(c) Copyright, Dr. Kevin S. McGrew, Institute for Applied Psychometrics (11-14-24)
Warning, may be TLDR for many. :). Also, I will be rereading again multiple times and may tweak minor (not substantive) errors and post updates….hey….blogging has an earthy quality to it:)
In a recent publication, Scott Decker, Joel Schneider, Okan Bulut and I (McGrew, 2023; click here to download and read) presented structural analysis of the WJ IV norm data using contemporary psychometric network analysis (PNA) methods. As noted in a clip from the article below, we recommended that intelligence test researchers, and particularly authors and publishers of the respective technical manuals for cognitive test batteries, needed to broaden the psychometric structural analysis of a test battery beyond the traditional (and almost exclusive) relieance on “common cause” factor analysis (EFA and CFA) methods to include PNA analysis…to compliment, not supplant factor based analyses.
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Our (McGrew et al., 2023) recommendation is consistent with some critics of intelligence test structural research (e.g., see Dombrowski et al., 2018, 2019; Farmer et al., 2020) who have cogently argued that most intelligence test technical manuals typically present only one of the major classes of possible structural models of cognitive ability test batteries. Interestingly, many school psychology scholars who conduct and report independent structural analysis of a test battery also do something similar…they often only present one form of structural analysis—-namely, bifactor g analyses.
In McGrew et al. (2023) we recommended future cognitive ability test technical manuals embrace a more ecumenical multiple method approach and include, when possible, most all major classes of factor analysis models, as well as PNA. A multiple-methods research approach in test manuals (and journal publications by independent researchers) can better inform users of the strengths and limitations of IQ test interpretations based on whatever conceptualization of psychometric general intelligence (including models with no such construct) underlies each type of dimensional analysis. Leaving PNA methods aside for now, the figure below presents the four major families of traditional CHC theoretical structural models. These figures are conceptual and are not intended to represent all nuances of factor models.
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Briefly, the four major families of traditional “common cause” CHC CFA structural models (Carroll, 2003; McGrew et al., 2023) vary primarily in the specification (or lack thereof) of a psychometric g factor. The different families of CHC models are conceptually represented in the figure above. In these conceptual representations the rectangles represent individual (sub)tests, the circles latent ability factors at different levels of breadth or generality (stratum levels as per Carroll, 1993), the path arrows the direction of influence (the effect) of the latent CHC ability factors on the tests or lower-order factors, and the single double headed arrow all possible correlations between all CHC broad CHC factors (in the Horn no-g model in panel D).
The classic hierarchical g model “places a psychometric g stratum III ability at the apex over multiple broad stratum II CHC abilities” (McGrew et al., 2023, p. 2). This model is most often associated with Carroll (1993; 2003) and is called (in panel A in the above figure) the Carroll hierarchical g broad CHC model. In this model the shared variance of subsets of moderately to highly correlated tests are first specified as 10 CHC broad ability factors (i.e., the measurement model; Gf, Gc, Gv, etc.). Next the covariances (latent factor correlations) among the broad CHC factors are specified as being the direct result of a higher-order psychometric g factor (i.e., the structural model).
The classic hierarchical g model “places a psychometric g stratum III ability at the apex over multiple broad stratum II CHC abilities” (McGrew et al., 2023, p. 2). This model is most often associated with Carroll (1993; 2003) and is called (in panel A in the above figure) the Carroll hierarchical g broad CHC model. In this model the shared variance of subsets of moderately to highly correlated tests are first specified as 10 CHC broad ability factors (i.e., the measurement model; Gf, Gc, Gv, etc.). Next the covariances (latent factor correlations) among the broad CHC factors are specified as being the direct result of a higher-order psychometric g factor (i.e., the structural model).
A sub-model under the Carroll hierarchical g broad CHC model includes three levels of factors—several first-order narrow (stratum I) factors, 10 second-order broad (stratum II) CHC factors, and the psychometric g factor (stratum III). This is called the Carroll hierarchical g broad+narrow CHC model in panel B in the figure above. In the above example, two first-order narrow CHC factors (auditory short-term storage-Wa; and auditory working memory capacity-Wc, which, in simple terms, is a factor defining auditory short-term memory tasks that also include heavy attentional control-based (AC as per Schneider & McGrew, 2018) active manipulation of stimuli—the essence of Gwm or working memory). For illustrative purposes, a narrow naming facility (NA) first-order factor, which has higher-order effects or influences from broad Gs and Gr is specified for evaluation. Wouldn’t you like to see the results of this hierarchical broad+narrow CHC model? Well……..stay tunned for the forthcoming WJ V technical manual (Q1 2025; LaForte, Dailey, & McGrew, 2025, in preparation) and your dream will come true.
The third model is the Horn no-g model (McGrew, et al., 2023). John Horn long argued that psychometric g was nothing more than a statistical abstraction or artifact (Horn, 1998; Horn & Noll, 1997; McArdle, 2007; McArdle & Hofner, 2014; Ortiz, 2015) and did not represent a brain or biologically based real cognitive ability. This is represented by the Horn no-g broad CHC model in panel D. The Horn no-g broad CHC model is like the Carroll hierarchical g broad CHC model, but the 10 broad CHC factor intercorrelations are retained instead of specifying a higher- or second-order psychometric g factor. In other words, the measurement models are the same but the structural models are different. In some respects the Horn no-g broad CHC model is like contemporary no-g psychometric network analysis models (see McGrew, 2023) that eschew the notion of a higher-order latent psychometric g factor to explain the positive definite correlation variance between individual tests (or first-order latent factors in the case of the Horn no-g model) in an intelligence battery (Burgoyne et al. 2022; Conway &Kovacs, 2015; Euler et al., 2023; Fried, 2020; Kan et al. 2019; Kievit et al. 2016; Kovacs & Conway, 2016, 2019; McGrew, 2023; McGrew et al., 2023; Protzko & Colom 2021a, 2021b, van der Maas et al. 2006, 2014, 2019). Over the past decade I’ve become more aligned with no-g psychometric network CHC models (e.g, process overlap theory or POT) or Horn’s no-g CHC model, and have, tongue-in-check, referred to the elusive psychometric g ability (not the psychometric g factor) as the “Loch Ness Monster of Psychology” (McGrew, 2021, 2022).
Three of these common cause CHC structural models (viz., Carroll hierarchical g broad CHC model, Carroll hierarchical g broad+narrow CHC, and Horn no-g broad CHC), as well as Dr. Hudson Golino and colleagues hierarchical exploratory graph analysis psychometric network analysis models (that topic is saved for another day), are to be presented in the structural analysis section of the forthcoming WJ V technical manual validity chapter. Stay tunned for some interesting analysis and interpretations in the “must read” WJ V technical manual. Yes….assessment professionals, a well written and thourough technical manual can be your BFF!
Finally, the fourth family of models, which McGrew et al. (2023) called g-centric models, are commonly known as bifactor g models. In the bifactor g broad CHC model (panel C in figure) the variance associated with a dominant psychometric g factor is first extracted from all individual tests. The residual (remaining) variance is modeled as 10 uncorrelated (orthogonal) CHC broad factors. The bifactor g model was excluded from the WJ V structural analysis. Why…..after I (McGrew et al., 2023) recommended that all four classes of traditional CHC structural analysis models should be presented in a test batteries technical manual????
Because…the complexity involved in specifying and evaluating bi-factor g models with 60 cognitive and achievement tests was found to be extremely complex and fraught with statistical convergence issues. Trust me…I tried hard and long to run bifactor g models for the WJ V norm data. It was possible to run bifactor g models separately on the cognitive and achievement sets of WJ V tests, but that does not allow for the direct comparison to the other three structural models that utilized all 60 cognitive and achievement tests in single CFA models. Instead, at of the time the WJ V technical manual analyses were being completed and are now being summarized, the Riverside Insights (RI) internal psychometric research team was tackling the complex issues involved in completing WJ V bifactor g models, first in the separate sets of cognitive and achievement tests. Stay tunned for future professional conference paper presentations, white papers, or journal article submissions by the RI research team.
Because…the complexity involved in specifying and evaluating bi-factor g models with 60 cognitive and achievement tests was found to be extremely complex and fraught with statistical convergence issues. Trust me…I tried hard and long to run bifactor g models for the WJ V norm data. It was possible to run bifactor g models separately on the cognitive and achievement sets of WJ V tests, but that does not allow for the direct comparison to the other three structural models that utilized all 60 cognitive and achievement tests in single CFA models. Instead, at of the time the WJ V technical manual analyses were being completed and are now being summarized, the Riverside Insights (RI) internal psychometric research team was tackling the complex issues involved in completing WJ V bifactor g models, first in the separate sets of cognitive and achievement tests. Stay tunned for future professional conference paper presentations, white papers, or journal article submissions by the RI research team.
Furthermore, the decision to not include bifactor g models does not suggest that the evaluation of WJ V bifactor g-centric CHC models is not important. As noted by Reynolds and Keith (2017), “bifactor models may serve as a useful mathematical convenience for partitioning variance in test scores” (p. 45; emphasis added). The bifactor g model pre-ordains “that the statistically significant lion’s share of IQ battery test variance must be of the form of a dominant psychometric g factor (Decker et al., 2021)” (McGrew, et al., 2023, p. 3). Of the four families of CHC structural models, the bifactor g model is the conceptual and statistical model that supports the importance of general intelligence (psychometric g) and the preeminence of the full-scale or global IQ score over broad CHC test scores (e.g., see Dobrowski et al., 2021; Farmer et al., 2021a, 2021b; McGrew et al., 2023)—a theoretical position inconsistent with the position of the WJ V senior author (yours truly) and with Dr. Richard Woodcock’s legacy (see additional footnote comments at the end). It is important to note that there is a growing body of research that has questioned the preference for bifactor g cognitive models based only on statistical fit indices, as structural model fit statistics frequently are biased in favor of bifactor solutions. Per Bonifay et al. (2017),“the superior performance of the bifactor model may be a symptom of ‘overfitting’—that is, modeling not only the important trends in data but also capturing unwanted noise” p. 184–185). For more on this, see Decker (2021), Dueber and Toland (2021), Eid et al., (2018), Greene et al. (2022), and Murray and Johnson(2013). See Dombroski et al. (2020) for a defense of some of the bifactor g criticisms.
Recognizing the wisdom of Box’s (1976) well known axiom that “all models are wrong, but some are useful” the WJ V technical manual authors (LaForte, Dailey, McGrew, 2025, in preparation) encourage independent researchers to use the WJ V norm data to evaluate and compare bifactor g CHC models with the models presented in forthcoming WJ V technical, as well as alternative models (e.g., PASS, process overlap theory, Cattell’s triadic Gf-Gc theory, etc.) suggested in the technical manual.
Footnote: Woodcock’s original (and enduring) position (Woodcock, 1978, 1997, 2002) regarding the validity and purpose of a composite IQ-type g score is at odds with the bifactor g CHC model. With the publication of the original WJ battery, Woodcock (1978) acknowledged the pragmatic predictive value of statistically partitioning cognitive ability test score variance into a single psychometric g factor, with the manifest total IQ score serving as a proxy for psychometric g. Woodcock stated “it is frequently convenient to use some single index of cognitive ability that will predict the quality of cognitive behavior, on the average, across a wide variety of real-life situations. This is the [pragmatic] rationale for using a single score from a broad-based test of intelligence” (p.126). However, Woodcock further stated that “one of the most common misconceptions about the nature of cognitive ability (particularly in discussions characterized by such labels as ‘IQ’ and ‘intelligence’) is that it is a single quality or trait held in varying degrees by individuals, something like [mental] height” (p. 126). In several publications Woodcock’s position regarding the importance of an overall general intelligence or IQ score was clear—“The primary purpose for cognitive testing should be to find out more about the problem, not to obtain an IQ” (Woodcock, 2002, p.6; also see Woodcock, 1997, p. 235). Two of the primary WJ III, WJ IV, and WJ V authors have conducted research or published articles (see Mather & Schneider, 2023; McGrew, 2023; McGrew et al., 2023) consistent with Woodcock’s position and have advocated for a Horn no-g or emergent property no-g CHC network model. Additionally, based on the failure to identify a brain-based biological g (i.e., neuro-g; Haier et al., 2024) in well over a century of research since Spearman first proposed g in the early 1900’s, McGrew (2020, 2021) has suggested that g may be the “Loch Ness Monster of psychology.” This does not imply that psychometric g is unrelated to combinations of different neurocognitive mechanisms, such as brain-wide neural efficiency and the ability of the whole-brain network, which is comprised of various brain subnetworks and connections via white matter tracts, to efficiently adaptively reconfigure the global network in response to changing cognitive demands (see Ng et al., 2024 for recent compelling research linking psychometric g to multiple brain network mechanisms and various contemporary neurocognitive theories of intelligence; NOTE…click link to download PDF of article and read sufficiently to impress your psychologist friends!!!!).
