Figure 1
Theoretical latent variable models.
Note. Panel A: One-factor model with a common cognitive control function. Panel B: Three-factor model with correlated functions. Panel C: Bi-factor model with separate shifting and updating functions. Latent variables are depicted as ellipses and the rectangles represent the dependent measures of the cognitive control tasks that load on these latent variables.
Table 1
Overview of the Cognitive Control Tasks and the Included Dependent Measures for each Task.
| COGNITIVE CONTROL FUNCTION | TASK | TASK DESCRIPTION | DEPENDENT MEASURE |
|---|---|---|---|
| Inhibition | Antisaccade | Participants must inhibit the tendency to make a saccade to a cue in order to detect the briefly presented target stimulus | The proportion of errors across all trials |
| Go No-Go | Participants must respond to squares of one color and withhold responses to the other color | The proportion of commission errors (i.e., errors on No-Go trials) | |
| Stroop | Participants must respond to the ink color of the color words while ignoring the meaning of the color word | Congruency effect: The difference between the median RTs on correct incongruent trials and the median RTs on correct congruent trials | |
| Shifting | Local-Global | Participants must switch between responding to either the global or the local feature of a stimulus | Switch cost: The difference between the median RTs on correct switch trials and the median RTs on correct no-switch trials |
| Plus-Minus | Participants need to add 3 to each two-digit number in the first block and subtract 3 in the second block. In the last block, they must switch between adding and subtracting 3 | Switch cost: The difference between the time to complete the alternating block and the average time to complete the addition and subtraction blocks | |
| Number-Letter | Participants must switch between classifying a number as even or odd and a letter as a vowel or consonant. A cue indicates which task the participant needs to perform | Switch cost: The difference between the median RTs on correct switch trials in the mixed block and the average median RTs on correct trials in the no-switch blocks | |
| Updating | Keep Track | Participants must remember every last word presented from each category of a serially presented word stream | The proportion of errors across all words |
| Letter Memory | Participants must remember the four last letters of a serially presented letter stream | The proportion of errors across all letters | |
| N-back | Participants must respond to a picture if it matches the picture 2 trials back | The proportion of omission errors (i.e., errors on Hit trials) |
Table 2
Normalized Multivariate Skewness and Kurtosis in Each Data Cleaning Step Using Mardia’s Skewness and Mardia’s Kurtosis Indices.
| DATASET | MARDIA’S SKEWNESS | MARDIA’S KURTOSIS |
|---|---|---|
| Untrimmed data | 1495.73 (p < .001) | 33.42 (p < .001) |
| Participant outliers removed | 779.02 (p < .001) | 16.40 (p < .001) |
| + Trial outliers removed | 756.67 (p < .001) | 15.23 (p < .001) |
| + Transformations executed | 256.28 (p < .001) | 6.24 (p < .001) |
| + Multivariate outliers removed | 158.90 (p = .619) | –1.64 (p = .101) |
[i] Note. The data were not multivariate normally distributed after each data cleaning step as indicated by significant p-values, except for the last step in the cleaning procedure where multivariate outliers were additionally removed.
Figure 2
Chi-square Q-Q plots.
Note. This figure shows the ordered Mahalanobis distances versus the estimated quantiles. Panel A: Chi-square Q-Q plot before all data cleaning steps. A clear deviation from the multivariate normal distribution can be observed. Panel B: Chi-square Q-Q plot after all data cleaning steps, indicating acceptable multivariate normality.
Table 3
Univariate Statistics for the Outcome Measures of the Nine Cognitive Control Tasks.
| TASK | SKEWNESS | KURTOSIS | SHAPIRO-WILK | RELIABILITY |
|---|---|---|---|---|
| Antisaccade | 0.47 | 0.30 | .95 | .81 |
| Go No-Go | 0.14 | 0.23 | .97 | .67 |
| Stroop | 0.37 | –0.01 | .99 | .20 |
| Local-Global | –0.08 | –0.30 | .99 | .20 |
| Plus-Minus | 0.23 | 0.14 | .99 | |
| Number-Letter | –0.07 | 0.23 | >.99 | .87 |
| Keep Track | –0.29 | 0.29 | .98 | |
| Letter Memory | –0.04 | –0.25 | .99 | .59 |
| N-back | –0.12 | –0.38 | .94 | .58 |
[i] Note. The univariate statistics are based on the data after cleaning procedures. Measures with a univariate normal distribution according to the Shapiro-Wilk test statistic values are shown in bold (p > .05). No reliability estimates could be obtained for the Plus-Minus task and the Keep Track task. The Plus-Minus task only delivers one RT per condition. For the Keep Track task, permutation-based split-half reliability was not possible with only 6 trials per participant. Cronbach’s alpha could also not be computed as the stimuli were randomly presented. For the Antisaccade and Letter Memory tasks, all trials were used to compute split-half reliability. For the Go No-Go and N-back tasks, only trials that were used to compute the dependent measures were used to compute split-half reliability (i.e., No-Go trials for the Go No-Go task and Hit trials for the N-back task). For the Stroop and Local-Global tasks, there were an equal number of trials of each condition in each split (i.e., congruent and incongruent trials for the Stroop task and switch and no-switch trials for the Local-Global task). For the Number-Letter task, the trials in the pure number and letter classification blocks and the switch trials in the mixed block were used to compute split-half reliability. There were an equal number of pure number and letter classification trials and switch trials in each split.
