Abstract
Modeling the combination of latent moderating and mediating effects is a significant issue in the social and behavioral sciences. Chen and Cheng (Structural Equation Modeling: A Multidisciplinary Journal 21: 94–101, 2014) generalized Jöreskog and Yang’s (Advanced structural equation modeling: Issues and techniques (pp. 57–88). Mahwah, NJ: Lawrence Erlbaum, 1996) constrained approach to allow for the concurrent modeling of moderation and mediation within the context of SEM. Unfortunately, due to restrictions related to Chen and Cheng’s partitioning scheme, their framework cannot completely conceptualize and interpret moderation of indirect effects in a mediated model. In the current study, the Chen and Cheng (abbreviated below as C & C) framework is extended to accommodate situations in which any two pathways that constitute a particular indirect effect in a mediated model can be differentially or collectively moderated by the moderator variable(s). By preserving the inherent advantage of the C & C framework, i.e., the matrix partitioning technique, while at the same time further generalizing its applicability, it is expected that the current framework enhances the potential usefulness of the constrained approach as well as the entire class of the product indicator approaches.
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Appendices
Appendix A: Expansions of \( {\boldsymbol{\upeta}}_{{\mathbf{F}}^{\ast }} \), \( {\boldsymbol{\upeta}}_{{\mathbf{S}}^{\ast }} \), \( {\mathbf{y}}_{{\mathbf{F}}^{\ast }}, \) and \( {\mathbf{y}}_{{\mathbf{S}}^{\ast }} \)
Before \( {\boldsymbol{\upeta}}_{{\mathbf{F}}^{\ast }} \), \( {\boldsymbol{\upeta}}_{{\mathbf{S}}^{\ast }} \), \( {\mathbf{y}}_{{\mathbf{F}}^{\ast }}, \) and \( {\mathbf{y}}_{{\mathbf{S}}^{\ast }} \) are discussed in the subsequent paragraph, it is necessary to gain familiarity with the notation of four basic types of matrices. The \( n \times n \) identity matrix will be denoted as Ι n and the \( mn \times mn \) commutation matrix will be indicated as K mn for \( m\ \ne\ n \) and K n for \( m = n \) (see definition 3.1 of Magnus and Neudecker 1979). The \( n\left(n + 1\right)/2 \times {n}^2 \) elimination matrix and \( {n}^2 \times n\left(n + 1\right)/2 \) duplication matrix will be denoted as L n and D n , respectively (see definitions 3.1a and 3.2a of Magnus and Neudecker 1980).
The expansions of \( {\boldsymbol{\upeta}}_{{\mathbf{F}}^{\ast }} \), \( {\boldsymbol{\upeta}}_{{\mathbf{S}}^{\ast }} \), \( {\mathbf{y}}_{{\mathbf{F}}^{\ast }}, \) and \( {\mathbf{y}}_{{\mathbf{S}}^{\ast }} \) can be obtained with the aid of several theorems and properties of the Kronecker product, vech and vec operators shown in Magnus and Neudecker (1980, 1988). The resulting forms of these expansions are expressed as below.
where \( {\boldsymbol{\upzeta}}_{{\mathbf{F}}^{\ast }}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}\left({\mathbf{F}}_{\mathbf{2}}{\boldsymbol{\upzeta}}_{\mathbf{F}} + {\boldsymbol{\upzeta}}_{\mathbf{F}} \otimes {\boldsymbol{\upzeta}}_{\mathbf{F}}\right) \),
(here “\( \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}} \)” is the symbol for “defined as”) in which \( {\mathbf{F}}_{\mathbf{1}} = [({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}}) \otimes ({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}}{)]}^{\hbox{-} 1} \), \( {\mathbf{F}}_{\mathbf{2}} = {\mathbf{I}}_f \otimes {\boldsymbol{\upalpha}}_{\mathbf{F}} + {\boldsymbol{\upalpha}}_{\mathbf{F}} \otimes {\mathbf{I}}_f \), \( {\mathbf{S}}_{\mathbf{1}} = [({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{FF}}) \otimes ({\mathbf{I}}_s - {\mathbf{B}}_{\mathbf{S}\mathbf{S}}{)]}^{\hbox{-} 1} \), \( {\mathbf{S}}_{\mathbf{2}} = {\boldsymbol{\upalpha}}_{\mathbf{F}} \otimes {\mathbf{B}}_{\mathbf{S}\mathbf{F}} \), \( {\mathbf{S}}_{\mathbf{3}} = {\boldsymbol{\upalpha}}_{\mathbf{F}} \otimes {\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }} \), \( {\mathbf{S}}_{\mathbf{4}} = {\mathbf{I}}_f \otimes [{\boldsymbol{\upalpha}}_{\mathbf{S}} + {\mathbf{B}}_{\mathbf{S}\mathbf{F}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1}{\boldsymbol{\upalpha}}_{\mathbf{F}} + {\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}{\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}({\boldsymbol{\upalpha}}_{\mathbf{F}} \otimes {\boldsymbol{\upalpha}}_{\mathbf{F}})] \), \( {\mathbf{S}}_{\mathbf{5}} = {\boldsymbol{\upalpha}}_{\mathbf{F}} \otimes {\mathbf{I}}_s \), \( {\mathbf{S}}_{\mathbf{6}} = {\mathbf{I}}_f \otimes [{\mathbf{B}}_{\mathbf{S}\mathbf{F}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1} + {\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}{\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}{\mathbf{F}}_{\mathbf{2}}] \), \( {\mathbf{S}}_{\mathbf{7}} = {\mathbf{I}}_f \otimes ({\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}{\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}) \), \( {\mathbf{E}}_{\mathbf{1}} = {\boldsymbol{\Lambda}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\upnu}}_{\mathbf{F}} + {\boldsymbol{\upnu}}_{\mathbf{F}} \otimes {\boldsymbol{\Lambda}}_{\mathbf{F}\mathbf{F}} \), \( {\mathbf{E}}_{\mathbf{2}} = {\boldsymbol{\Lambda}}_{\mathbf{FF}} \otimes {\boldsymbol{\Lambda}}_{\mathbf{FF}} \), \( {\mathbf{E}}_{\mathbf{3}} = {\mathbf{I}}_p \otimes [{\boldsymbol{\upnu}}_{\mathbf{F}} + {\boldsymbol{\Lambda}}_{\mathbf{F}\mathbf{F}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1}{\boldsymbol{\upalpha}}_{\mathbf{F}}] + [{\boldsymbol{\upnu}}_{\mathbf{F}} + {\boldsymbol{\Lambda}}_{\mathbf{F}\mathbf{F}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1}{\boldsymbol{\upalpha}}_{\mathbf{F}}] \otimes {\mathbf{I}}_p \), \( {\mathbf{E}}_{\mathbf{4}} = {\mathbf{I}}_p \otimes ({\boldsymbol{\Lambda}}_{\mathbf{FF}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{FF}})}^{\hbox{-} 1}) \), \( {\mathbf{E}}_{\mathbf{5}} = ({\boldsymbol{\Lambda}}_{\mathbf{FF}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{FF}})}^{\hbox{-} 1}) \otimes {\mathbf{I}}_p \), \( {\mathbf{A}}_{\mathbf{1}} = {\boldsymbol{\Lambda}}_{\mathbf{FF}} \otimes {\boldsymbol{\upnu}}_{\mathbf{S}} \), \( {\mathbf{A}}_{\mathbf{2}} = {\boldsymbol{\upnu}}_{\mathbf{F}} \otimes {\boldsymbol{\Lambda}}_{\mathbf{SS}} \), \( {\mathbf{A}}_{\mathbf{3}} = {\boldsymbol{\Lambda}}_{\mathbf{FF}} \otimes {\boldsymbol{\Lambda}}_{\mathbf{SS}} \), \( {\mathbf{A}}_{\mathbf{4}} = {\mathbf{I}}_p \otimes [{\boldsymbol{\upnu}}_{\mathbf{S}} + {\boldsymbol{\Lambda}}_{\mathbf{S}\mathbf{S}}{({\mathbf{I}}_s - {\mathbf{B}}_{\mathbf{S}\mathbf{S}})}^{\hbox{-} 1}({\boldsymbol{\upalpha}}_{\mathbf{S}} + {\mathbf{B}}_{\mathbf{S}\mathbf{F}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1}{\boldsymbol{\upalpha}}_{\mathbf{F}} + {\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}{\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}({\boldsymbol{\upalpha}}_{\mathbf{F}} \otimes {\boldsymbol{\upalpha}}_{\mathbf{F}}))] \), \( {\mathbf{A}}_{\mathbf{5}} = [{\boldsymbol{\upnu}}_{\mathbf{F}} + {\boldsymbol{\Lambda}}_{\mathbf{F}\mathbf{F}}{({\mathbf{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1}{\boldsymbol{\upalpha}}_{\mathbf{F}}] \otimes {\mathbf{I}}_q \), \( {\mathbf{A}}_{\mathbf{6}} = {\mathbf{I}}_p \otimes [{\boldsymbol{\Lambda}}_{\mathbf{S}\mathbf{S}}{({\mathbf{I}}_s - {\mathbf{B}}_{\mathbf{S}\mathbf{S}})}^{\hbox{-} 1}({\mathbf{B}}_{\mathbf{S}\mathbf{F}}{({\mathrm{I}}_f - {\mathbf{B}}_{\mathbf{F}\mathbf{F}})}^{\hbox{-} 1} + {\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}{\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}{\mathbf{F}}_{\mathbf{2}})] \), \( {\mathbf{A}}_{\mathbf{7}} = ({\boldsymbol{\Lambda}}_{\mathbf{FF}}{\left({\mathbf{I}}_f - {\mathbf{B}}_{\mathbf{FF}}\right)}^{\hbox{-} 1}) \otimes {\mathbf{I}}_q \), \( {\mathbf{A}}_{\mathbf{8}} = {\mathbf{I}}_p \otimes ({\boldsymbol{\Lambda}}_{\mathbf{SS}}{({\mathbf{I}}_s - {\mathbf{B}}_{\mathbf{SS}})}^{\hbox{-} 1}) \) and \( {\mathbf{A}}_{\mathbf{9}} = {\mathbf{I}}_p \otimes ({\boldsymbol{\Lambda}}_{\mathbf{S}\mathbf{S}}{({\mathbf{I}}_s - {\mathbf{B}}_{\mathbf{S}\mathbf{S}})}^{\hbox{-} 1}{\mathbf{B}}_{\mathbf{S}{\mathbf{F}}^{\ast }}{\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}) \).
Here, \( {\boldsymbol{\upzeta}}_{{\mathbf{F}}^{\ast }} \) and \( {\boldsymbol{\upzeta}}_{{\mathbf{S}}^{\ast }} \) are vectors of disturbance terms of \( {\boldsymbol{\upeta}}_{{\mathbf{F}}^{\ast }} \) and \( {\boldsymbol{\upeta}}_{{\mathbf{S}}^{\ast }}, \) while \( {\boldsymbol{\upvarepsilon}}_{{\mathbf{F}}^{\ast }} \) and \( {\boldsymbol{\upvarepsilon}}_{{\mathbf{S}}^{\ast }} \) are vectors of measurement errors of \( {\mathbf{y}}_{{\mathbf{F}}^{\ast }} \) and \( {\mathbf{y}}_{{\mathbf{S}}^{\ast }} \). Meanwhile, F 1 , F 2 , S 1 to S 7 , E 1 to E 5 , and A 1 to A 9 are all constant matrices.
Appendix B: Partitioned Matrices Ψ and Θ
The disturbance covariance matrix Ψ is partitioned into a \( 5 \times 5 \) array of submatrices as expressed below:
where \({{ {\boldsymbol{\Psi}}_{{\mathbf{F}}^{\ast}\mathbf{F}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}{\mathbf{F}}_{\mathbf{2}}{\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}}} \), \( {\boldsymbol{\Psi}}_{{\mathbf{F}}^{\ast}\mathbf{S}}\underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}{\mathbf{F}}_{\mathbf{2}}{\left({\boldsymbol{\Psi}}_{\mathbf{SF}}\right)}^{\mathrm{T}} \), \( {\boldsymbol{\Psi}}_{{\mathbf{F}}^{\ast}\mathbf{T}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}{\mathbf{F}}_{\mathbf{2}} {\left({\boldsymbol{\Psi}}_{\mathbf{TF}}\right)}^{\mathrm{T}} \), \( {\boldsymbol{\Psi}}_{{\mathbf{F}}^{\ast }{\mathbf{F}}^{\ast }}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{1}}{\mathbf{L}}_f{\mathbf{F}}_{\mathbf{1}}\left[{\mathbf{F}}_{\mathbf{2}}{\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}{\mathbf{F}}_{\mathbf{2}}^{\mathrm{T}} + \left({\mathbf{I}}_{f^2} + {\mathbf{K}}_{ff}\right)\left({\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}} \otimes {\boldsymbol{\Psi}}_{\mathbf{F}\mathbf{F}}\right)\right]{\mathbf{F}}_{\mathbf{1}}^{\mathrm{T}}{\mathbf{L}}_f^{\mathrm{T}}{\mathbf{W}}_{\mathbf{1}}^{\mathrm{T}} \),
(here the symbol “\( {\Delta}_{n \times m} \)” is used to transform a \( nm \times 1 \) vector into an \( n \times m \) matrix).
The measurement error covariance matrix Θ is partitioned into a \( 5 \times 5 \) array of submatrices as expressed below:
where \( {\boldsymbol{\Theta}}_{{\mathbf{F}}^{\ast}\mathbf{F}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{3}}{\mathbf{L}}_p{\mathbf{E}}_{\mathbf{3}}{\boldsymbol{\Theta}}_{\mathbf{F}\mathbf{F}} \), \( {\boldsymbol{\Theta}}_{{\mathbf{F}}^{\ast}\mathbf{S}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{3}}{\mathbf{L}}_p{\mathbf{E}}_{\mathbf{3}}{\left({\boldsymbol{\Theta}}_{\mathbf{SF}}\right)}^{\mathrm{T}} \), \( {\boldsymbol{\Theta}}_{{\mathbf{F}}^{\ast}\mathbf{T}}\ \underset{{\mkern6mu}}{\underset{={\mkern1mu}}{\Delta}}\ {\mathbf{W}}_{\mathbf{3}}{\mathbf{L}}_p{\mathbf{E}}_{\mathbf{3}}{\left({\boldsymbol{\Theta}}_{\mathbf{TF}}\right)}^{\mathrm{T}} \),
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Chen, SP. (2015). A General SEM Framework for Integrating Moderation and Mediation: The Constrained Approach. In: van der Ark, L., Bolt, D., Wang, WC., Douglas, J., Chow, SM. (eds) Quantitative Psychology Research. Springer Proceedings in Mathematics & Statistics, vol 140. Springer, Cham. https://doi.org/10.1007/978-3-319-19977-1_17
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