which resembles the word it has just heard. This subtest measures Lexical Knowledge at stratum I and is a measure of Crystallized Intelligence at stratum II.
- Mazes; the child has to go through a maze with a stick as fast as they can. Because of the speed factor this subtest is a measure of Spatial Scanning at stratum I, which falls under Broad Visual Perception at stratum II.
- Analogies; the child has to complete verbal analogies that are stated as follows: A:B is like C: … (there are four options to choose from). The constructors of this subtest tried to avoid measuring Lexical Knowledge, by including only those words that are highly frequently used in ordinary life. All words in the analogy items are accompanied by illustrations, so as to reduce the verbal aspect of the task to a minimum. This subtest is a measure of Induction at stratum I which makes it a measure of Fluid Intelligence at stratum II.
- Quantity; in this multiple-choice test the child has to make comparisons between pictures, differing in volume, length, weight, and surface. This subtest is a measure of Quantitative Reasoning at stratum I, which measures Fluid Intelligence at stratum II.
- Disks; the child has to use pins to put disks with two, three, or four holes on a board as fast as possible until three layers of disks are on the board. This subtest is a measure of Spatial Relations at stratum I, which measures Broad Visual Perception at stratum II.
- Learning Names; the child has to memorize the names of different butterflies and cats using pictures presented on cardboard. This subtest measures Associative Memory at stratum I, which makes it a measure of General Memory and Learning at stratum II.
- Hidden Figures; the child has to discover which of six figures is hidden in a complex drawing. This subtest is a measure of Flexibility of Closure at stratum I, which makes it a measure of Broad Visual Perception at stratum II.
- Idea Production; the child has to name as many words, objects, or situations as possible that can be associated with a broad category within a certain time span, for example: “What can you eat?” This subtest is a measure of Ideational Fluency at stratum I, which is a measure of Broad Retrieval Ability at stratum II.
- Storytelling; the child has to tell as much as possible about a picture on a board and what could happen to the persons or objects in the picture. The total score is composed of both quantitative measures (number of words, number of relations, did or did not the child tell a plot, etc.) and qualitative measures (did the child grasp the central meaning of the story). This subtest consists of different elements and measures at stratum I: Naming Facility and Ideational Fluency, Sequential Reasoning, and to some extent Communication Ability. These stratum-I abilities are measures of Broad Retrieval Ability, Fluid Intelligence, and Crystallized Intelligence, respectively, at stratum II.

- Information; the child has to verbally answer all kinds of general questions, some of which have several possible correct answers. This subtest measures General Information, which is a measure of Crystallized Intelligence at stratum II.
- Picture Completion; the child has to find out which essential part of a picture is missing, within a given time. This subtest measures Closure Speed at stratum I, which makes it a measure of Broad Visual Perception at stratum II.
- Similarities; the child has to find a similarity between two objects or concepts. There are several correct answers. This subtest is a measure of Induction at stratum I, which makes it a measure of Fluid Intelligence at stratum II.
- Picture Arrangement; the child has to order a series of pictures in such a way that the pictures form a comprehensive story within a given time. This subtest is a measure of General Sequential Reasoning at stratum I, which makes it a measure of Fluid Intelligence at stratum II.
- Arithmetic; the child has to solve arithmetic problems. These arithmetic problems are verbally presented: Four boys have 72 fish. They divided the fish, and everybody gets the same amount. How many fish does each boy get? This subtest is a measure of Crystallized Intelligence at stratum II.
- Block Design; using blocks, the child has to replicate a pattern presented on a card. This subtest is a measure of Visualization at stratum I, which measures Broad Visual Perception at stratum II.
- Vocabulary; the child has to give the meaning of a presented word. This subtest measures Lexical Knowledge at stratum I, which makes it a measure of Crystallized Intelligence at stratum II.
- Object Assembly; the child has to put different pieces of cardboard together to copy a given figure within a given time. This subtest is a measure of Visualization at stratum I, which makes it a measure of Broad Visual Perception at stratum II.
- Comprehension; the child has to answer different questions in which they have to give their insight and judgment about everyday-life issues. This subtest measures General Knowledge, which is a measure of Crystallized Intelligence at stratum II and is a measure of General Sequential Reasoning at stratum I, which is a measure of Fluid Intelligence at stratum II.
- Coding; the child has to put a sign in a series of figures (code A) or under a series of numbers (code B). The sign belonging to the figure of number that was presented to the child earlier. This subtest is a measure of Visual Memory as stratum I, which falls under General Memory and Learning at stratum II.
- Digit Span; the child has to repeat a series of numbers in the sequence presented to them auditorily (Forward Digit Span) or in reverse order starting with the last number they heard back to the first number (Backward Digit Span). This subtest is a measure of Memory Span at stratum I, which makes it a measure of General Memory and Learning at stratum II.
- Mazes; the child has to trace the way out of a maze presented on paper with a pencil within a given time. The child is not allowed to enter a dead end. This subtest is a measure of Spatial Scanning at stratum I, which falls under Broad Visual Perception at stratum II.

