Fig. 3: Latent factor models of cognitive control (CC).

Proposed CC functions are represented as latent variables (depicted with ellipses) that predict variation in performance on specific tasks (rectangles) chosen to measure those abilities. Factor loadings are depicted with single-headed arrows between the factors and nine measured tasks. The short arrows indicate residual variances, the unique variance in each task that is unrelated to the CC factors, attributable to measurement error as well as reliable task-specific variation. a In a correlated factors model, tasks are predicted by CC factors that are allowed to be correlated, and unity and diversity are represented in the correlations between factors (represented with curved double-headed arrows). The numbers shown are the average correlations and the range of correlations from six studies using a similar battery (Ns = 137–786). b A higher-order “Common” CC factor can also be used to model the correlations among the factors [39, 158]. This higher-order factor predicts the lower-order factors, and they correlate to the extent to which they are jointly predicted by the common factor. In such models, the diversity is captured by the residuals of these factors after the variance due to the common factor is removed (inhibiting-specific, updating-specific, and shifting-specific variances). The numbers shown indicate the average and range of factor loadings for the Common CC factor, and the corresponding averages and ranges for the residual variances for inhibiting, updating, and shifting factors (i.e., the variance not explained by the common factor), derived from the correlations in panel a. *indicates the standardized loadings were bound at 1, and the residual variances bound at zero. c Alternative model structures (called nested factors models or bifactor models) can be used to capture unity and diversity factors more directly. In these models, all tasks load on a common factor, but also load on orthogonal specific factors. These models thus partition each factor into variance that is common across all tasks and variance that is unique to tasks assessing particular processes. Although these alternative parameterizations typically do not result in appreciably different fits to the data, they can make it more convenient to examine relationships to other constructs of interest: Because the unity and diversity components are represented with orthogonal latent variables rather than in the correlations between factors or with residual variances, it is straightforward to discern whether a construct is related to the unity vs. diversity components.