# Advancing the Understanding of the Factor Structure of Executive Functioning

^{1}

^{2}

^{3}

^{*}

*g*and Its Underlying Executive Processes)

## Abstract

**:**

## 1. Introduction

#### 1.1. Relationship between EF and Relevant Cognitive Constructs

#### 1.2. The Current Study

## 2. Methods

#### 2.1. Participants

#### 2.2. Measures

#### 2.3. Procedures

## 3. Statistical Analyses

#### 3.1. Data Trimming and Transformation

#### 3.2. Data Analyses

^{2}), the standardized root mean square residual (SRMR), the root mean square error of approximation (RMSEA), and the comparative fit index (CFI). Values of SRMR < .08, RMSEA < .06, and CFI > .95 were taken as indication of adequate model fit (Hu and Bentler 1999). We also used Akaike information criterion (AIC) and Bayesian information criterion (BIC) while comparing between the models. A smaller AIC or BIC score favors a better model. We reported standardized loadings of each indicator on its corresponding latent factor. All models were estimated using Amos 24.

## 4. Results

#### 4.1. Preliminary Data Analysis

#### 4.2. Five-Factor Model

^{2}(94) = 130.10, p = .008; CFI = .95; RMSEA = .04; SRMR = .05, AIC = 214.10; BIC = 221.86. All path coefficients from the indicators to the corresponding latent variables in this model were moderate to high (shifting: λ = .38 to λ = .67; updating: λ = .47 to λ = .86; WMC: λ = .55 to λ = .76; relational integration: λ = .24 to λ = .73; divided attention: λ = .29 to λ = .92), and were all significant (all p < .01). Correlations between the latent variables were also moderate to high (ranging from r = .37 to r = .74). The shifting factor shared the lowest variance with the other constructs, whereas relational integration and updating shared 55% of the variance.

#### 4.3. Hierarchical Model

^{2}(99) = 136.44, p = .008; CFI = .95; RMSEA = .04; SRMR = .05; AIC = 210.44; BIC = 332.85. General cognitive ability accounted for 24% of the shifting variance, 61% of the updating variance, 41% of the WMC, 96% of the relational integration, and 52% of the divided attention variance. Thus, there was both significant shared variance for the five cognitive variables, as well as significant unique variances on each of the cognitive abilities.

#### 4.4. Additional Model: Two-Layer Six-Factor Model

^{2}(97) = 168.42, p < .001; CFI= .91; RMSEA= .06; SRMR= .06; AIC = 246.42; BIC = 375.44. All path coefficients in this model were significant, except for the paths from inhibition and WMC to shifting and from inhibition to updating. The correlation between WMC and inhibition was high (r = .81). Notably, the relational integration factor seemed to be almost isomorphic with WMC (λ = .86).

#### 4.5. Model Comparison

^{2}(3) = 38.32, p = .0001) and an equally good one compared to the hierarchical model (∆χ

^{2}(5) = 6.34, p = .275). However, the hierarchical model is more parsimonious, as underlined by the information criteria (AIC and BIC).

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Factor Loadings for the Exploratory Factor Analysis of EF Variables

Measures | Factors | |
---|---|---|

1 | 2 | |

Number–letter | .56 | |

Color–shape | .35 | |

Category switch | .77 | |

Keep track | .72 | |

Letter memory | −.19 | .59 |

Nonverbal n-back | .63 | |

Antisaccade | .22 | .49 |

Stopsignal | .25 | |

Correlation | ||

Factor 1 | - | |

Factor 2 | .34 | - |

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**Figure 1.**Correlated cognitive latent variables models. The proportion of residual variance of each indicator was calculated by subtracting the variance of the indicator from 1. All parameters were statistically significant (p < .05). WMC = working memory capacity; RI = relational integration; DA = divided attention.

**Figure 2.**Hierarchical model of cognitive latent variables. The proportion of residual variance of each indicator was calculated by subtracting the variance of the indicator from 1. All parameters were statistically significant (p < .05). WMC = working memory capacity; RI = relational integration; DA = divided attention.

**Figure 3.**Two-layer six-factor model. The proportion of residual variance of each indicator was calculated by subtracting the variance of the indicator from 1. Not significant (p < .05) paths are indicated as the dotted line. WMC = working memory capacity; RI = relational integration; DA = divided attention.

Tasks | Authors | Task Description | Dependent Variables |
---|---|---|---|

Shifting | Friedman et al. (2016) | ||

Number–letter | When a number–letter pair appears in the top half of the matrix, participants have to classify the number as odd or even; but when the pair appears in the bottom half of the matrix, they should classify the letter as vowel or consonant. | Switch cost: the difference between the mean reaction time (RT) of correct switch trials and the mean RT of correct repeat (nonswitch) trials in random mixed blocks | |

Color–shape | Participants need to classify the color (green vs. red) or the geometric shape (circle vs. triangle) of the target stimulus. | ||

Category switch | Participants are instructed to switch back and forth regarding the dimension of animacy (living or nonliving) or size of the target stimulus (smaller or larger than a soccer ball). | ||

Updating | Friedman et al. (2016) | ||

Keep track | Participants remember the last exemplar of each of the five target categories. | Accuracy (i.e., the proportion of correct trials) | |

Letter memory | Participants remember the last four letters in the letter string. | ||

Nonverbal n-back | Schellig et al. (2011) | Participants identify the stimulus if the stimulus matches the stimulus n-times back. | The average of the z-scores across the 2-back and 3-back tasks |