#### 2.9.3. g Loadings

#### 2.9.4. Computation of g Scores and “True” g Scores

#### 2.9.5. Combinations of Subtests

#### 2.9.6. Correlations of g Scores and “True” g Scores

#### 2.9.7. Scatter Plot

#### 2.9.8. Computation of the Correction Value

## 3. Results

_{harmonic}for each comparison, the ninth column reports the N on which the computation of g loadings was based, the tenth column reports the reliability value of the g vector, the eleventh column reports the reliability of the Glass’ d vector, the twelfth column reports the amount of restriction of range, expressed in u, and the thirteenth column reports the age of the subjects. It is clear that with one exception all the correlations are positive and many times Spearman’s hypothesis is strongly supported.

_{r}), the correlation meta-analytically corrected for four statistical artifacts (rho-4), the standard deviation of rho-4 (SD

_{rho}

_{-4}), the correlation meta-analytically corrected for five statistical artifacts (rho-5), the percentage of variance explained by four artifacts (%VE), and the 80% credibility interval (80% CI). The first four corrections for statistical artifacts are correction for sampling error, correction for unreliability of the g vector, correction for unreliability of the d vector, and correction for restriction of range, respectively; the fifth correction is the correction for imperfectly measuring the construct of g.

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Three scatter plots of the reliability of the vector of g loadings and sample size each for different range of N.

**Figure 3.**RAKIT and Dutch WISC-R combined: scatter plot of correlations of g score from artificial test battery with “true” g against number of subtests and regression line.

Subtest | g | |
---|---|---|

Dutch names | English names | |

Rakit | ||

Figuur Herkennen | Figure Recognition | 0.53 |

Exclusie | Exclusion | 0.47 |

Geheugenspan | Memory Span | 0.56 |

Woordbetekenis | Verbal Meaning | 0.39 |

Doolhoven | Mazes | 0.61 |

Analogieën | Analogies | 0.63 |

Kwantiteit | Quantity | 0.51 |

Schijven | Disks | 0.49 |

Namen Leren | Learning Names | 0.62 |

Verborgen figuren | Hidden Figures | 0.20 |

Ideeënproduktie | Idea Production | 0.25 |

Vertelplaat | Storytelling | 0.58 |

WISC-R | ||

Informatie | Information | 0.61 |

Onvolledige tekeningen | Picture Completion | 0.52 |

Overeenkomsten | Similarities | 0.57 |

Plaatjes ordenen | Picture Arrangement | 0.70 |

Rekenen | Arithmetic | 0.64 |

Blokpatronen | Block Design | 0.56 |

Woordenschat | Vocabulary | 0.43 |

Figuurleggen | Object Assembly | 0.27 |

Begrijpen | Comprehension | 0.35 |

Substitutie | Coding | 0.44 |

Cijferreeksen | Digit Span | 0.44 |

Doolhoven | Mazes | 0.53 |

**Table 2.**Overview of studies with correlations between the vector of g loadings and the vector of standardized Latin-American Hispanic/White differences on the subtests of an IQ battery.

Study | Group | Test Battery | r | N_{subtests} | N_{White} | N_{Hispanic} | N_{harmonic} | N_{g} | r_{gg} | r_{dd} | u | Mean Age (Range) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Carretta (1997) [49] | Hispanic | AFOQT | 0.36 | 16 | 212,238 | 12,647 | 47,743 | 212,238 | 1.00 | 1.00 | 0.68 | 21 (18–27) |

Reynolds, Willson, & Ramsey (1999) [50] | Mexican-American | WISC-R | 0.77 | 12 | 2200 | 223 | 810 | 2200 | 0.98 | 0.87 | 0.77 | 10.12 (6–16) |

Valencia & Rankin (1986) [51] | Mexican-American | K-ABC | 0.70 | 13 | 100 | 100 | 200 | 1500 | 0.96 | 0.57 | 0.64 | 11 (10–12.5) |

Hartmann, Kruuse, & Nyborg (2007) [14] | Hispanic | Various tests | 0.71 | 16 | 3556 | 181 | 689 | 3556 | 0.98 | 0.84 | 0.99 | 19.9 (17–25) |

Hartmann, Kruuse, & Nyborg (2007) [14] | Hispanic | ASVAB | 0.74 | 10 | 6947 | 1704 | 5473 | 6947 | 0.99 | 1.00 | 0.59 | 19.6 (15–24) |

Snyder ^{2} (1991) [52] | Hispanic | WISC-R | 0.81 | 11 | 64 | 64 | 128 | 1800 | 0.96 | 0.45 | 0.68 | 10.5 ^{1} (6.5–14.5) |

