Choosing Between the Bi-Factor and Second-Order Factor Models: A Direct Test Using Latent Variable Modeling

ABSTRACT

This paper is concerned with the process of selecting between the increasingly popular bi-factor model and the second-order factor model in measurement research. It is indicated that in certain settings widely used in empirical studies, the second-order model is nested in the bi-factor model and obtained from the latter after imposing appropriate parameter constraints. These restrictions can be directly tested within the framework of the latent variable modeling methodology employing widely circulated software. The outlined model selection procedure provides a readily applied means of choosing between the two models of growing interest to measurement scholars, and is illustrated using numerical data.

KEYWORDS:

• Bi-factor model
• confirmatory factor analysis
• model choice
• nested models
• second-order factor model

Acknowledgments

We are grateful to T. Asparouhov for valuable discussions on simulation software. We thank C. DiStefano for helpful comments on the bi-factor model.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. The constraint of uncorrelated local factors in the bi-factor model is not essential for the discussed nested model relationship and testing procedure. The latter is in fact applicable in the manner discussed below also with (i) correlated local factors in M1, and/or (ii) correlated observed residuals (measurement error terms), and/or (iii) correlated first-order (latent) residuals in M2, on the earlier made assumption of overall model identification. The assumption of uncorrelated local factors in M1 is advanced merely for convenience and simplicity of the following discussion, and contributes to avoiding possible cases of empirical under-identification (cf. Rindskopf, 1984). These assumptions are relaxed in the discussion and conclusion section, and the model comparison procedure of this article is correspondingly extended in Appendix 3.

2. In certain measurement research settings (studies or cases), one or more observed measures (a) may have no loading on a local factor in M1, or (b) may load only on the second-order factor in M2, or (c) there may be cross-loadings for two local or first-order factors, unlike the general setup described in this section of the main text. These settings are covered in the discussion and conclusion section, and the pertinent generally applicable extension of the discussed model choice procedure is provided in Appendix 3.

3. If for a given pair of identified models M1 and M2 a measure Y_ij loads in M2 both on the second-order factor ξ (with a loading K_ij say) and a first-order factor η_j (with a loading λ_ij say), then in M1 that measure Y_ij would load on the global factor with a loading λ_ij π_j + K_ij (1 ≤ i ≤ q_j; 1 ≤ j ≤ l). In that case, K_ij/λ_ij needs to be subtracted from the respective ratio of global to local loading for M1 in Equation (6)(6) $v_{1\mathit{j}}/{\gamma }_{1\mathit{j}}=v_{2\mathit{j}}/{\gamma }_{2\mathit{j}}=\mathit{\hspace{0.17em}}\dots \mathit{\hspace{0.17em}}=v_{\mathit{qj},\mathit{j}}/{\gamma }_{\mathit{qj},\mathit{j}}(={\pi }_j)$(6) , in order to obtain the relevant restrictions nesting M2 in M1. With this minor modification (with respect to that measure), the outlined model choice procedure remains applicable as well. This nesting model testing extension, like some of those in Appendix 3, appear not to be available in the extant literature (cf. Mansolf & Reise, 2017; Yung et al., 1999).