Modeling Ability Differentiation in the Second-Order Factor Model

Abstract

In this article we present factor models to test for ability differentiation. Ability differentiation predicts that the size of IQ subtest correlations decreases as a function of the general intelligence factor. In the Schmid–Leiman decomposition of the second-order factor model, we model differentiation by introducing heteroscedastic residuals, nonlinear factor loadings, and a skew-normal second-order factor distribution. Using marginal maximum likelihood, we fit this model to Spanish standardization data of the Wechsler Adult Intelligence Scale (3rd ed.) to test the differentiation hypothesis.

Keywords:

• heteroscedasticity
• nonlinear factor analysis
• nonnormality
• marginal maximum likelihood
• Schmid–Leiman factor analysis
• bi-factor model
• ability differentiation
• intelligence

Notes

¹As Molenaar et al. (2010) show, the effect on the factor distribution and the factor loadings cannot be combined, as this results in an empirically underidentified model. However, the effect on the residuals could be combined with each of these two effects. We elaborate on this later.

²We considered this as sufficient as it is generally recommended to use 10 (Vermunt & Magidson, 2005) or 15 (L. K. Muthén & Muthén, 2007) quadrature points per dimension. We also tried 25 quadrature points, but parameters estimates changed by less than .01 (relative to the 15-point solution).

³Confidence intervals are based on the likelihood profile (Neale & Miller, 1997).

⁴The only difference is that in Model 2 subtest DS is associated with a significant estimate of β _j1, whereas in Model 3 this parameter is not significant anymore.

⁵Model 4 is not nested under Model 3, so a comparison in terms of a likelihood ratio is not possible.

⁶We equated the factor loadings of both indicators of the PS factor, as without this restriction the model is unidentified.