Table 4
Descriptive Statistics (Mean Error Rates and Median Reaction Times) for the Outcome Measures of the Nine Cognitive Control Tasks.
| TASK | DEPENDENT MEASURE | MEAN (SD) | RANGE |
|---|---|---|---|
| Antisaccade | Mean ERR | 0.04 (0.05) | [0.00;0.30] |
| Go No-Go | Mean ERR NoGo trials (commission errors) | 0.11 (0.08) | [0.00;0.45] |
| Stroop | Median RT Congruency Effect | 95.93 (99.40) | [–99.12;408.06] |
| Local-Global | Median RT Switch Cost | 349.00 (195.77) | [–117.11;904.12] |
| Number-Letter | Median RT Switch Cost | 148.51 (114.89) | [–14.22;623.48] |
| Plus-Minus | Time Switch Cost | 14.03 (17.55) | [–35.91;77.22] |
| Keep Track | Mean ERR | 0.38 (0.10) | [0.11;0.67] |
| Letter Memory | Mean ERR | 0.12 (0.06) | [0.01;0.30] |
| N-back | Mean ERR Hit trials (omission errors) | 0.09 (0.08) | [0.00;0.43] |
[i] Note. The error-rate based measures were arcsine transformed. The Stroop task and Number-Letter task were square root and cube root transformed, respectively. For interpretability, all values presented in this table were back-transformed to the original scale. All RT-based measures are shown in milliseconds, with the exception of the Plus-Minus task, which is shown in seconds. Error-rates are shown as proportions. Means, standard deviations, and ranges were calculated based on individual participants’ mean error rates or median reaction times.
Table 5
Pearson Correlation Matrix between the Nine Cognitive Control Tasks.
| TASK | ANTISACCADE (I) | GO NO-GO (I) | STROOP (I) | LOCAL-GLOBAL (S) | PLUS-MINUS (S) | NUMBER-LETTER (S) | KEEP TRACK (U) | LETTER MEMORY (U) | N-BACK (U) |
|---|---|---|---|---|---|---|---|---|---|
| Antisaccade (I) | – | ||||||||
| Go No-Go (I) | .13* | – | |||||||
| Stroop (I) | .10 | .07 | – | ||||||
| Local-Global (S) | .07 | .01 | –.07 | – | |||||
| Plus-Minus (S) | .10 | .10 | .01 | .11 | |||||
| Number-Letter (S) | .08 | .03 | .17** | .18** | .15* | – | |||
| Keep Track (U) | .11* | .11 | .12 | –.03 | .15* | .15* | – | ||
| Letter Memory (U) | .10 | .17** | .12 | .07 | .20** | .07 | .29*** | – | |
| N-back (U) | .22*** | .06 | .13* | .05 | .01 | .23*** | .20** | .12 | – |
[i] Note. I = Inhibition, S = Shifting, U = Updating. * p < .05, ** p < .01, *** p < .001.
Table 6
Fit Indices of Fitted Latent Variable Models.
| MODEL | ONE-FACTOR | THREE-FACTOR | Bi-FACTOR | Bi-FACTOR WITH VARIANCE CONSTRAINT |
|---|---|---|---|---|
| χ2 | 40.08 (p = .050) | 34.65 (p = .074) | 27.25 (p = .163) | 27.34 (p = .198) |
| CFI | .85 | .88 | .93 | .94 |
| GFI | .97 | .97 | .98 | .98 |
| AGFI | .94 | .94 | .95 | .95 |
| RMSEA | .04 | .04 | .04 | .03 |
| SRMR | .05 | .05 | .04 | .04 |
| AIC | 5650.95 | 5651.52 | 5650.12 | 5648.22 |
| BIC | 5714.12 | 5725.21 | 5734.94 | 5728.93 |
[i] Note. CFI = Comparative Fit Index; GFI = Goodness of Fit Index; AGFI = Adjusted Goodness of Fit Index; RMSEA = Root Mean Square Error of Approximation; SRMR = Standardized Root Mean Square Residual; AIC = Akaike Information Criterion; BIC = Bayesian Information Criterion.
Figure 3
Factor loadings of the one-factor model.
Note. The standardized factor loadings are shown on the longer arrows. The residual variances of each task due to measurement error and idiosyncratic task requirements are shown adjacent to the arrows next to each task. For all parameters, boldface type indicates p < .05. The latent variable is depicted as an ellipse and the rectangles represent the dependent measures of the cognitive control tasks that load on this latent variable.
Figure 4
Factor loadings of the full three-factor model.
Note. The standardized factor loadings are shown on the straight, single-headed arrows. The residual variances of each task due to measurement error and idiosyncratic task requirements are shown adjacent to the single-headed arrows next to each task. The double-headed arrows indicate the correlations between the latent variables. For all parameters, boldface type indicates p < .05. Latent variables are depicted as ellipses and the rectangles represent the dependent measures of the cognitive control tasks that load on these latent variables.
Figure 5
Factor loadings of the bi-factor model.
Note. The standardized factor loadings are shown on the longer arrows. The residual variances of each task due to measurement error and idiosyncratic task requirements are shown adjacent to the arrows next to each task. For all parameters, boldface type indicates p < .05. Latent variables are depicted as ellipses and the rectangles represent the dependent measures of the cognitive control tasks that load on these latent variables.
Figure 6
Factor loadings of the bi-factor model with the variance of the Letter-Memory task set to zero.
Note. The standardized factor loadings are shown on the longer arrows. The residual variances of each task due to measurement error and idiosyncratic task requirements are shown adjacent to the arrows next to each task. For all parameters, boldface type indicates p < .05. Latent variables are depicted as ellipses and the rectangles represent the dependent measures of the cognitive control tasks that load on these latent variables.