INTRODUCTION 

In light of the growing popularity of unitary theories of executive control, it becomes increasingly important to test the validity of unitary control.  This study is an attempt to adjudicate between these two sets of theories (Page 1).  Unitary theories posit that a single mechanism (e.g., guided activation, adaptive neural coding, the Central Executive, executive attention), often thought to be mediated by prefrontal cortex, implements all executive control processes (e.g., inhibition, updating, shifting attention).  An alternative account of executive control is that it is the emergent property of a number of discrete mechanisms, each implementing one or a set of such processes (e.g., Baddeley, 1996; Miyake et al., 2000).  A similar unitary/emergent question applies to the understanding of general fluid intelligence, and interestingly, some have suggested (and indeed, demonstrated) a relationship between measures of general fluid intelligence and measures of executive attention, a putative unitary control mechanism.

Both neuropsychology and neuroimaging evidence can be found for both accounts of control, though the discrete processes view has received more support in recent years.  Behavioral evidence is mixed.  Evidence for both accounts and the attendant inferential problems are summarized in the table on Pages 2 and 3. 

METHODS

 A Confirmatory Tetrad Analysis (CTA; Page 4) allows us to test a special case of structural equations models (SEM) depicted on Page 5.  This is in fact the statistical model implied by completely unitary theories of executive control—one sees here that a unitary mechanism implements a variety of control process—and can be contrasted with the model on Page 6, which depicts the case of control processes implemented by discrete mechanisms. 

The CTA tests whether the structure of observed covariances corresponds to the structure implied by a unitary model.  A “tetrad” consists of a product of covariances between control processes (e.g., σ12σ34 on Page 5).  In the unitary model, a unitary mechanism “causes” (or is manifested in) each control process, and as a result, each covariance is to a great degree a function of this mechanism, φ [σ_ij = f(φl_il_j)].  Any arbitrary product of covariances (i.e., a tetrad), too, should reflect this unitary mechanism [σ12σ34 = f(φl₁l₂l₃l₄)].  The unitary model implies tetrad equivalence resulting from such a contribution of φ.  It follows that tetrad difference values (e.g., σ₁₂σ₃₄ - σ₁₃σ₂₄) should be equivalent to zero.  If this is the case, when tetrads matched on variable inclusion are subtracted, one will observe what Bollen & Ting (1993) call “vanishing tetrads.”  The CTA involves formulating a vector of tetrad differences using all non-redundant tetrad permutations, and a chi-square test determines whether the vector value is different from zero. 

Under the unitary model (Page 5), the chi-square test will be non-significant: the tetrad vector will be no different from zero.  Under the discrete mechanisms model (Page 6), the tetrads should not be equivalent.  That is, the covariances should not reflect a unitary control mechanism, φ.  Thus, arbitrary combinations of covariances to form tetrads will not produce equivalent tetrads.  A significant chi-square test will reveal that the tetrad difference vector is not equivalent to zero. 

Page 7 depicts the theoretical model that was tested in the present study.  On the right are five putative executive control processes, chosen following a literature review.  The included processes were those that are commonly associated with executive control and are discussed as distinct.  On the left are two measures thought to reflect a unitary control mechanism.  Tests of general fluid intelligence are thought to reflect g while working memory span (WMS) tasks are thought to reflect (mostly) the cognitive primitive executive attention as well as short-term memory (STM) processes.  We tested this model in two ways: with tests of fluid intelligence approximating the unitary mechanism; and with WMS (partialling out STM) approximating the unitary mechanism. 

N.B.: The study originally focused on a single task performance measure for each executive control process.  Four of these measures were of reaction time (RT) costs (i.e., RT on experimental trial – RT on control trials).  Merely one error measure was used (i.e., in this case, errors of omission [misses]).  We subsequently extracted multiple performance measures—for example, both an attention shifting cost (RT cost measure) and number of errors on trials of the attention-shifting task (Error measure).

Results from 121 subjects are reported here.  Page 11 presents a correlation matrix.  Results of interest are:

• OSPAN (a measure of WMS) correlates well with g-comp (a composite of general fluid intelligence test scores); LSPAN (a measure of STM span) does not correlate highly with g-comp.
• In correlations with performance on tasks involving executive control, OSPAN and g-comp behaved similarly: both were well correlated with error measures of performance.  LSPAN, however, was correlated with only half of the error measures, but most of the RT cost measures.

