Testing Measurement Invariance in the Target Rotated Multigroup Exploratory Factor Model

Abstract

We propose a method to investigate measurement invariance in the multigroup exploratory factor model, subject to target rotation. We consider both oblique and orthogonal target rotation. This method has clear advantages over other approaches, such as the use of congruence measures. We demonstrate that the model can be implemented readily in the freely available Mx program. We present the results of 2 illustrative analyses, one based on artificial data, and the other on real data relating to personality in male and female psychology students.

Notes

¹Actually Lawley and Maxwell (1971) discussed the constraint Off-diag(λ′θ⁻¹λ) = 0. For purposes of identification, Off-diag(λ′λ) = 0 serves just as well (Jöreskog, 1978).

²Sometimes additional equality constraints are placed on Ψ _i or Φ _i . However we do not consider these, as they do not follow from Mellenbergh's (1989) definition of measurement invariance. In the presence of latent mean differences, the equality of Ψ _i or Φ _i would seem to require a theoretical account of how mean differences in the common factors can arise in the absence of differences in the covariance matrices of the common factors. We note that the assumption of homoskedasticity in multivariate analysis of variance (i.e., equality of covariance matrices) represents a statistical, not a substantive, constraint.

^a Revised by adding five covariances among residuals (four in the female sample and one in the male sample).

^b Revised by removing the equality constraints on the intercepts of two facets (A2 and A5).

³The correlated residual concerns the following facet pairs: self-discipline (C29) and ambition (C28), reliability (C3) and sincerity (A2), energy (E4) and changes (O4), and modesty (A5) and ideas (O5).

⁴The correlated residual concerns the facet pair sociability (E2) and adventurism (E5).

^a Intercepts of facets A2 sincerity and A5 modesty were unconstrained over sex.

⁵This is an oblique confirmatory oblique factor model in which all untargeted (cross-) factor loadings are fixed to zero, but with the same additional free parameters as in Model 5 in Table 3.

⁶An anomynous reviewer maintained that a confirmatory approach is possible, stating that “if NEO researchers would consistently try to replicate cross-loadings of the various NEO items, we would eventually get to the point where we would understand and a priori expect the large number of specific NEO items to cross-load.” Clearly, the aim of this article is to present an exploratory, target rotated, multigroup factor model. Our illustration seems fitting in the light of McCrae et al. (1996), but we fully accept that a confirmatory approach might be possible at some point in the future.

⁷Full results are available on request.

⁸Mx is freely available (see http://www.vcu.edu/mx).