# The Unity and Diversity of Executive Functions and Their Contributions to Complex Frontal Lobe Tasks: A Latent Variable Analysis

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11 UNITY AND DIVERSITY OF EXECUTIVE FUNCTIONS 59 the second major goal was to specify their relative contributions to more complex tests commonly used to assess executive functioning. With respect to the first goal, we used confirmatory factor analysis (CFA) to specify the degree to which the three postulated executive functions are separable or share the same underlying ability or mechanism. CFA is similar to the EFA technique more commonly used in the field (the term factor analysis with no modifier typically refers to EFA, rather than CFA). One major difference, however, is that, whereas EFA finds the one underlying factor model that best fits the data, CFA allows researchers to impose a particular factor model and then see how well that statistical model fits the data (Kline, 1998). In other words, with EFA, one lets the observed data determine the underlying factor model a posteriori (this characteristic of EFA is part of the reason the factors extracted with this method do not necessarily have clear interpretations), whereas with CFA, one derives a factor model or models a priori on the basis of theoretical considerations and then evaluates its fit to the data. Thus, CFA is a highly theory-driven multivariate analysis technique and serves as a valuable tool for specifying the organization of executive functions. We used CFA to compare models with one, two, or three factors. Figure 1A illustrates the theoretical model that provided the basis for our analysis (called the full, three-factor model). Ellipses in the figure represent the three target latent variables (i.e., Shifting, Updating, and Inhibition), whereas rectangles represent the manifest variables (i.e., individual tasks) that were used to tap the specific functions, as indicated by the straight, single-headed arrows. The curved, double-headed arrows represent correlations among the latent variables. If it is necessary to postulate three separable factors (one for each target executive function), then this full, three-factor model should provide an excellent fit to the data, and the correlations among the three latent variables will provide an estimation of the degree to which the three target functions are related to one another. In contrast, if the three executive functions essentially tap the same underlying construct and hence should be considered unitary, then a model with one factor (created by fixing all of the correlations among the three latent variables to 1.0 so that this alternative model is nested within the full, three-factor model) should provide an excellent fit to the data, a fit statistically no worse than the full, three-factor model (because the full, three-factor model has more free parameters than the onefactor model, the absolute fit of the one-factor model cannot exceed that of the three-factor model). Similarly, if two of the target executive functions (but not the third one) tap a common underlying ability, then a model with two factors (i.e., a model that fixes the correlation between the two unitary executive functions to 1.0 but lets the other two correlations vary freely) should provide a fit to the data that is statistically as good as the full, threefactor model. Finally, if the executive functions are completely independent,

12 60 MIYAKE ET AL. FIG. 1. (A) The theoretical full, three-factor model used for the confirmatory factor analysis (CFA). The ellipses represent the three executive functions (latent variables), and the rectangles represent the individual tasks (manifest variables) that were chosen to tap the specific executive functions, as indicated by the straight, single-headed arrows. The curved double-headed arrows represent correlations among the latent variables. Both models depict three latent constructs, namely, Shifting, Updating, and Inhibition, which are hypothesized to be correlated but separable. (B) A generic model for the structural equation modeling (SEM) analysis. This model is identical to the CFA model with the addition of a manifest variable on the right side that represents a complex executive function measure. In this particular model (the full model), the manifest variable on the right has paths from all three latent variables to estimate the contribution of each to performance on the executive task.

20 68 MIYAKE ET AL. a small value for the χ 2 statistic indicates no statistically meaningful difference between the covariance matrix generated by the model and the observed matrix, suggesting a satisfactory fit. The SRMR also assesses badness of fit, as it is the square root of the averaged squared residuals (i.e., differences between the observed and predicted covariances). Lower values of the SRMR indicate a closer fit, with values less than.08 indicating a relatively close fit to the data (Hu & Bentler, 1998). The other fit indices (AIC, CFI, and IFI) are typically used to measure goodness of fit. AIC is a modified version of the χ 2 statistic that takes into consideration the complexity of the evaluated model (in terms of degrees of freedom) and penalizes more complex models (i.e., models with fewer degrees of freedom). Lower values of AIC (including negative values) indicate better fit. In contrast, for CFI and IFI, higher values indicate better fit, as these indices quantify the extent to which the tested model is better than a baseline model (e.g., one with all covariances set to zero). Typically, IFI and CFI values that exceed.90 or.95 are considered good fits (the values of IFI can exceed 1.0). In addition to these commonly used indices, we also examined specific indications of fit, such as the magnitudes of asymptotically standardized residuals, in comparing different alternative models. None of the models we endorse in our discussion of the CFA and SEM results had large residuals, according to the criteria recommended by Jöreskog and Sörbom (1989). To examine if one model was significantly better than another, we conducted χ 2 difference tests on nested models. This test entails subtracting the χ 2 for the fuller model from the χ 2 for the nested model with a fewer number of free parameters or larger degrees of freedom (degrees of freedom are calculated with an analogous subtraction). If the resulting χ 2 is statistically significant, then the fuller model provides a significantly better fit. For these and all other statistical tests reported here, we used an alpha level of.05. RESULTS AND DISCUSSION Preliminary Data Analysis A summary of descriptive statistics for the nine measures used to tap the three target executive functions (i.e., Shifting, Updating, and Inhibition) is presented in Table 1. All of the measures had relatively low skewness and kurtosis coefficients, and the normalized multivariate kurtosis (Mardia, 1970) for all nine measures was also quite low:.19 (the normalized multivariate kurtosis did not exceed.95 for any of the SEM analyses). Internal reliability estimates for the tasks used in the CFA were calculated using either Cronbach s alpha or the split-half (odd even) correlation adjusted by the Spearman-Brown prophecy formula. As seen in Table 1, the reliability estimates for the tasks were relatively low (except for number letter and stopsignal), a characteristic often reported for executive tasks (Denckla, 1996; Rabbitt, 1997b). The zero-order correlations among the nine measures, provided in Appendix A, were generally low (.34 or lower), consistent with the results from previous individual differences studies of executive functions. It is important to point out, however, that the correlations among the nine measures were not uniformly low; rather, the tasks considered to tap the same executive function tended to show significant correlations with one another, while correlating not as strongly with the tasks considered to tap the other executive functions, thus showing some signs of convergent and discriminant validity.

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