On Page 12, the results of interest are: 

• Measures from tasks of various instantiations of executive control are fairly intercorrelated, consistent with the “universal positive correlation” described by Spearman (1904).
• However, Error measures are not well correlated with RT cost measures.  (Most Error X Error and RT Cost X RT Cost correlations are significant.)

Results of CTA:

 Two tests (one with g-comp, one with OSPAN and LSPAN) were carried out using the five planned measures of performance on tasks involving executive control (Page 13).  In both cases, tetrads vanished: the unitary statistical model could not be rejected, consistent with the existence of a unitary control mechanism.  A factor analysis yielding a satisfactory 1-factor solution provided further, orthogonal evidence for a unitary account (Page 14). 

However, when multiple dimensions of task performance were considered (Page 15), a different case emerged.  As reported on Page 16, the unitary model was rejected (i.e., the CTA yielded a tetrad vector significantly different from zero)—but only when g-comp was included in the model as a proxy for unitary control.  In other words, when both Error and RT cost measures were included in the model, the covariance structure did not match that implied by a unitary model.  A factor analysis confirms this non-unitary structure.  The structure matrix on Page 17 illustrates that 2 factors provide the best fit, and that these factors discriminate between error and RT cost measures of executive control task performance.  Interestingly, OSPAN loads on the first (“error”) but not the second (“RT cost”) factor—and the opposite is true for LSPAN!

DISCUSSION

Page 18:

• When a mixture of error and RT cost measures are specified to assess executive control, the unitary model is rejected.  However, in the case of the CTA (but not for factor analyses), rejecting the unitary model requires that the fluid intelligence composite g-comp is included in the model.  This is further evidence that executive attention (as assessed by OSPAN) is not “isomorphic to…general intelligence” (Engle, 2002, p. 22).
• However, what does this imply for the main research question: Is executive control implemented by a unitary mechanism or by discrete mechanisms?  We can make at least two inferences:
  • Two separate control mechanisms need to be specified for “speed-related performance” and “accuracy-related performance”

Or, more likely

• The commonalities shared by RT-cost measures (or those shared by error measures) may obscure any independence of executive control processes.  In other words, behavioral investigations of the age-old question of unitary-versus-non-unitary control may not be fruitful if differences/similarities between control processes are eclipsed by differences/similarities in the response-based behavioral measures used!
  • This may help reconcile the apparent discrepancy between neuropsychological/neuroimaging findings of dissociation of processes and behavioral findings that are consistent with a unitary account.

Page 19:

• Some previous theoretical accounts treat “simple span” measures (i.e., measures of short-term memory like digit-, letter-, or word-span) as involving no more than storage and rehearsal processes.  However, in the present study we clearly illustrate a correspondence between “simple span” and executive control—and that these relationships are largely independent of working memory span.  Furthermore, Page 20 illustrates how our simple span measure (LSPAN) corresponds to raw RT scores—both for control and experimental trials.  In this way, LSPAN mirrors g-comp—but OSPAN does not (see above comment about isomorphism).  Certainly there may be more to STM span than “pure” storage.  Our findings urge further examination of the higher-level functions associated with measures of STM span.

Answers to Frequently Asked Questions:

Q: You said that the covariance between two measures of executive control processes is a function of φ (the unitary mechanism) under the unitary model.  Is not the covariance also a function of idiosyncratic commonality between these measures (e.g., the same mode of response)? 

A: Yes.  However, including all permutations of covariance products removes the idiosyncratic commonality, leaving only the contribution of φ.  Thus, the omnibus test for the CTA (the chi-square test of the tetrad vector value from zero) is not affected by the idiosynchracies that affect bivariate relationships. 

A: This is simply a term used to describe tetrad equivalence.  Since a unitary model implies that
     Tetrad1 = Tetrad2, taking the difference between two tetrads that contain the same variables (but
     represented through different covariance combinations) should make these tetrads cancel out, or
     “vanish.” 

Q: What was the power to detect a non-unitary structure for the    non-significant tetrad tests? 

A: With 121 subjects, we had > .80 power to detect at least a medium-large effect (chi-square effect size w = ~.4) and good power to detect a medium-sized effect (w = ~.3).

Q: What is the status of this project?  Are these data published anywhere? 

A: We continue to collect data for this project in order to achieve satisfactory power to be confident about the results of the confirmatory tetrad analysis (CTA).  These data are not yet published; if you would like to receive a copy of future publications related to this project, please contact: