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The development of factor models is inextricably tied to the history of intelligence research. One of the most commonly-cited scholars in the field is John Carroll, whose three-stratum theory of cognitive ability has been one of the most influential models of cognitive ability in the past 20 years. Nonetheless, there is disagreement about how Carroll conceptualized the factors in his model. Some argue that his model is best represented through a higher-order model, while others argue that a bi-factor model is a better representation. Carroll was explicit about what he perceived the best way to represent his model, but his writings are not always easy to understand. In this article, I clarify his position by first describing the details and implications of bi-factor and higher-order models then show that Carroll's published views are better represented by a bi-factor model.

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The Wechsler Preschool and Primary Scale of Intelligence--Fourth Edition (WPPSI-IV; Wechsler, 2012) represents a substantial departure from its predecessor, including omission of 4 subtests, addition of 5 new subtests, and modification of the contents of the 5 retained subtests. Wechsler (2012) explicitly assumed a higher-order structure with general intelligence (g) as the second-order factor that explained all the covariation of several first-order factors but failed to consider a bifactor model. The WPPSI-IV normative sample contains 1,700 children aged 2 years and 6 months through 7 years and 7 months, bifurcated into 2 age groups: 2:6-3:11 year olds (n = 600) and 4:0-7:7 year olds (n = 1,100). This study applied confirmatory factor analysis to the WPPSI-IV normative sample data to test the fit of a bifactor model and to determine the reliability of the resulting factors. The bifactor model fit the WPPSI-IV normative sample data as well as or better than the higher-order models favored by Wechsler (2012). In the bifactor model, the general factor accounted for more variance in every subtest than did its corresponding domain-specific factor and the general factor accounted for more total and common variance than all domain-specific factors combined. Further, the domain-specific factors exhibited poor reliability independent of g (i.e., ωh coefficients of .05 to .33). These results suggest that only the general intelligence dimension was sufficiently robust and precise for clinical use.

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