Dalliard (2013) [16] | Hispanic | DAS-II | 0.70 | 13 | 864 | 432 | 1152 | 2952 | 0.98 | 1.00 | 0.73 | (5–17) |

Taylor & Richards (1991) [53] | Hispanic | WISC-R | 0.72 | 10 | 1200 | 100 | 369 | 1200 | 0.96 | 0.70 | 0.56 | 8.3 (6–11) |

Sandoval (1979) [54] | Mexican-American/Hispanic | WISC-R | 0.55 | 10 | 351 | 349 | 700 | 1200 | 0.96 | 0.85 | 0.50 | 8 (5–11) |

Kaufman, McLean & Kaufman (1995) [42] | Hispanic | KAIT | 0.56 | 8 | 1535 | 138 | 502 | 1535 | 0.96 | 0.75 | 0.45 | 11–94 |

Dean^{2} (1979) [55] | Mexican-American | WISC-R | 0.75 | 10 | 60 | 60 | 120 | 2200 | 0.98 | 0.44 | 0.68 | 10 |

Naglieri, Rojahn, & Matto (2007) [56] | Hispanic | CAS | −0.47 | 12 | 1956 | 244 | 868 | 155 | 0.78 | 0.92 | 0.82 | 8.3 (5–17) |

Kane (2007) [15] | Hispanic | UNIT | 0.42 | 6 | 77 | 77 | 154 | 77 | 0.48 | 0.50 | 1.07 | 10.5 |

Flemmer & Roid (1997) [43] | Hispanic | Leiter-R | 0.28 | 7 | 258 | 62 | 181 | 410 | 0.88 | 0.54 | 0.31 | 11–21 |

_{subtests}is number of subtests in the intelligence battery; N

_{White}= sample size for Whites; N

_{Hispanics}= sample size for Latin-American Hispanics; N

_{harmonic}is computed using the formula $\frac{4}{\frac{1}{n1}+\frac{1}{n2}}$ where n1 and n2 are the number of participants in groups n1 and n2, respectively; N

_{g}= sample size for g vector; r

_{gg}is the reliability of the g vector; r

_{dd}is the reliability of the Glass’d vector; u indicates the restriction of range. AFOQT = Air Force Officer Qualification Test, WISC-R = Wechsler Intelligence Test for Children-Revised, K-ABC = Kaufman Assessment Battery for Children, ASVAB = Armed Services Vocational Aptitude Battery, DAS-II = Differential Ability Scales-II, KAIT = Kaufman Adolescent and Adult Intelligence Test, CAS = Cognitive Assessment System, UNIT = Universal Nonverbal Intelligence Test, Leiter-R = Leiter International Performance Scale-Revised.

^{1}Estimated;

^{2}Referral sample; see text for explanation.

**Table 3.**Meta-analytical results for correlations between g loadings and Latin-American Hispanic/White differences.

Studies Included | K | Total N | r | SD_{r} | rho-4 | SD_{rho-4} | rho-5 | %VE | 80% CI |
---|---|---|---|---|---|---|---|---|---|

All studies | 14 | 154 | 0.55 | 0.333 | 0.75 | 0.368 | 0.80 | 24.3 | 0.27–1.22 |

All studies minus outlier | 13 | 142 | 0.63 | 0.158 | 0.86 | 0 | 0.91 | 199.7 | 0.86–0.86 |

_{r}= standard deviation of observed correlations; rho-4 = correlation meta-analytically corrected for four artifacts; SD

_{rho-4}= standard deviation of correlation meta-analytically corrected for four artifacts; rho-5 = correlation meta-analytically corrected for five artifacts; %VE = percentage of variance accounted for by four artifacts; 80% CI = 80% credibility interval, computed using SD

_{rho-4}.

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**MDPI and ACS Style**

te Nijenhuis, J.; van den Hoek, M.; Dragt, J.
A Meta-Analysis of Spearman’s Hypothesis Tested on Latin-American Hispanics, Including a New Way to Correct for Imperfectly Measuring the Construct of *g*. *Psych* **2019**, *1*, 101-122.
https://doi.org/10.3390/psych1010008

**AMA Style**

te Nijenhuis J, van den Hoek M, Dragt J.
A Meta-Analysis of Spearman’s Hypothesis Tested on Latin-American Hispanics, Including a New Way to Correct for Imperfectly Measuring the Construct of *g*. *Psych*. 2019; 1(1):101-122.
https://doi.org/10.3390/psych1010008

**Chicago/Turabian Style**

te Nijenhuis, Jan, Michael van den Hoek, and Joep Dragt.
2019. "A Meta-Analysis of Spearman’s Hypothesis Tested on Latin-American Hispanics, Including a New Way to Correct for Imperfectly Measuring the Construct of *g*" *Psych* 1, no. 1: 101-122.
https://doi.org/10.3390/psych1010008