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BSS, Ψ and FF ΨSS) from the structural part of the basic model. In contrast,
constraints of the submatrices of ν, Λ and Θ are simply represented as either the null matrix or the identity matrix due to the fact that yF is equivalent to ηF.
In summary, with the basic model having been established in Section 1 of the current chapter, the model specification of the F-F fundamental framework was then developed by following the above-described three steps. Importantly, this three-step sequence can serve as a template for developing the model specification of not only the R-F fundamental framework in the following section but also the F-F/R-R, R-F/R-R, F-F/R-F and F-F/R-F/R-R integrated frameworks in the next chapter.
Section 3 The R-F Fundamental Framework
In this section, the model specification of the R-F fundamental framework will be developed by following the same three-step sequence utilized in establishing the F-F fundamental framework. In particular, with respect to the R-F framework, the first step is to derive the expansions of the nonlinear vectors ηF, η and ~S yF. The
second step is to formulate the structural equation matrix representation of the R-F framework by combining the expansions of ηF and η into the structural part of ~S the basic model (i.e., η[ ηTF | ηTS | ηTT |ηFT |η~ST]T) and the expansion of yF into the measurement part of the basic model (i.e., y[ yFT |yTT |yFT]T). The final step is to develop the constraint specification for the R-F framework based on the normality
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assumption of the basic model shown in Expression 15 as well as the structural equation matrix representation of the R-F framework.
Step 1: Expanding the Nonlinear Vectors
It is readily apparent that the expanded form of ηF in the R-F fundamental framework is identical to that of the afore-described F-F fundamental framework due to the fact that the same partitioning scheme is applied on the basic model in each case. Recall that the derivation of ηF is described in Appendix A, with the resulting expanded form shown in Equation 16 (see page 66). Meanwhile, the expansions of
~
η and S yF, whose general forms are defined as WMvec(ηFηST) and )
vech( F TF
Y y y
W , are obtained by applying several theorems and properties of the Kronecker product, vech and vec operators (Magnus & Neudecker, 1980, 1988).
Regarding the expansion of yF, it should be noted that the optional null setting that was previously placed on the factor loading matrices ΛFS and Λ from Equation FT 14 (see page 63) is changed to an obligatory null, making yF purely a function of
ηF and thus reducing the complexity of the derivation of yF. The expanded forms of η and ~S yF are shown below, with the details of the derivations appearing in Appendix C.
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ηS~ WMM1M2ηFWMM1M3DfWFTηFζS~, (27) yFWYLpvec(νFνFT )WYLpE1ηFWYLpE2DfWFTηFεF, (28) where ζS~ WMM1[M4ζS M5(ζSζF)],
εF WYLp[E3εF E4(εF ζF)E5(ζF εF)εF εF], in which M1 (Is BSS)1If , M2 αSIf, M3 BSFIf ,
) ) ((I FF -1 F
4 I B α
M s f , M5 Is (If BFF)-1,
FF F F FF
1 Λ ν ν Λ
E , E2 ΛFF ΛFF,
p f
f p
p
p ν ν I I Λ B α Λ B α I
I
E3 F F ( FF(I FF)-1 F )( FF(I FF)-1 F ) , )
) (I
( FF FF -1
4 I Λ B
E p f and E5 ( ΛFF(If BFF)-1 )Ip. Here, ζS~ and εF are defined as vectors of disturbance terms of η and ~S
measurement errors of yF, respectively, while M to 1 M5 and E to 1 E5 are all constant matrices. It should be made clear that the connections between η and ~S ηF in Equation 27 and between yF and ηF in Equation 28 were created by imposing
two conditions. First, the column positions of the non-zero elements in the term (WMM1M3Df ) from Equation 27 must match the column positions of the non-zero
elements in W . Second, the column positions of the zero elements in the term F (WYLpE2Df) from Equation 28 must match the column positions of the zero elements in W . F
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Step 2: Formulating the Structural Equation Matrix Representation
The R-F fundamental framework is developed following a procedure similar to that of the F-F fundamental framework; that is, by integrating the expansions of ηF,
~
η and S yF with their corresponding vectors from the basic model. The structural and measurement parts of the R-F fundamental framework, respectively composed of
1 Equations 29 and 30.
, from Equations 16 and 27 (see pages 66 and 75), all submatrices in the fourth and fifth rows of B are set to null, with the exception that BS~F and BS~F are set to
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to estimate latent nonlinear effects.,
Here, y, a 31 partitioned vector of observed indicators, is associated with the 1
3 partitioned vectors of intercepts and measurement errors ν and ε as well as
the 35 partitioned factor loading matrix Λ. In accordance with Equation 28 (see page 75), ΛFF, ΛFS, ΛFT, ΛFF and ΛFS~ are set to WYLpE1, 0, 0,
default, but can potentially be specified as non-null in the unlikely event that a study necessitates ηF or η being measured by ~S ΛFS.
Step 3: Implementing the Constraint Specification
Having developed the generalized R-F framework as shown in Equations 29 and 30, we now proceed to examine the specification of constraints in the context of these two equations. The six partitioned matrices (α, B, Ψ, ν, Λ and Θ) along with their respective constraints are described in Equations 31 to 36. Constraints embedded in α, ν, Ψ and Θ are derived based on the normality assumption of Expression
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15 (see page 64) as well as extensions of several properties of the multivariate normal distribution (Isserlis, 1918; Ghazal & Neudecker, 2000), the details of which can be found in Appendix D. Note that as ηF is the same in the R-F framework as it is in
specified as null. The resultant form of α is expressed as
. of εF to avoid a violation of the assumption inherent to SEM that expected values of
measurement errors are set to null. The resultant form of ν is expressed as
.
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The partitioned coefficient matrix B and Λ are taken directly from Equations 29 and 30 (see pages 76 and 77) to be expressed by Equations 33 and 34, respectively.
.
The disturbance covariance matrix Ψ and the measurement error covariance matrix Θ are partitioned into 5 and 5 3 array of submatrices to be expressed 3 by Equations 35 and 36, respectively.
,
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Here, constraints associated with ζF, ζS~ and εF are composed of particular
combinations of submatrices from the basic model.
On the whole, constraints embedded in the six resultant matrices are represented as either the null matrix or a constraint matrix which is a function of one or more of the submatrices associated with η and F ηS (i.e., α , F αS, B , FF B , SF B , SS Ψ FF and Ψ ) and/or submatrices associated with SS y (i.e., F ν , F Λ and FF Θ ) from the FF basic model.
In summary, the model specification of the R-F fundamental framework was developed by following the same three-step template established in the previous section with a few modifications to reflect differences between the R-F and F-F frameworks. It should further be pointed out that, in light of the above discussion from Sections 2 and 3, the process of constraint specification is neatly incorporated into the F-F and R-F fundamental frameworks. Importantly, the forms of the derived constraint matrices remain the same regardless of the number of latent interaction and/or quadratic effects and product indicators selected.