Inhibition | Friedman et al. (2016) Kaiser et al. (2010) | ||

Antisaccade | Participants have to look in the opposite direction of visual cues to detect a briefly presented target. | The proportion of correct target discrimination responses across three antisaccade blocks. | |

Stop signal | Participants have to categorize and respond to stimuli until a stop signal appears for withholding a response. | The mean stop signal delay is subtracted from the median RT on go trials | |

WMC | Oswald et al. (2015) | ||

Operation span | Participants have to solve a series of math problems while remembering letters in correct serial order. | The partial-credit score | |

Reading span | Participants have to identify whether the sentences are meaningful while remembering letters in correct serial order. | ||

Symmetry span | Participants have to identify whether the patterns are symmetrical while remembering the correct presentation order of red squares in the 4×4 matrix. | ||

Relational integration | von Bastian and Oberauer (2013) | The dependent variable is the discriminability index (d′), reflecting the sensitivity of target detection. It is computed by relating the hit rate and false alarm rate (d′ = z (hit rate)—z (false alarm rate)). | |

Numerical version | Participants have to respond when three identical last digits appear either in a row, column, or diagonal line in a 3 × 3 matrix. | ||

Verbal version | Participants are asked to respond when three rhyming words are shown either in a row, column, or diagonal line within the 3 × 3 matrix. | ||

Figural version | Participants are asked to respond when four black dots form a square in a 3 × 3 matrix. | ||

Divided attention | Sturm (2008) | ||

Unimodal version | Participants have to monitor two visual stimulus presentation conditions. Whenever the same shape (either square or circle) gets noticeably lighter twice in a row, participants should respond. | The logarithmic mean RT of the given responses | |

Crossmodal version | Participants are required to monitor one visual and one auditory stimulus presentation conditions. Whenever the square gets noticeably lighter or the sound gets noticeably softer twice in a row, participants are asked to respond. |

Tests | Mean | SD | Skewness | Kurtosis | Reliability |
---|---|---|---|---|---|

Executive functioning | |||||

Shifting | |||||

Number–letter | 457.88 | 157.64 | −0.75 | 0.25 | .89 ^{c} |

Color–shape | 828.06 | 275.50 | −0.80 | 0.92 | .92 ^{c} |

Category switch | 592.94 | 186.20 | −0.96 | 1.30 | .83 ^{c} |

Updating | |||||

Keep track | 0.75 | 0.10 | −0.68 | 0.20 | .72 ^{b}/.73 ^{d} |

Letter memory | 0.69 | 0.19 | −0.41 | −0.37 | .59 ^{b}/.59 ^{d} |

Nonverbal n-back | |||||

Nonverbal 2-back ^{a} | 1.27 | 0.10 | −0.37 | 0.73 | .84 ^{b} |

Nonverbal 3-back ^{a} | 1.22 | 0.09 | −0.36 | 0.11 | .86 ^{b} |

Inhibition | |||||

Antisaccade | 0.65 | 0.17 | −0.62 | 0.03 | .94 ^{a} |

Stop signal | 165.93 | 55.48 | 0.60 | 0.25 | .94 ^{e} |

WMC | |||||

Operation span | 0.82 | 0.19 | −1.39 | 1.56 | .72 ^{b}/.73 ^{d} |

Reading span | 0.66 | 0.23 | −0.66 | −0.11 | .73 ^{b}/.73 ^{d} |

Symmetry span | 0.65 | 0.20 | −0.56 | −0.03 | .55 ^{b}/.55 ^{d} |

Relational integration | |||||

Numerical | 2.43 | 0.73 | −0.22 | −0.13 | .77 ^{c} |

Verbal | 2.51 | 0.71 | 0.00 | −0.31 | .72 ^{c} |

Figural | 2.48 | 0.42 | −0.59 | 0.36 | .59 ^{c} |

Divided attention | |||||

Unimodal | 481.60 | 151.06 | −1.36 | 2.07 | .96 ^{b} |

Crossmodal | 492.11 | 171.42 | −0.82 | 0.39 | .96 ^{b} |

^{a}Scores were arcsine transformed, and then converted into z-scores.

^{b}Cronbach’s Alpha.

^{c}Split-half reliability.

^{d}McDonald’s Omega.

^{e}Reliability for difference scores. WMC = working memory capacity.

Model | χ^{2} | df | CFI | RMSEA | SRMR | AIC | BIC |
---|---|---|---|---|---|---|---|

A. Five-factor EF model | 130.10 | 94 | .95 | .04 | .05 | 214.10 | 353.05 |

B. Hierarchical model | 136.44 | 99 | .95 | .04 | .05 | 210.44 | 332.85 |

C. Two-layer six-factor model | 168.42 | 97 | .91 | .06 | .06 | 246.42 | 375.44 |

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**MDPI and ACS Style**

Himi, S.A.; Bühner, M.; Hilbert, S.
Advancing the Understanding of the Factor Structure of Executive Functioning. *J. Intell.* **2021**, *9*, 16.
https://doi.org/10.3390/jintelligence9010016

**AMA Style**

Himi SA, Bühner M, Hilbert S.
Advancing the Understanding of the Factor Structure of Executive Functioning. *Journal of Intelligence*. 2021; 9(1):16.
https://doi.org/10.3390/jintelligence9010016

**Chicago/Turabian Style**

Himi, Samsad Afrin, Markus Bühner, and Sven Hilbert.
2021. "Advancing the Understanding of the Factor Structure of Executive Functioning" *Journal of Intelligence* 9, no. 1: 16.
https://doi.org/10.3390/jintelligence9010016